Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations
GGlobal Nonlinear Stability of Schwarzschild Spacetimeunder Polarized Perturbations
Sergiu Klainerman and J´er´emie SzeftelDecember 24, 2018 a r X i v : . [ g r- q c ] D ec AbstractWe prove the nonlinear stability of the Schwarzschild spacetime under axially symmetricpolarized perturbations, i.e. solutions of the Einstein vacuum equations for asymptoticallyflat 1 + 3 dimensional Lorentzian metrics which admit a hypersurface orthogonal spacelikeKilling vectorfield with closed orbits. While building on the remarkable advances made inlast 15 years on establishing quantitative linear stability, the paper introduces a series ofnew ideas among which we emphasize the general covariant modulation (GCM) procedurewhich allows us to construct, dynamically, the center of mass frame of the final state. Themass of the final state itself is tracked using the well known Hawking mass relative to awell adapted foliation itself connected to the center of mass frame.Our work here is the first to prove the nonlinear stability of Schwarzschild in a restrictedclass of nontrivial perturbations. To a large extent, the restriction to this class of pertur-bations is only needed to ensure that the final state of evolution is another Schwarzschildspace. We are thus confident that our procedure may apply in a more general setting. ontents Z -frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 CONTENTS Z -polarized S - surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.5 Invariant S -foliations . . . . . . . . . . . . . . . . . . . . . . . . . . 572.1.6 Schwarzschild spacetime . . . . . . . . . . . . . . . . . . . . . . . . 622.2 Main Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2.1 Main equations for general S -foliations . . . . . . . . . . . . . . . . 632.2.2 Null Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . 672.2.3 Hawking mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.2.4 Outgoing Geodesic foliations . . . . . . . . . . . . . . . . . . . . . . 702.2.5 Additional equations . . . . . . . . . . . . . . . . . . . . . . . . . . 852.2.6 Ingoing geodesic foliation . . . . . . . . . . . . . . . . . . . . . . . . 862.2.7 Adapted coordinates systems . . . . . . . . . . . . . . . . . . . . . 862.3 Perturbations of Schwarzschild and invariant quantities . . . . . . . . . . . 942.3.1 Null frame transformations . . . . . . . . . . . . . . . . . . . . . . . 942.3.2 Schematic notation Γ g and Γ b . . . . . . . . . . . . . . . . . . . . . 982.3.3 The invariant quantity q . . . . . . . . . . . . . . . . . . . . . . . . 992.3.4 Several identities for q . . . . . . . . . . . . . . . . . . . . . . . . . 1002.4 Invariant wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.4.2 Wave equations for α , α , and q . . . . . . . . . . . . . . . . . . . . 104 ONTENTS ( ext ) M . . . . . . . . . . . . . . . . . . . . . . . . . 1153.2.2 Main norms in ( int ) M . . . . . . . . . . . . . . . . . . . . . . . . . 1193.2.3 Combined norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.2.4 Initial layer norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.3 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.3.1 Smallness constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.3.2 Statement of the main theorem . . . . . . . . . . . . . . . . . . . . 1223.4 Bootstrap assumptions and first consequences . . . . . . . . . . . . . . . . 1253.4.1 Main bootstrap assumptions . . . . . . . . . . . . . . . . . . . . . . 1253.4.2 Control of the initial data . . . . . . . . . . . . . . . . . . . . . . . 1263.4.3 Control of averages and of the Hawking mass . . . . . . . . . . . . 1263.4.4 Control of coordinates system . . . . . . . . . . . . . . . . . . . . . 1283.4.5 Pointwise bounds for high order derivatives . . . . . . . . . . . . . . 1293.4.6 Construction of a second frame in ( ext ) M . . . . . . . . . . . . . . . 1303.5 Global null frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.5.1 Extension of frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.5.2 Construction of the first global frame . . . . . . . . . . . . . . . . . 1323.5.3 Construction of the second global frame . . . . . . . . . . . . . . . 134 CONTENTS
ONTENTS q (Theorem M1) 239 M by τ . . . . . . . . . . . . . . . . . . . . . . . . 2415.1.2 Assumptions for Ricci coefficients and curvature . . . . . . . . . . . 2425.1.3 Structure of nonlinear terms . . . . . . . . . . . . . . . . . . . . . . 2445.1.4 Main quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2455.2 Proof of Theorem M1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2505.2.1 Flux Decay Estimates for q . . . . . . . . . . . . . . . . . . . . . . 2505.2.2 Proof of Theorem M1 . . . . . . . . . . . . . . . . . . . . . . . . . . 2515.2.3 Proof of Proposition 5.2.2 . . . . . . . . . . . . . . . . . . . . . . . 2535.3 Improved weighted estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 257 CONTENTS q . . . . . . . . . . . . . . . . . . . . . . 2825.4.3 Proof of Theorem 5.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 2845.4.4 Proof of Proposition 5.2.4 . . . . . . . . . . . . . . . . . . . . . . . 2885.4.5 Proof of Proposition 5.2.5 . . . . . . . . . . . . . . . . . . . . . . . 289 α AND α (Theorems M2, M3) 293 ( ext ) M . . . . . . . . . . . . . . . . . . . . 2936.1.2 A transport equation for α . . . . . . . . . . . . . . . . . . . . . . . 2946.1.3 Estimates for transport equations in e . . . . . . . . . . . . . . . . 2976.1.4 Decay estimates for α . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.1.5 End of the proof of Theorem M2 . . . . . . . . . . . . . . . . . . . 3086.2 Proof of Theorem M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3096.2.1 Estimate for α in ( int ) M . . . . . . . . . . . . . . . . . . . . . . . . 3096.2.2 Estimate for α on Σ ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 3116.2.3 Proof of Proposition 6.2.2 . . . . . . . . . . . . . . . . . . . . . . . 3136.2.4 Proof of Lemma 6.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 3176.2.5 Proof of Proposition 6.2.6 . . . . . . . . . . . . . . . . . . . . . . . 320 ONTENTS ∗ . . . . . . . . . . . . . . . . . . . . . . . 3277.1.2 Main Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3297.1.3 Basic Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3327.1.4 Main equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3357.1.5 Equations involving q . . . . . . . . . . . . . . . . . . . . . . . . . . 3357.1.6 Additional equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3397.2 Structure of the proof of Theorem M4 . . . . . . . . . . . . . . . . . . . . . 3437.3 Decay estimates on the last slice Σ ∗ . . . . . . . . . . . . . . . . . . . . . . 3467.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3467.3.2 Differential identities involving GCM conditions on Σ ∗ . . . . . . . 3497.3.3 Control of the flux of some quantities on Σ ∗ . . . . . . . . . . . . . 3507.3.4 Estimates for the (cid:96) = 1 modes on Σ ∗ . . . . . . . . . . . . . . . . . 3607.3.5 Decay of Ricci and curvature components on Σ ∗ . . . . . . . . . . . 3707.4 Control in ( ext ) M , Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3767.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3767.4.2 Proposition 7.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3797.4.3 Estimates for ˇ κ, ˇ µ in ( ext ) M . . . . . . . . . . . . . . . . . . . . . . 3807.4.4 Estimates for the (cid:96) = 1 modes in ( ext ) M . . . . . . . . . . . . . . . 3817.4.5 Completion of the Proof of Proposition 7.4.5 . . . . . . . . . . . . . 3840 CONTENTS ( ext ) M , Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . 3887.5.1 Estimate for η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3897.5.2 Crucial Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3907.5.3 Proof of Proposition 7.5.1, Part I . . . . . . . . . . . . . . . . . . . 3997.5.4 Proof of Proposition 7.5.1, Part II . . . . . . . . . . . . . . . . . . . 4037.6 Conclusion of the Proof of Theorem M4 . . . . . . . . . . . . . . . . . . . 4077.7 Proof of Theorem M5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 ρ . . . . . . . . . . . . . . . . . . . . . . . . . 4438.4.2 Control of (cid:3) g ( r ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4458.4.3 End of the proof of Proposition 8.3.5 . . . . . . . . . . . . . . . . . 4518.5 Proof of Proposition 8.3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4538.5.1 A wave equations for α + Υ α . . . . . . . . . . . . . . . . . . . . . 4538.5.2 End of the proof of Proposition 8.3.6 . . . . . . . . . . . . . . . . . 462 ONTENTS α and Υ α . . . . . . . . . . . . . . . . . . . . . . . . . . 4648.6.2 Control of α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4668.6.3 End of the proof of Proposition 8.3.7 . . . . . . . . . . . . . . . . . 4698.7 Proof of Proposition 8.3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4708.7.1 r -weighted divergence identities for Bianchi pairs . . . . . . . . . . 4718.7.2 End of the proof of Proposition 8.3.8 . . . . . . . . . . . . . . . . . 4808.8 Proof of Proposition 8.3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4868.8.1 Proof of Proposition 8.8.1 . . . . . . . . . . . . . . . . . . . . . . . 4888.8.2 Weighted estimates for transport equations along e in ( ext ) M . . . 4988.8.3 Several identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5058.8.4 Proof of Proposition 8.8.2 . . . . . . . . . . . . . . . . . . . . . . . 5098.8.5 Proof of Proposition 8.8.3 . . . . . . . . . . . . . . . . . . . . . . . 5168.9 Proof of Proposition 8.3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5258.9.1 Weighted estimates for transport equations along e in ( int ) M . . . 5268.9.2 Proof of Proposition 8.9.1 . . . . . . . . . . . . . . . . . . . . . . . 5288.10 Proof of Proposition 8.3.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 S surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 5372 CONTENTS a . . . . . . . . . . . . . . . . . . . . . 5639.3.3 Transversality conditions . . . . . . . . . . . . . . . . . . . . . . . . 5649.3.4 Equation for the (cid:96) = 1 mode of d/ S ,(cid:63) a . . . . . . . . . . . . . . . . . 5659.4 Existence of GCM spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 5669.4.1 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5709.4.2 Comparison of the Hawking mass . . . . . . . . . . . . . . . . . . . 5729.4.3 Iteration procedure for Theorem 9.4.2 . . . . . . . . . . . . . . . . . 5739.4.4 Boundedness of the iterates . . . . . . . . . . . . . . . . . . . . . . 5769.4.5 Convergence of the Iterates . . . . . . . . . . . . . . . . . . . . . . 5779.5 Proof of Proposition 9.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5819.6 Proof of Proposition 9.4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5949.6.1 Pull-back of the main equations . . . . . . . . . . . . . . . . . . . . 5949.6.2 Basic Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5969.6.3 Proof of the estimates (9.6.5), (9.6.6) . . . . . . . . . . . . . . . . . 6059.7 A Corollary to Theorem 9.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 6099.8 Construction of GCM hypersurfaces . . . . . . . . . . . . . . . . . . . . . . 614 ONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6179.8.2 Extrinsic properties of Σ . . . . . . . . . . . . . . . . . . . . . . . 6199.8.3 Construction of Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
10 REGGE-WHEELER TYPE EQUATIONS 649 m and r . . . . . . . . . . . . . . . . . . . 65210.1.4 Deformation tensors of the vectorfields R, T, X . . . . . . . . . . . . 65310.1.5 Basic Integral Identities . . . . . . . . . . . . . . . . . . . . . . . . 65810.1.6 Main Morawetz Identity . . . . . . . . . . . . . . . . . . . . . . . . 66010.1.7 A first estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66410.1.8 Improved lower bound in ( ext ) M . . . . . . . . . . . . . . . . . . . 67010.1.9 Cut-off Correction in ( int ) M . . . . . . . . . . . . . . . . . . . . . . 67810.1.10 The red shift vectorfield . . . . . . . . . . . . . . . . . . . . . . . . 68510.1.11 Combined Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 68910.1.12 Lower bounds for Q . . . . . . . . . . . . . . . . . . . . . . . . . . 69610.1.13 First Morawetz Estimate . . . . . . . . . . . . . . . . . . . . . . . . 69810.1.14 Analysis of the error term E (cid:15) . . . . . . . . . . . . . . . . . . . . . . 70510.1.15 Proof of Theorem 10.1.1 . . . . . . . . . . . . . . . . . . . . . . . . 70610.2 Dafermos-Rodnianski r p - weighted estimates . . . . . . . . . . . . . . . . . 71010.2.1 Vectorfield X = f ( r ) e . . . . . . . . . . . . . . . . . . . . . . . . . 7134 CONTENTS X = f ( r ) e . . . . . . . . . . . . . . . . . . . . 71310.2.3 Proof of Theorem 10.2.1 . . . . . . . . . . . . . . . . . . . . . . . . 72310.3 Higher Weighted Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 73110.3.1 Wave equation for ˇ ψ . . . . . . . . . . . . . . . . . . . . . . . . . . 73110.3.2 The r p weighted estimates for ˇ ψ . . . . . . . . . . . . . . . . . . . . 73210.4 Higher Derivative Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 73810.4.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 73810.4.2 Strategy for recovering higher order derivatives . . . . . . . . . . . 73810.4.3 Commutation formulas with the wave equation . . . . . . . . . . . 73910.4.4 Some weighted estimates for wave equations . . . . . . . . . . . . . 75310.4.5 Proof of Theorem 5.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . 75810.4.6 Proof of Theorem 5.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . 76510.5 More weighted estimates for wave equations . . . . . . . . . . . . . . . . . 770 A APPENDIX TO CHAPTER 2 777
A.1 Proof of Proposition 2.2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 777A.2 Proof of Proposition 2.2.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . 780A.3 Proof of Lemma 2.2.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783A.4 Proof of Proposition 2.2.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 787A.5 Proof of Proposition 2.2.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . 792A.6 Proof of Proposition 2.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 796A.7 Proof of Lemma 2.3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810A.8 Proof of Corollary 2.3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813
ONTENTS α . . . . . . . . . . . . . . . . . . . . . 841A.14.2 Commutation lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 843A.14.3 Main commutation . . . . . . . . . . . . . . . . . . . . . . . . . . . 850A.14.4 Proof of Theorem 2.4.7 . . . . . . . . . . . . . . . . . . . . . . . . . 859 B APPENDIX TO CHAPTER 8 863
B.1 Proof of Proposition 8.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 863
C APPENDIX TO CHAPTER 9 871
C.1 Proof of Lemma 9.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871
D APPENDIX TO CHAPTER 10 885
D.1 Horizontal S -tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885D.1.1 Mixed tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886D.1.2 Invariant Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 887D.1.3 Comparison of the Lagrangians . . . . . . . . . . . . . . . . . . . . 887D.1.4 Energy-Momentum tensor . . . . . . . . . . . . . . . . . . . . . . . 889D.2 Standard Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8906 CONTENTS
D.3 Vectorfield X f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891D.4 Proof of Proposition 10.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 894 hapter 1INTRODUCTION The goal of the book is to prove the nonlinear stability of the Schwarzschild spacetimeunder axially symmetric polarized perturbations, i.e. solutions of the Einstein vacuumequations (1.2.1) for asymptotically flat 1 + 3 dimensional Lorentzian metrics which admita hypersurface orthogonal spacelike Killing vectorfield Z with closed orbits. Recall thatthe Schwarzschild metric g m of mass m > g m = − (cid:18) − mr (cid:19) dt + (cid:18) − mr (cid:19) − dr + r (cid:0) dθ + sin θdϕ (cid:1) . This class of perturbations allows us to restrict our analysis to the case when the finalstate of evolution is itself a Schwarzschild spacetime. This is not the case in general, asa typical perturbation of Schwarzschild may approach a member of the Kerr family withsmall angular momentum.The simplest version of our main theorem can be stated as follows.
Theorem 1.1.1 (Main Theorem (first version)) . The future globally hyperbolic develop-ment of an axially symmetric, polarized , asymptotically flat initial data set, sufficientlyclose (in a specified topology) to a Schwarzschild initial data set of mass m > , hasa complete future null infinity I + and converges in its causal past J − ( I + ) to anothernearby Schwarzschild solution of mass m ∞ close to m . See section 2.1.1 for a precise definition of axial symmetry and polarization. This property is preservedby the Einstein equations, i.e. if the data is axially symmetric, polarized, so is its development.
CHAPTER 1. INTRODUCTION
Our theorem is an important step in the long standing effort to prove the full nonlinearstability of Kerr spacetimes K ( a, m ), in the sub-extremal regime | a | < m . We give asuccinct review below of some of the most important results which have been obtained sofar in this direction. Consider solutions to the Einstein vacuum equations (EVE), R αβ = 0 (1.2.1)with R αβ the Ricci curvature of a four dimensional, Lorentzian manifold ( M , g ). Solutionsof the equations are invariant under general diffeomorphisms of Φ : M −→ M , i.e. if g verifies EVE so does its pull back Φ g . We recall that an initial data set (Σ (0) , g (0) , k (0) )consists of a 3 dimensional manifold Σ (0) together with a complete Riemannian metric g (0) and a symmetric 2-tensor k (0) which verify a well known set of constraint equations(see for instance the introduction in [17]). A Cauchy development of an initial data setis a globally hyperbolic space-time ( M , g ), verifying EVE together with an embedding i : Σ (0) −→ M such that i ∗ ( g (0) ) , i ∗ ( k (0) ) are the first and second fundamental forms of i (Σ (0) ) in M . A well known foundational result in GR associates a unique maximal, globalhyperbolic, future development to all sufficiently regular initial data sets, see [13], [14] .We further restrict the discussion to asymptotically flat initial data sets, i.e. assume thatoutside a sufficiently large compact set K , Σ (0) \ K is diffeomorphic to the complementof the unit ball in R and admits a system of coordinates in which g (0) is asymptoticallyeuclidean and k (0) vanishes at an appropriate order at infinity.EVE admits a remarkable two parameter family of explicit solutions, the Kerr spacetimes K ( a, m ), 0 ≤ a ≤ m , which are stationary, axisymmetric and asymptotically flat. In theusual Boyer-Lindquist coordinates they take the form, g a,m = − q ∆Σ ( dt ) + Σ (sin θ ) q (cid:16) dϕ − amr Σ dt (cid:17) + q ∆ ( dr ) + q ( dθ ) , (1.2.2)where ∆ = r + a − mr,q = r + a (cos θ ) , Σ = ( r + a ) q + 2 mra (sin θ ) = ( r + a ) − a (sin θ ) ∆ . (1.2.3) See also [50] for a modern treatment. .2. THE KERR FAMILY m > g m = − (cid:18) − mr (cid:19) dt + (cid:18) − mr (cid:19) − dr + r dσ S . (1.2.4)Though the metric seems singular at r = 2 m ( r = r + , the largest root of ∆( r ) = 0, inthe case of Kerr) it turns out that one can glue together two regions r > m and tworegions r < m of the Schwarzschild metric to obtain a metric which is smooth along thenull hypersurface E = { r = 2 m } called the Schwarzschild event horizon. The portion of r < m to the future of the hypersurface t = 0 is a black hole whose future boundary r = 0 is singular. The region r > m , free of singularities, is called the domain of outercommunication. The more general family of Kerr solutions, which are both stationary andaxially symmetric, possesses (in addition to well defined event horizons, black holes anddomains of outer communication) Cauchy horizons ( r = r − for the smallest root r = r − of∆( r ) = 0) inside the black hole region across which predictability seems to fail . Again,one can easily check, from the precise nature of the Kerr metric, that the region outsidethe event horizon, i.e. outside the Kerr black hole, is free of singularities . Note thatthe Kerr spacetimes K ( a, m ) possess two Killing vectorfields; the stationary vectorfield T = ∂ t , which is timelike in the asymptotic region, away from the horizon, and the axialsymmetric Killing vectorfield Z = ∂ ϕ . In the particular case of Schwarzschild, T is alsoorthogonal to the hypersurfaces t =const.Here are some other important properties of the Kerr family. • The Kerr solution has a remarkable algebraic feature, encoded in the so called Petrovtype D property, according to which it admits, at every point a pair of null vectors( l, l ), normalized by the condition g ( l, l ) = −
2, called principal null vectors, suchthat all components of the Riemann curvature tensor vanish identically except forthe two independent components R ( l, l, l, l ) , (cid:63) R ( l, l, l, l )with (cid:63) R the Hodge dual of R . • In addition to the symmetries provided by the Killing vectorfields T and Z , the Kerrsolution possesses a nontrivial Killing tensor i.e. a symmetric 2-covariant tensor C (the Carter tensor) verifying, D ( α C βγ ) = 0 . I.e. various smooth extensions are possible. The generalization of this observation to all but an exceptional set of initial data is the celebrated weak cosmic censorship conjecture of Penrose. CHAPTER 1. INTRODUCTION • The Kerr family is distinguished among all stationary solutions of EVE by thevanishing of a four tensor called the Mars-Simon tensor, see [42].
The nonlinear stability of the Kerr family is one of the most pressing issues in mathe-matical GR today. Roughly, the problem is to show that all spacetime developments ofinitial data sets, sufficiently close to the initial data set of a Kerr spacetime, behave inthe large like a (typically another) Kerr solution. This is not only a deep mathematicalquestion but one with serious astrophysical implications. Indeed, if the Kerr family wouldbe unstable under perturbations, black holes would be nothing more than mathematicalartifacts. Here is a more precise formulation of the conjecture.
Conjecture (Stability of Kerr conjecture).
Vacuum initial data sets, sufficiently close toKerr initial data, have a maximal development with complete future null infinity and withdomain of outer communication which approaches (globally) a nearby Kerr solution. So far, the only space-time for which full nonlinear stability has been established is theMinkowski space, corresponding to the particular case a = m = 0. The result was firstproved in [17], see also [37], [39], [6] and [28]. Theorem 1.3.1 (Global stability of Minkowski) . Any asymptotically flat initial data set,which is sufficiently close to the trivial one, has a regular, complete, maximal develop-ment . Here are, very schematically, the main ideas in the proof of stability of Minkowski space.(I) Perturbations radiate and decay sufficiently fast (just fast enough!) to insureconvergence.(II) Interpret the Bianchi identities as a Maxwell like system. This is an effective, invariant , way to treat the hyperbolic character of the equations. This means, roughly, that observers which are far away from the black hole may live forever. The complete result in [17] also provides very precise information about the decay of the curvaturetensor along null and timelike directions as well as many other geometric information concerning thecausal structure of the corresponding spacetime. Of particular interest are peeling properties i.e. theprecise decay rates of various components of the curvature tensor along future null geodesics. .3. STABILITY OF KERR approximate
Killing and conformalKilling symmetries of the equations, see [33], [34], [35], [16].(ii) Generalized energy estimates using both the Bianchi identities and the approximateKilling and conformal Killing vector fields.(iii) The null condition identifies the deep mechanism for nonlinear stability, i.e. thespecific structure of the nonlinear terms enables stability despite the slow decayrate of the perturbations, see [32], [34], [15].(iv) Complex boot-strap argument according to which one makes educated assumptionsabout the behavior of the space-time and then proceeds to show that they are infact satisfied. This amounts to a conceptual linearization , i.e. a method by whichthe equations become, essentially, linear without actually linearizing them.There are three, related, major obstacles in passing from the stability of Minkowski tothat of the Kerr family.1. The first can be understood in the general framework of nonlinear hyperbolic ordispersive equations. Given a nonlinear equation N [ φ ] = 0 and a stationary solu-tion φ we have two notions of stability, orbital stability , according to which smallperturbations of φ lead to solutions φ which remain close, in some norm (typically L based ) for all time, and asymptotical stability , according to which the perturbedsolutions converge, as t → ∞ , to a nearby stationary solution. Note that the sec-ond notion is far stronger, and much more precise, than the first and that orbitalstability can only be established (without appealing to the the stronger version)only for equations with very weak nonlinearities. For quasilinear equations, suchas the Einstein field equations, a proof of stability requires, necessarily, a proof ofasymptotic stability. This must then be based on a detailed understanding of thedecay properties of the linearized equations.One is thus led to study the linearized equations N (cid:48) [ φ ] ψ = 0, with N (cid:48) [ φ ] theFr´echet derivative of N at φ , which, in many important cases, are hyperbolic It is irrelevant whether a specific linearization procedure needs to be implemented; what is importanthere is to identify the linear mechanism for decay, such as the Maxwell system in the case of the stabilityof Minkowski space mentioned above. In the case of EVE the linearized equations are linear hyperbolic only after we mod out the linearizedversion of general coordinate transformations. CHAPTER 1. INTRODUCTION systems with variable coefficients that typically present instabilities. In the excep-tional situation, when nonlinear stability can ultimately be established, one can tieall the instability modes of the linearized system to two properties of the nonlinearequation:(a) The presence of a continuous , family of other stationary solutions of N [ φ ] =0 near φ .(b) The presence of a continuous family of diffeomorphisms of the backgroundmanifold which map, by pull back, solutions to solutions.For a typical stationary solution φ , both properties exist and generate nontrivialsolutions of the linearized equation N (cid:48) [ φ ] ψ = 0. In the case of relatively simplescalar nonlinear equations, where the symmetry group of the equation is small, aneffective strategy of dealing with this problem (known under the name of modulationtheory) has been developed, see for example [44], [46]. In the case of the Einsteinequations this problem is compounded by the large invariance group of the equations,i.e. all diffeomorphisms of the spacetime manifold. To deal with both problems andestablish stability one has to • Track the parameters ( a f , m f ) of the final Kerr spacetime. • Track the coordinate system (gauge condition) relative to which we have decayfor all linearized quantities. Such a coordinate system cannot be imposed a-priori, it has to emerge dynamically in the construction of the spacetime.2. As described earlier, the fundamental insight in the stability of the Minkowski spacewas that we can treat the Bianchi identities as a Maxwell system in a slightlyperturbed Minkowski space by using the vectorfield method. This cannot workfor perturbations of Kerr due to the fact that some of the null components of thecurvature tensor are non-trivial in Kerr.3. Even if we can establish a useful version of linearization (i.e. one which addresses theabove mentioned problems), there are still major obstacles in understanding theirdecay properties. Indeed, when one considers the simplest, relevant, linear equationon a fixed Kerr background, i.e. the wave equation (cid:3) g ψ = 0 (often referred toas the poor’s man linearization of EVE), one encounters serious difficulties evento prove the boundedness of solutions for the most reasonable, smooth, compactlysupported, data. Below is a very short description of these. In the case of the stability of Kerr we have a 2 parameter family of solutions K ( a, m ). This is responsible of the fact that a small perturbation of the fixed stationary solution φ may notconverge to φ but to another nearby stationary solution. In the case of EVE, any diffeomorphism has that property. With respect to the so called principal null directions. .3. STABILITY OF KERR • The problem of trapped null geodesics.
This concerns the existence of nullgeodesics neither crossing the event horizon nor escaping to null infinity,along which solutions can concentrate for arbitrary long times. This leads todegenerate energy estimates which require a very delicate analysis. • The trapping properties of the horizon.
The horizon itself is ruled by nullgeodesics, which do not communicate with null infinity and can thus concen-trate energy. This problem was solved by understanding the so called red-shifteffect associated to the event horizon, which more than counteracts this typeof trapping. • The problem of superradiance.
This is essentially the failure of the stationaryKilling field T = ∂ t to be everywhere timelike in the domain of outer communi-cations and, thus, the failure of the associated conserved energy to be positive.Note that this problem is absent in Schwarzschild and, in general, for axiallysymmetric solutions. • Superposition problem.
This is the problem of combining the estimates in thenear region, close to the horizon, (including the ergoregion and trapping) withestimates in the asymptotic region, where the spacetime looks Minkowskian.4. The full linearized system of EVE around Kerr, usually referred to as the linearizedgravity system (LGS), whatever its formulation, presents far more difficulties beyondthose mentioned above concerning the poor man’s linear scalar wave equation onKerr, see the discussion below.Historically, two versions of LGS have been considered.(a) At the level of the metric itself, i.e. if G denotes the Einstein tensor, G αβ = R αβ − Rg αβ , G (cid:48) ( g ) δ g = 0 . (1.3.1)(b) Via the Newman-Penrose (NP) formalism, based on null frames.In what follows we review the main known results concerning solutions to the linearizedequations on a Kerr background. In the Schwarzschild case, these geodesics are located on the so-called photon sphere r = 3 m . CHAPTER 1. INTRODUCTION
The first important results concerning both items (3) and (4) above were obtained byphysicists based on the classical method of separation of variables and formal mode anal-ysis. In the particular case where g is the Schwarzschild metric, the linearized equations(1.3.1) can be formally decomposed into modes, by associating t-derivatives with mul-tiplication by iω and angular derivatives with multiplication by l , i.e. the eigenvaluesof the spherical laplacian. A similar decomposition, using oblate spheroidal harmonics,can be done in Kerr. The formal study of fixed modes from the point of view of metricperturbations as in (1.3.1) was initiated by Regge-Wheeler [49] who discovered the mas-ter Regge-Wheeler equation for odd-parity perturbations. This study was completed byVishveshwara [56] and Zerilli [60]. A gauge-invariant formulation of metric perturbations was then given by Moncrief [47]. An alternative approach via the Newman-Penrose (NP)formalism was first undertaken by Bardeen-Press [5]. This latter type of analysis waslater extended to the Kerr family by Teukolsky [55] who made the important discoverythat the extreme curvature components, relative to a principal null frame, satisfy decou-pled, separable, wave equations. These extreme curvature components also turn out tobe gauge invariant in the sense that small perturbations of the frame lead to quadraticerrors in their expression. The full extent of what could be done by mode analysis, inboth approaches, can be found in Chandrasekhar’s book [11]. Chandrasekhar also intro-duced (see [12]) a transformation theory relating the two approaches. More precisely, heexhibits a transformation which connects the Teukolsky equations to the Regge-Wheelerone. This transformation was further elucidated and extended by R. Wald [57] and re-cently by Aksteiner and al [2]. The full mode stability, i.e. lack of exponentially growingmodes, for the Teukolsky equation on Kerr is due to Whiting [59] (see also [51] for astronger quantitive version). Note that mode stability is far from establishing even boundedness of solutions to thelinearized equations. To achieve that and, in addition, to derive realistic decay estimatesone needs an entirely different approach based on a far reaching extension of the classicalvectorfield method used in the proof of the nonlinear stability of Minkowski [17]. Thenew vectorfield method compensates for the lack of enough Killing and conformal Killingvectorfields on a Schwarzschild or Kerr background by introducing new vectorfields whosedeformation tensors have coercive properties in different regions of spacetime, not nec- Method based on the symmetries of Minkowski space to derive uniform, robust, decay for nonlinearwave equations, see [33], [34], [35], [16]. .3. STABILITY OF KERR
Kerr ( a, m ), (cid:3) g a,m ψ = 0 . (1.3.2)The starting and most demanding part of the new method is the derivation of a global,simultaneous, Energy–Spacetime Morawetz estimate which degenerates in the trappingregion. This task is somewhat easier in Schwarzschild, or for axially symmetric solutionsin Kerr, where the trapping region is restricted to a smooth hypersurface. The firstsuch estimates, in Schwarzschild, were proved by Blue and Soffer in [7], [8] followed bya long sequence of further improvements in [10], [19], [45] etc. See also [31] and [53] fora vectorfield method treatment of the axially symmetric case in Kerr with applicationsto nonlinear equations. In the absence of axial symmetry the derivation of an Energy-Morawetz estimate in
Kerr ( a, m ), | a/m | (cid:28) | a | < m is even more subtle and was recently achieved by Dafermos, Rodnianski and Shlapentokh-Rothman [24] by combining mode decomposition with the vectorfield method.Once an Energy-Morawetz estimate is established one can commute with the time trans-lation vectorfield and the so called Red Shift vectorfield, first introduced in [19], to deriveuniform bounds for solutions. The most efficient way to also get decay, and solve the superposition problem , is due to Dafermos and Rodnianski, see [20], based on the pres-ence of a family of r p -weighted , quasi-conformal vectorfields defined in the far r region ofspacetime . A first quantitative (i.e. which provides precise decay estimates) proof of the linearstability of Schwarzschild spacetime has recently been established by Dafermos, Holzegeland Rodnianksi in [22], via the NP formalism (expressed in a double null foliation ). It isimportant to note that while the Teukolsky equation (in the NP formalism) is separable,and thus amenable to mode analysis, it is not Lagrangian and thus cannot be treated These replace the scaling and inverted time translation vectorfields used in [33] or their correspondingdeformations used in [17]. A recent improvement of the method, relevant to our work here, allowing oneto derive higher order decay can be found in [4]. A somewhat weaker version of linear stability of Schwarzschild was subsequently proved in [30] byusing the original, direct, Regge-Wheeler, Zerilli approach combined with the vectorfield method andadapted gauge choices. This is possible in Schwarzschild where the principal null directions are integrable. CHAPTER 1. INTRODUCTION by direct energy type estimates. To overcome this difficulty [22] relies on a new physicalspace version of the Chandrasekhar transformation [12], which takes solutions of theTeukolsky equations to solutions of Regge-Wheeler, which is manifestly both Lagrangianand coercive. After quantitative decay has been established for this latter equation, basedon the new vectorfield method, the physical space form of the transformation allows oneto derive quantitative decay for solutions of the original Teukolsky equation. Once decayestimates for the Teukolsky equation have been established, the remaining work in [22] isto bound all other curvature and Ricci coefficients associated to the double null foliation.This last step requires carefully chosen gauge conditions along the event horizon of thefixed Schwarzschild background. This final gauge is itself then quantitatively bounded interms of the initial data, giving thus a comprehensive statement of linear stability.
In the passage from linear to nonlinear stability of Schwarzschild one has to overcomemajor new difficulties. Some are similar to those encountered in the stability of Minkowski[17] such as,1. Need of an appropriate geometric setting which takes into account the decay andpeeling properties of the curvature. In [17] this was achieved with the help of thefoliation of the perturbed spacetime given by two optical functions ( int ) u and ( ext ) u and a maximal time function t . The exterior optical function ( ext ) u , which wasinitialized at infinity, was essential to derive the decay and peeling properties alongnull directions while ( int ) u , initialized on a timelike axis, was responsible for coveringthe interior, non-radiative, back scattering, decay.2. The peeling and decay estimates have to be derived by some version of the geometricvectorfield method which relates decay to generalized energy type estimates.3. The peeling and decay estimates mentioned above should be sufficiently strong tobe able to deal with the error terms generated by the vectorfield method. For thisto happen, the error terms need to exhibit an appropriate null structure.The new main difficulties are as follows:1. One needs a procedure which allows to take into account the change of mass anddetect its final value. Note also that we need to restrict the nature of the per- .3. STABILITY OF KERR recoilproblem , does not occur in linear theory.Here is a very short summary of how we solve these new challenges in our work.1. We resolve the first difficulty by restricting our analysis to axially symmetric, polar-ized perturbations and by tracking the mass using a quantity, called the quasi-localHawking mass, for which we derive simple propagation equations which establishmonotonicity of the mass up to errors which are quadratic with respect to the per-turbations.2. We resolve the second difficulty by making use of the fact that the extreme com-ponents of the curvature are, up to quadratic terms, invariant under null frame8 CHAPTER 1. INTRODUCTION transformations. As in [22], we also make use of a transformation, similar to thatof Chandrasekhar mentioned above, which maps the extreme components of thecurvature to a new quantity q , defined up to quadratic errors, that verifies a Regge-Wheeler type equation. Once we manage to control q , i.e. to derive quantitativedecay estimates for it, we can also control, in principle , the two extreme curvatureinvariants α and α , the first by inverting the Chandrasekhar transformation andthe second by using a variant of the Teukolsky- Starobinski identities. One is thenleft with the arduous task of recovering all other null components of the curvaturetensor and all connection coefficients.3. The third difficulties manifests itself in the most sensitive part of the entire ar-gument, i.e. in the task of deriving quantitative decay estimates for q by makinguse of the Regge-Wheeler type equation it verifies. To do this we rely on the newvectorfield method as outlined in subsection1.3.3 above. The main new difficultiesare:(i) The vectorfield method introduces new error terms, not present in linear the-ory. To estimate these terms we need precise decay informations, off the finalSchwarzschild space, for all connection coefficients and curvature of the per-turbation.(ii) The most difficult terms are those due to the quadratic errors made in thederivation of the Regge-Wheeler equation for q . As in the proof the stabilityof the Minkowski space the precise rates of decay for various curvature andconnection coefficients, i.e. the peeling properties of the perturbation, and thethe precise structure of these error terms is of fundamental importance.4. We solve the fourth and most important new difficulty by a procedure we call General Covariant Modulation (GCM). This procedure, which takes advantage ofthe full covariance of the Einstein equations, allows us to construct the perturbedspacetime by a continuity argument involving finite GCM admissible spacetimes M as represented in Figure 1.1. The past boundaries C ∪ C are incoming andoutgoing null hypersurfaces on which the initial perturbation is prescribed. Thefuture boundaries consists of the union A ∪ C ∗ ∪ C ∗ ∪ Σ ∗ where A and Σ ∗ arespacelike, C ∗ is incoming null, C ∗ outgoing null. The boundary A is chosen so that,in the limit when M converges to the final state, is included in the perturbed blackhole. The spacelike boundary Σ ∗ plays a fundamental role in our construction asseen below. The spacetime M also contains a timelike hypersurface T which divides M into an exterior region we call ( ext ) M and an interior one ( int ) M . We say that M is a GCM admissible spacetime if it verifies the following properties. Provided that one can deal with the nonlinear terms. In the linear setting this was partially achieved in [23]. .3. STABILITY OF KERR H + C ⇤ C ⇤ AC C ⌃ ⇤ I +( ext ) M ( int ) MT H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ AC C ⌃ ⇤ I +( ext ) M ( int ) MT H + C ⇤ C ⇤ AC C ⌃ ⇤ I +( ext ) M ( int ) MT H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T Figure 1.1: The GCM admissible space-time M (i) The far region ( ext ) M is foliated by a geodesic foliation induced by an outgoingoptical function u initialized on Σ ∗ (ii) The near region ( int ) M is foliated by a geodesic foliation induced by an incom-ing optical function u initialized at T such that its level sets on T coincidewith those of u .(iii) The foliation induced on Σ ∗ is such that specific geometric quantities takeSchwarzschildian values. We refer to these as GCM conditions. These condi-tions are dynamically reset in the continuation process on which our proof isbased.(iv) The area radius r ( u ) of the spheres of constant u along Σ ∗ is far greater thanthe corresponding value of u . This condition allows us to simplify somewhatthe null structure and Bianchi equations induced on Σ ∗ and corresponds to theexpectation that the spacelike hypersurfaces Σ ∗ converges to the null infinityof the final state of the perturbation.5. The GCM conditions together with the control derived on q , α and α mentionedearlier allows us to control all null connection and curvature coefficients along on Σ ∗ ,i.e. to derive appropriated decay estimates for them. These estimates can then be0 CHAPTER 1. INTRODUCTION transported to ( ext ) M using the the full scope of the null structure and null Bianchiidentities associated to the outgoing geodesic foliation.6. The decay estimates in ( ext ) M can then be used as initial condition along the time-like hypersurface T for the incoming foliation of ( int ) M . These allows us to alsoderive appropriate decay estimates for all null connection and curvature coefficientsof the foliation induced by u .7. The precise decay estimates derived in 5 are sufficiently strong to allow us to controlall error terms generated in the process of estimating q , as mentioned in 3.Note that in Figure 1.1, one starts with initial conditions on the union of null hypersurfaces C ∪C rather than an initial spacelike hypersurface Σ (0) . One can justify this simplificationbased on the results of [37], [38], see Remark 3.3.1. The full red line H + represents thefuture event horizon of the perturbed Schwarzschild. The line T represents the timelikehypersurface separating ( int ) M from ( ext ) M . In deriving decay estimates the precisechoice of T is irrelevant. A choice, however, needs to be made in order to avoid a derivativeloss for our top energy estimates .The spacetime is constructed by a continuity argument, i.e. we assume that the spacetimeterminating at C ∗ ∪ C ∗ saturates a given bootstrap assumption ( BA ) and show, by along sequence of a-priori estimates which take advantage of the smallness of the initialperturbation, that ( BA ) can be improved and the spacetime extended past C ∗ ∪ C ∗ ∪ Σ ∗ .Our work here is the first to prove the nonlinear stability of Schwarzschild in a restrictedclass of nontrivial perturbations, i.e. perturbations for which new ideas, such as our GCMprocedure are needed. To a large extent, the restriction to this class of perturbations isonly needed to ensure that the final state of evolution is another Schwarzschild space. Weare thus confident that our procedure may apply in a more general setting. We would liketo single out two other recent important contributions to nonlinear stability of black holes.In the context of asymptotically flat Einstein vacuum equations the result of Dafermos-Holzegel-Rodnianski [23] constructs a class of Kerr black hole solutions starting fromfuture infinity while Hintz-Vasy [26] prove the nonlinear stability of Kerr-de Sitter,for small angular momentum, in the context of the Einstein vacuum equations with anontrivial positive cosmological constant. Though the two results are very different theyshare in common the fact that the perturbations they treat decay exponentially. Thismakes the analysis significantly easier than in our case when the decay is barely enoughto control the nonlinear terms. See [17] for a similar situation. See also [27] for the stability of Kerr-Newman de Sitter. .4. ORGANIZATION The paper is organized as follow. In Chapter 2 we introduce the main quantities, equationsand basic tools needed later. It is our main reference kit providing all main null structureand null Bianchi equations, in general null frames, in the context of axially symmetricpolarized spacetimes. Though we work with the reduced equations, i.e the equationsreduced by the symmetries, most of the work in the paper does not really depend of thereduction. Besides insuring that the final state is a Schwarzschild space the reductiononly plays a significant role in the GCM construction.Chapter 3, the heart of the paper, contains the precise version of our main theorem,its main conclusions as well as a full strategy of its proof, divided in nine supportingintermediate results, Theorems M0 – M8 . We also give a short description of the proof ofeach theorem.In the other chapters of this paper we give complete proofs of Theorems, M0 – M8 and afull description of our GCM procedure.The reader versed in the formalism of null structure and Bianchi equations, as discussedin [17], is encouraged to glance fast over Chapter 2, to get familiarized with the notation,and then move directly to Chapter 3. This work would be inconceivable without the remarkable advances made in the last sixtyyears on black holes. The works of Regge-Weeler, Carter, Teukolsky, Chandrasekhar,Wald etc., made during the so called golden age of black hole physics in the sixties andseventies, have greatly influenced our understanding of invariant quantities and the waveequations they satisfy. The advances made in the last fifteen years, quoted earlier, whichhave led to the development of new mathematical methods to derive the decay of waveson black holes spacetimes, are even more immediately relevant to our work. In particularwe would like to single out the direct influence of Dafermos-Holzegel-Rodnianski [22] inthe gestation of our own ideas in this paper. Finally, the work on the nonlinear stabilityof the Minkowski space in [17], a milestone in the mathematical GR, has significantlyinstructed our work here. We would like to thank E. Giorgi for her careful proofreadingof various sections of the manuscript. Various discussions we had with S. Aksteiner werevery useful. Finally we thank our wives Anca and Emilie for their incredible patience, See also [37]. CHAPTER 1. INTRODUCTION understanding and support during our many years of work on this project.The first author has been supported by the NSF grant DMS 1362872. He would liketo thank the mathematics departments of Paris 6, Cergy-Pontoise and IHES for theirhospitality during his many visits in the last six years. The second author is supportedby the ERC grant ERC-2016 CoG 725589 EPGR. hapter 2PRELIMINARIES
We consider vacuum, four dimensional, simply connected, axially symmetric spacetimes( M , g , Z ) with g Lorentzian and Z an axial Killing vectorfield on M . We denote by A the axis of symmetry, i.e. the points on M for which X := g ( Z , Z ) = 0. In the case ofinterest for us we assume dX (cid:54) = 0 and that A is a smooth manifold of codimension 2. TheErnst potential of the spacetime is given by, σ µ := D µ ( − Z α Z α ) − i ∈ µβγδ Z β D γ Z δ . The 1-form σ µ dx µ is closed and thus there exists a function σ : M → C , called the Z -Ernst potential, such that σ µ = D µ σ. Note also that D µ g ( Z , Z ) = 2 G µλ Z λ = −(cid:60) ( σ µ )where G µν = D µ Z ν . Hence we can choose the potential σ such that (cid:60) σ = − X . By astandard calculation one can show that, (cid:3) σ = − X − D µ σ D µ σ. Definition 2.1.1.
An axially symmetric Lorentzian manifold ( M , g , Z ) is said to be po-larized if the Ernst potential σ is real, i.e. σ = − X . In that case the metric g can bewritten in the form, g = Xdϕ + g ab dx a dx b (2.1.1)334 CHAPTER 2. PRELIMINARIES where X and g are independent of ϕ . We refer to the orbit space M / Z as the reduced spaceand the metric g = g ab dx a dx b as the reduced metric. Note that the reduced space ( M / Z , g ) is smooth away from the axis A . Moreover the scalar X verifies the wave equation, (cid:3) g X = X − D µ X D µ X. (2.1.2)We denote by R , resp. R the curvature tensor of the spacetime metric g , respectively g ,and by (cid:3) g , resp (cid:3) g the d’Alembertian with respect to g and resp. the reduced metric g .We also denote by Γ the Christoffel symbols of g and by Γ the ones of g . Note that theonly non-vanishing Christoffel symbols are: Γ ϕϕb = 12 X − ∂ b X, Γ aϕϕ = − g as ∂ s X, Γ abc = Γ abc . (2.1.3)One can easily prove the following. Proposition 2.1.2.
The scalar curvature R of the reduced metric g of an axially sym-metric polarized Einstein vacuum spacetime vanishes identically . Moreover, setting Φ := log X we find, R ab = D a D b Φ + D a Φ D b Φ , (cid:3) g Φ = − D a Φ D a Φ . (2.1.4) Also, R aϕb ϕ = − X − D a D b X + 14 X − D a XD b X = − R ab , R acb ϕ = 0 , (2.1.5) R abc d = R abc d , and, R abcd = g ac R bd + g bd R ac − g ad R bc − g bc R ad . (2.1.6) Finally, when applied to Z -invariant functions, (cid:3) g = (cid:3) g + g ab ∂ a Φ ∂ b . (2.1.7) Remark 2.1.3.
The wave equation in (2.1.4) is equivalent to (cid:3) g Φ = 0 . (2.1.8) Remark 2.1.4.
Schwarzschild spacetime is axially symmetric polarized with, X = r (sin θ ) , Φ = log( r ) + log(sin θ ) . This is an easy consequence of the equation (2.1.2). .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES We consider orthonormal frames e , e , e θ = e , e ϕ = X − / Z , with X := g ( Z , Z ), whichare Z - equivariant , i.e. [ Z , e α ] = 0. From now on, the index ϕ is referring to the framerather than the coordinates Lemma 2.1.5.
Setting (Λ α ) βγ := g ( D α e γ , e β ) we have, (Λ ϕ ) aϕ = − D a Φ , (Λ ϕ ) ab = (Λ a ) bϕ = 0 , ∀ a = 0 , , , (2.1.9) and, D a e b = D a e b , D a e ϕ = 0 , D ϕ e a = (Λ ϕ ) ϕa e ϕ = ( D a Φ) e ϕ , D ϕ e ϕ = (Λ ϕ ) aϕ e a = − D a Φ e a . (2.1.10) Proof.
Straightforward verification.
Lemma 2.1.6.
We have, D s R abcd = D s R abcd , D s R ϕbcd = 0 , D s R ϕbϕd = − D s R bd , D ϕ R abcd = 0 , D ϕ R ϕbcd = D s Φ R sbcd + D c Φ R bd − D d Φ R bc , D ϕ R ϕbϕd = 0 . Proof.
Straightforward verification.
Definition 2.1.7.
We say that a spacetime tensor U is Z -invariant if L Z U = 0 and Z -invariant polarized if its contractions to an odd number of e ϕ = X − / Z vanish identically. Proposition 2.1.8.
All higher covariant derivatives of the Riemann curvature tensor R of an axially symmetric polarized spacetime ( M , g , Z ) are Z -invariant, polarized.Proof. The statement has been already verified above for both R and DR . It sufficesto show that, given an arbitrary Z -invariant, polarized tensor U , its covariant derivative DU is also Z -invariant, polarized. The invariance is immediate. To show polarization6 CHAPTER 2. PRELIMINARIES we consider all frame components of DU with respect to our adapted equivariant frame e , e , e , e ϕ . Assume first that the components of DU contain only one e ϕ . These are, D ϕ U a , D a U bϕc with various combinations of horizontal indices a, b, c . Now, in view of the polarizationproperty of U and the relations D a e b = D a e b , D a e ϕ = 0 we easily deduce, D a U bϕc = e a U bϕc − U D a bϕc − U b D a ϕc − U bϕc = 0 . Similarly, since e ϕ ( U a ) = X − / Z ( U a ) = X − / L Z U a = 0 and D ϕ e a is proportional to e ϕ , D ϕ U a = e ϕ ( U a ) − U D ϕ e a = 0 . Similarly we can check that the contraction of DU with any odd number of e ϕ must bezero.In what follows we shall refer to Z -invariant, polarized tensors as simply Z -polarized. We denote by A the axis of symmetry of Z , i.e. the set of zeroes of X = g ( Z , Z ). Sincewe assume dX (cid:54) = 0, A is a smooth timelike submanifold of dimension 2. In view of thedefinition of axial symmetry every trajectory of Z is closed and intersects A at one point.The following regularity result at A holds true. Lemma 2.1.9.
At the axis of symmetry A we have, g µν ∂ µ X∂ ν X X = e g µν ∂ µ Φ ∂ ν Φ −→ . (2.1.11) Proof.
This is a classical result, see for example [43]. We provide a proof for the con-venience of the reader. We introduce a coordinates system ( x , x , x , x ) centered at apoint q = (0 , , ,
0) on the axis such that the Christoffel symbols of the metric vanishat q and ∂ x | q and ∂ x | q are tangent to the axis at q . In particular, in this coordinatessystem, the matrix ∂ α Z µ ( q ) is given by ∂ α Z µ ( q ) = (cid:18) A (cid:19) , where A is an antisymmetric matrix. Note that we used the fact that Z vanishes on theaxis, that q belongs to the axis, and that ∂ α Z µ ( q ) is antisymmetric since Z is Killing. .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES x ( ϕ ) denotes an orbit of Z close to q , and y = ( x , x ), we have in particular fromTaylor formula dydϕ = Ay + O ( y ) . Hence exp( − ϕA ) y ( ϕ ) = y (0) + O ( ϕy )and since y (2 π ) = y (0) in view of the 2 π -periodicity of the orbits of Z , we inferexp( − πA ) y (0) = y (0) + O ( y ) . As y (0) can be taken arbitrarily small, we infer that exp(2 πA ) is the 2 × A is antisymmetric and non zero, its eigenvalues necessarily are i and − i , and hence A T A = I . This yields A α µ A γ ν A αν = A α µ ( A T A ) γα = A γµ and hence ∂ α Z µ ( q ) ∂ γ Z ν ( q ) ∂ α Z ν ( q ) = ∂ γ Z µ ( q ) . Finally, since Z vanishes on the axis, and since the coordinates system we use in thislemma has vanishing Christoffel symbols at q , we have as | x | goes to 0 g µν ∂ µ X∂ ν X X = Z µ D α Z µ Z ν D α Z ν Z µ Z µ = ∂ β Z µ ( q ) x β ∂ α Z µ ( q ) ∂ γ Z ν ( q ) x γ ∂ α Z ν ( q ) ∂ β Z µ ( q ) x β ∂ γ Z µ ( q ) x γ + O ( x ) . Together with the previous identity, we infer near any point q on the axis g µν ∂ µ X∂ ν X X −→ . This concludes the proof of the lemma.We note that Z -polarized, smooth, vectorfields are automatically tangent to A . This isthe content of the following. Lemma 2.1.10.
Any, regular (i.e. smooth) Z -polarized vectorfield U is tangent to theaxis A . CHAPTER 2. PRELIMINARIES
Proof.
Let U a polarized Z -invariant regular vectorfield. Since it is Z -invariant, we have0 = [ Z , U ] = Z α D α U − U α D α Z . Since Z = 0 on the axis and U is regular (hence bounded on the axis) we infer that U α D α Z = 0 on A . In view of (2.1.10), U α D α Z = U ( e φ ) e ϕ , and since e ϕ is unitary, we infer that U ( X / ) = U ( e φ ) = 0 on A and hence U ( X ) = 0 when X = 0. Corollary 2.1.11.
Let u be a smooth regular optical function, i.e. g αβ D α u D β u = 0 ,which is Z -invariant, i.e. Z ( u ) = 0 . Then its associated null geodesic generator L = − g αβ ∂ α u∂ β is Z -invariant, polarized, tangent to the axis of symmetry A .Proof. It is easy to check that L is Z -invariant, polarized. It must therefore be tangentto A in view of Lemma 2.1.10. S - surfaces Throughout our work we shall deal various Z - polarized, S - foliations i.e. foliations givenby compact 2- surfaces S with induced metrics of the form, g/ = γdθ + Xdϕ , γ = γ ( θ ) > , θ ∈ [0 , π ] . (2.1.12)Here γ and X are independent of ϕ and Φ = log X vanishes on the poles θ = 0 and θ = π . The regularity condition (2.1.11) takes the form,lim sin θ → (cid:16) e θ ( e Φ ) (cid:17) = 1 (2.1.13)where e θ is the unit vector, e θ := γ − / ∂ θ . We denote the induced covariant derivative ∇ / and define the volume radius of S by theformula | S | = 4 πr .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES | S | is the volume of the surface using the volume form of the metric g/ . Note alsothat the area element on S is given by √ γe Φ dθdϕ. In this subsection we record some basic general formulas concerning these surfaces. Weconsider adapted orthonormal frames e θ , e ϕ = X − / Z = X − / ∂ ϕ . Note that in view of (2.1.10) we have, ∇ / ϕ e ϕ = − ( e θ Φ) e θ , ∇ / ϕ e θ = ( e θ Φ) e ϕ , ∇ / θ e θ = ∇ / θ e ϕ = 0 . (2.1.14)In what follows, we consider Z -invariant polarized tensors tangent to S or simply polarized k -tensors on S .In view of Lemma 2.1.10, a regular Z -polarized tensor on S must vanish on the axis ofsymmetry i.e. at θ = 0 and θ = π . More precisely we have, Lemma 2.1.12.
The following facts hold true for Z -polarized tensors on S .1. If U is a -form then, on the axis of symmetry of Z , (i.e. for θ = 0 and θ = π ), U θ := U ( e θ ) = 0
2. For a covariant -tensor, then, on the axis of symmetry of Z , (i.e. for θ = 0 and θ = π ), U θθ = U ϕϕ = 0 . Similar statements can be deduced for higher order tensors.Proof.
Immediate consequence of Lemma 2.1.10.
Lemma 2.1.13.
The Gauss curvature K of the metric (2.1.12) can be expressed in termsof the polar function Φ := log X by the formula, (cid:52) / Φ = − K. (2.1.15) Proof.
Direct calculation using the form of the g/ metric in (2.1.12). Note that the component U ϕ must automatically vanish on S . Note that the components U θϕ , U ϕθ must automatically vanish on S . CHAPTER 2. PRELIMINARIES
Basic operators on S We recall (see [17] chapter 2) the following operations which preserve the space of fullysymmetric traceless tensors:
Definition 2.1.14.
We denote by S k the set of k -covariant polarized tensors which arefully symmetric and traceless, i.e. which verify, f A ...A k = f ( A ...A k ) , g/ A A f A A ...A k = 0 . We define the following operators on S k -tensors.1. The operator D / k which takes S k into S k − is the divergence operator, ( D / k f ) A ,...A k : = ( div / f ) A ,...A k := g/ AB ∇ / B f AA ,...A k .
2. The operator D (cid:63) / k which takes S k − into S k is the fully symmetrized, traceless, co-variant derivative operator , ( D (cid:63) / k f ) A ...A k : = (cid:40) −∇ / A f, k = 1 , − k ∇ / ( A f A ...A k ) + k ( k − g/ ( A A ( div / f ) A ...A k ) , k ≥ .
3. The operator (cid:52) / k takes S k to S k , ( (cid:52) / k f ) A ...A k := g/ BC ∇ / B ∇ / C f A ...A k . Remark 2.1.15.
Note that if f ∈ S k then curl / f := ∈ BC ∇ / B f CA ...A k = 0 . Lemma 2.1.16.
Given f ∈ S k , k ≥ , we have the identity, ∇ / B f A ...A k = − ( D (cid:63) / k +1 f ) BA ...A k + 1 k g/ ( BA ( D / k f ) A ...A k ) . (2.1.16) In other words the covariant derivatives of any tensor in S k can be expressed as a linearcombination of D (cid:63) / k +1 f and g/ ⊗ D / k − f .Proof. The proof follows easily from definitions and the vanishing of the curl / . For exam-ple, if k = 2,3 ∇ / B f A A = ( ∇ / B f A A + ∇ / A f A B + ∇ / A f BA )+ ( ∇ / B f A A − ∇ / A f BA ) + ( ∇ / B f A A − ∇ / A f A B )= − D (cid:63) / f ) BA A − g/ A A ( D / f ) B − g/ A B ( D / f ) A − g/ A A ( D / f ) B ]= − (cid:20) ( D (cid:63) / f ) BA A − g/ ( BA ( D / f ) A ) (cid:21) . For an arbitrary k -tensor, f ( A ...A k ) = (cid:80) σ ∈ Π f A σ (1) ...A σ ( k ) . In the particular case when k = 1 we get( D (cid:63) / f ) A = −∇ / A f and when k = 2 we get D (cid:63) / f AB = − ( ∇ / A f B + ∇ / B f A − g/ AB div / f ). .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES D (cid:63) / k is the formal adjoint of D / k , i.e., (cid:90) S ( D / k f ) g = (cid:90) S f ( D (cid:63) / k g ) . It is also easy to check that the kernels of D / k are trivial for all k ≥ D (cid:63) / : S −→ S consists of constants on S while the kernel of D (cid:63) / consists of constant multiple of co-vectors f with f θ = Ce Φ . Moreover, D (cid:63) / · D / = −(cid:52) / + K, D / · D (cid:63) / = −(cid:52) / , D (cid:63) / · D / = − (cid:52) / + K, D / · D (cid:63) / = −
12 ( (cid:52) / + K ) . (2.1.17)Similar identities also hold for higher k . Using (2.1.17) one can also prove the following(see also Chapter 2 in [17]). Proposition 2.1.17.
Let ( S, g/ ) be a compact manifold with Gauss curvature K . Wehave, i.) The following identity holds for vectorfields f ∈ S , (cid:90) S (cid:0) |∇ / f | + K | f | (cid:1) = (cid:90) S | D / f | . ii.) The following identity holds for symmetric, traceless tensors in S , (cid:90) S (cid:0) |∇ / f | + 2 K | f | (cid:1) = 2 (cid:90) S | D / f | . iii.) The following identity holds for scalars f ∈ S , (cid:90) S |∇ / f | = (cid:90) S | D (cid:63) / f | . iv.) The following identity holds for vectors f ∈ S , (cid:90) S (cid:0) |∇ / f | − K | f | (cid:1) = 2 (cid:90) S | D (cid:63) / f | . Proof.
All statements appear in [17].2
CHAPTER 2. PRELIMINARIES
Proposition 2.1.18. we have for f ∈ S , (cid:90) S (cid:0) |∇ / f | + K |∇ / f | (cid:1) = (cid:90) S |(cid:52) / f | . Moreover, under mild assumptions on the curvature such as K = 1 r + O (cid:16) (cid:15)r (cid:17) , re θ ( K ) = O (cid:16) (cid:15)r (cid:17) , for any f ∈ S k , k ≥ , (cid:90) S (cid:0) |∇ / f | + r − |∇ / f | (cid:1) (cid:46) (cid:90) S |(cid:52) / k f | + O ( (cid:15) ) r − (cid:90) S | f | . Proof.
Follows from the standard Bochner identity on S . Reduced PictureLemma 2.1.19.
The following relations hold true between the spacetime picture and thereduced one.1. Let (1+3) f ∈ S k sucht that (1+3) f θ...θ = f . Then, ( D / k (1+3) f ) θ...θ = e θ ( f ) + ke θ (Φ) f. (2.1.18)
2. If f ∈ S we have, d (cid:63) / f = − e θ ( f ) .
3. If (1+3) f ∈ S k − , k ≥ , such that (1+3) f θ...θ = f we have, D (cid:63) / k (1+3) f ) θ...θ = − e θ ( f ) + ( k − e θ (Φ) f. (2.1.19)
4. Let (1+3) f ∈ S k sucht that (1+3) f θ...θ = f . Then, (cid:52) / k (1+3) f θ ...θ k = e θ ( e θ f ) + e θ (Φ) e θ f − k (cid:0) e θ (Φ) (cid:1) f. Proof.
The proof follows easily from the definitions of D / k , D (cid:63) / k , (cid:52) / k and the formulae(2.1.14). We check below the formula (2.1.18). − ( D (cid:63) / k (1+3) f ) θ...θ = e θ f −
12 ( D / k − f ) θ...θ = e θ ( f ) −
12 ( e θ f + ( k − e θ (Φ) f )= 12 ( e θ f − ( k − e θ (Φ) f )as desired. .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES Definition 2.1.20.
We say that a scalar f is a reduced k -scalar on S if there is a Z -invariant, polarized, k -covector (1+3) f ∈ S k such that, f = (1+3) f θ...θ . We denote by s k the set of k reduced scalars. • Given a k reduced scalar f , reduced from (1+3) f we define, |∇ / f | = |∇ / (1+3) f | , |∇ / l f | = |∇ / l (1+3) f | . • Given a k -reduced scalar f on S we define, d/ k f := e θ ( f ) + ke θ (Φ) f. • Given a ( k − -reduced scalar f ∈ S k − we define, d (cid:63) / k f := − e θ ( f ) + ( k − e θ (Φ) f. • Given a k -reduced scalar f ∈ s k we define, (cid:52) / k f := e θ ( e θ f ) + e θ (Φ) e θ f − k (cid:0) e θ (Φ) (cid:1) f. In view of Lemma 2.1.19 we have, d/ k f = ( D / k (1+3) f ) θ...θ and, d (cid:63) / k f = (cid:40) ( D (cid:63) / k (1+3) f ) θ...θ , k = 1 , D (cid:63) / k (1+3) f ) θ...θ , k ≥ . Clearly d/ k takes k -reduced scalars into ( k − -reduced scalars, d (cid:63) / k takes ( k − -reducedones into k -reduced and (cid:52) / k takes k -reduces scalars into k -reduced scalars. Remark 2.1.21.
Note that, in view of Lemma 2.1.12, any reduced scalar in s k , for k ≥ ,must vanish on the axis of symmetry of Z , i.e. at the two poles. Remark 2.1.22.
The operator d/ k and d (cid:63) / k can only be applied to k -reduced, resp ( k − -reduced scalars. Thus whenever we write a sequence of operators involving d/ k , d (cid:63) / k weunderstand from the context to which type of k -reduced scalars they are applied, see forexample the proposition below. The same remark applies to (cid:52) / k . CHAPTER 2. PRELIMINARIES
Remark 2.1.23.
Note that for given reduced scalar f ∈ s k and h ∈ s we can write, he θ ( f ) = 12 h ( d/ k f − d (cid:63) / k +1 f ) . The term h d/ k f is the reduced form of a tensor product of (1+3) h with D / k (1+3) f while h d (cid:63) / k +1 f is the reduced form of a contraction between (1+3) h and D (cid:63) / k +1 (1+3) ψ This can beformalized precisely using Lemma 2.1.16. The Remark will be useful in what follows, forexample in Lemma 2.2.14.
Remark 2.1.24.
The duality between the operators d/ k and d (cid:63) / k follows in view of theduality of D / k and D (cid:63) / k . It can also be interpreted directly in terms of the area element √ γe Φ dθdϕ , (cid:90) S ( d/ k f g − g d (cid:63) / k g ) da S = (cid:90) S e θ ( f g ) + e θ (Φ) f g = (cid:90) π (cid:90) π ( e θ ( f g ) + e θ (Φ) f g ) √ γe Φ dθdϕ = (cid:90) π (cid:90) π ∂ θ ( e Φ f g ) dθdϕ = 0 . Proposition 2.1.25.
The following identities hold true, d (cid:63) / k d/ k = −(cid:52) / k + kK,d/ k d (cid:63) / k = −(cid:52) / k − − ( k − K. (2.1.20) In particular for k = 1 , d (cid:63) / d/ = −(cid:52) / + K, d/ d (cid:63) / = −(cid:52) / , d (cid:63) / d/ = −(cid:52) / + 2 K, d/ d (cid:63) / = −(cid:52) / − K. Moreover, note the following commutation formulas d/ k d (cid:63) / k − d (cid:63) / k − d/ k − = − k − K, − d/ k (cid:52) / k + (cid:52) / k − d/ k = K d/ k − ke θ ( K ) , − d (cid:63) / k (cid:52) / k − + (cid:52) / k d (cid:63) / k = (2 k − K d (cid:63) / k + ( k − e θ ( K ) . Proof.
We have, for a k reduced scalar f , − d (cid:63) / k d/ k f = ( e θ − ( k − e θ (Φ))( e θ ( f ) + ke θ (Φ) f )= e θ ( e θ ( f )) + ke θ (Φ) e θ f + k ( e θ e θ Φ) f − ( k − e θ (Φ))( e θ ( f ) + ke θ (Φ) f )= e θ ( e θ ( f )) + e θ (Φ) e θ f + k ( e θ e θ Φ) f − k ( k − (cid:0) e θ (Φ) (cid:1) . In view of Lemma 2.1.13 we have, since Φ is a scalar − K = (cid:52) / Φ = e θ e θ (Φ) + (cid:0) e θ (Φ) (cid:1) . .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES − d (cid:63) / k d/ k f = e θ ( e θ ( f )) + e θ (Φ) e θ f + k (cid:16) − K − (cid:0) e θ (Φ) (cid:1) (cid:17) f − k ( k − (cid:0) e θ (Φ) (cid:1) = e θ ( e θ ( f )) + e θ (Φ) e θ f − kKf − k (cid:0) e θ (Φ) (cid:1) = (cid:52) / k f − kKf. Similarly, for a ( k − f , − d/ k d (cid:63) / k f = ( e θ + ke θ (Φ))( e θ ( f ) − ( k − e θ (Φ) f )= e θ ( e θ ( f )) + ke θ (Φ) e θ f − ( k − e θ e θ Φ) f − ( k − e θ (Φ) e θ ( f ) − k ( k − (cid:0) e θ (Φ) (cid:1) f = e θ ( e θ ( f )) + e θ (Φ) e θ f − ( k − (cid:16) − K − (cid:0) e θ (Φ) (cid:1) (cid:17) − k ( k − (cid:0) e θ (Φ) (cid:1) f = e θ ( e θ ( f )) + e θ (Φ) e θ f + ( k − K − ( k − (cid:0) e θ (Φ) (cid:1) f = (cid:52) / k − f + ( k − K. Next, we check the commutation formulas. We have d/ k d (cid:63) / k − d (cid:63) / k − d/ k − = −(cid:52) / k − − ( k − K − (cid:16) − (cid:52) / k − + ( k − K (cid:17) = − k − K from which we infer d/ k ( −(cid:52) / k ) = d/ k ( d (cid:63) / k d/ k − kK )= d/ k d (cid:63) / k d/ k − kK d/ k − ke θ ( K )= (cid:16) d (cid:63) / k − d/ k − − k − K (cid:17) d/ k − kK d/ k − ke θ ( K )= (cid:16) − (cid:52) / k − + ( k − K (cid:17) d/ k − ( k − K d/ k − ke θ ( K )and hence − d/ k (cid:52) / k + (cid:52) / k − d/ k = K d/ k − ke θ ( K ) . Also, we have d (cid:63) / k ( −(cid:52) / k − ) = d (cid:63) / k ( d/ k d (cid:63) / k + ( k − K )= d (cid:63) / k d/ k d (cid:63) / k + ( k − K d (cid:63) / k + ( k − e θ ( K )= (cid:16) − (cid:52) / k + kK (cid:17) d (cid:63) / k + ( k − K d (cid:63) / k + ( k − e θ ( K )and hence − d (cid:63) / k (cid:52) / k − + (cid:52) / k d (cid:63) / k = (2 k − K d (cid:63) / k + ( k − e θ ( K )as desired.6 CHAPTER 2. PRELIMINARIES
A remarkable identity
First, note the following observation which follows immediately from the form of d (cid:63) / . Lemma 2.1.26.
The kernel of d (cid:63) / is spanned by e Φ . The above lemma, in connection with a Poincar´e inequality for d (cid:63) / , see (2.1.35), will resultin the need of a specific treatment for the projection of some of the quantities on the kernelof d (cid:63) / . This motivates the following definition. Definition 2.1.27 (The (cid:96) = 1 mode) . For a -reduced scalar f , the (cid:96) = 1 mode denotesits projection on the kernel of d (cid:63) / , i.e. (cid:90) S f e Φ . For a -reduced scalar f , the (cid:96) = 1 mode denotes the projection of e θ ( f ) on the kernel of d (cid:63) / , i.e. (cid:90) S e θ ( f ) e Φ . Remark 2.1.28.
The above definition is motivated by the fact that, in Schwarzschild,this corresponds to the projection on the (cid:96) = 1 spherical harmonic . We are now ready to state the following remarkable identity which will play a crucial rolelater in the paper.
Lemma 2.1.29 (Vanishing of the (cid:96) = 1 mode of the Gauss curvature) . The (cid:96) = 1 modeof K vanishes identically, i.e. (cid:90) S e θ ( K ) e Φ = 0 . (2.1.21) Proof.
To prove (2.1.21) we write, − (cid:90) S e θ ( K ) e Φ = (cid:90) S d (cid:63) / ( K ) e Φ = (cid:90) S K d/ ( e Φ ) = 2 (cid:90) S Ke θ (Φ) e Φ . In general, there are 3 spherical harmonics corresponding to (cid:96) = 1, but only one is axially symmetric.This is why we have only one projection instead of 3 in our case. .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES (cid:52) / Φ = e θ ( e θ (Φ)) + e θ (Φ) − (cid:90) S e θ ( K ) e Φ = 2 (cid:90) S (cid:52) / Φ e θ (Φ) e Φ = 2 (cid:90) S (cid:0) e θ ( e θ (Φ)) + e θ (Φ) (cid:1) e θ (Φ) e Φ = (cid:90) S d/ (cid:0) ( e θ Φ) (cid:1) e Φ = − (cid:90) S ( e θ Φ) d (cid:63) / ( e Φ ) = 0as desired . Poincar´e inequalities on 2-spheres
Proposition 2.1.17 takes the following reduced form,
Proposition 2.1.30.
The following identities hold true for reduced k -scalars f ∈ s k . i.) If f ∈ s , (cid:90) S (cid:0) |∇ / f | + Kf (cid:1) = (cid:90) S | d/ f | . (2.1.22) ii.) If f ∈ s , (cid:90) S (cid:0) |∇ / f | + 4 Kf (cid:1) = 2 (cid:90) S | d/ f | . (2.1.23) iii.) If f ∈ s , (cid:90) S |∇ / f | = (cid:90) S | d (cid:63) / f | . (2.1.24) iv.) If f ∈ s , (cid:90) S (cid:0) |∇ / f | − Kf (cid:1) = (cid:90) S | d (cid:63) / f | . (2.1.25) v.) If f ∈ s , (cid:90) S |∇ / f | + (cid:90) S K |∇ / f | = (cid:90) S |(cid:52) / f | . (2.1.26) Under mild assumptions on the Gauss curvature K , such as K = 1 r + O (cid:16) (cid:15)r (cid:17) , re θ ( K ) = O (cid:16) (cid:15)r (cid:17) . Note that the boundary term which appears from the last integration by parts has the form( ∂ θ Φ) e ( π ) − ( ∂ θ Φ) e (0) and hence vanishes in view of the regularity condition (2.1.13), see alsothe computation in Remark 2.1.24. CHAPTER 2. PRELIMINARIES
We also have for f ∈ s k , k ≥ , (cid:107)∇ / f (cid:107) L ( S ) + r − (cid:107)∇ / f (cid:107) L ( S ) (cid:46) (cid:107)(cid:52) / k f (cid:107) L ( S ) + (cid:15)r − (cid:107) f (cid:107) L ( S ) . (2.1.27) Proof.
The proof of the above statements can be either derived from their space-timeversion or checked directly.
Lemma 2.1.31.
The following relations hold between Z -polarized S -tensors and reducedscalars . • If f ∈ s |∇ / f | = | e θ f | , |∇ / f | = | e θ ( e θ f ) | + | e θ Φ e θ f | . • If f ∈ s , |∇ / f | = | e θ f | + | e θ (Φ) | | f | . • If f ∈ s , |∇ / f | = 2 (cid:0) | e θ f | + 4 | e θ (Φ) | | f | (cid:1) . Proof. If f ∈ s , |∇ / f | = ∇ A ∇ / B f ∇ / A ∇ / B f = |∇ / θ ∇ / θ f | + |∇ / ϕ ∇ / ϕ f | = | e θ ( e θ f ) | + | e θ Φ e θ f | . If f ∈ s is reduced from a Z invariant, polarized vector F , |∇ / f | = ∇ A F B ∇ / A F B = |∇ / θ F θ | + |∇ / ϕ F ϕ | = | e θ f | + | e θ Φ f | . If f ∈ s is reduced from a symmetric, traceless Z -invariant, polarized tensor F = (1+3) f we have, |∇ / f | = |∇ / θ F θθ | + 2 |∇ / θ F ϕθ | + |∇ / θ F ϕϕ | + |∇ / ϕ F θθ | + 2 |∇ / ϕ F ϕθ | + |∇ / ϕ F ϕϕ | = |∇ / θ F θθ | + |∇ / θ F ϕϕ | + 2 |∇ / ϕ F ϕθ | Note that the expressions on the left of the inequalities below should be interpreted as applying tothe spacetime tensor from which f is reduced. .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES ∇ / θ F θθ = e θ f = −∇ / θ F ϕϕ , ∇ / ϕ F ϕθ = e θ Φ F θθ − e θ Φ F ϕϕ = 2 e θ Φ f. Thus, |∇ / f | = 2 | e θ f | + 8( e θ Φ f ) as desired. Proposition 2.1.32 (Poincar´e) . The following inequalities hold for k -reduced scalars.1. If f ∈ s , (cid:90) S |∇ / f | ≥ (cid:90) S K | d (cid:63) / f | . (2.1.28)
2. If f ∈ s (cid:90) S |∇ / f | ≥ (cid:90) S Kf . (2.1.29)
3. If f ∈ s , (cid:90) S |∇ / f | ≥ (cid:90) S Kf . (2.1.30) Proof.
We first prove the result for f ∈ s . According to Lemma 2.1.31,2 − |∇ / f | = | e θ f | + 4 | e θ (Φ) | | f | = ( e θ f − e θ (Φ) f ) + 4 f ( e θ f ) e θ (Φ)= ( e θ f − e θ (Φ) f ) + 2 e θ ( f ) e θ (Φ) . Hence, 2 − (cid:90) S |∇ / F | da S = (cid:90) S ( e θ f − e θ (Φ) f ) da S + 2 (cid:90) S e θ ( f ) e θ (Φ) √ γe Φ dθdϕ. Now, (cid:90) S e θ ( f ) e θ (Φ) √ γe Φ dθdϕ = (cid:90) π (cid:90) π ∂ θ ( f ) e θ (Φ) e Φ dθdϕ = − (cid:90) π (cid:90) π f e θ ( e Φ e θ Φ) √ γdθdϕ = − (cid:90) π (cid:90) π (cid:0) e θ e θ Φ + ( e θ Φ) (cid:1) f e Φ √ γdθdϕ = (cid:90) S Kf da S . CHAPTER 2. PRELIMINARIES
Hence, (cid:90) S |∇ / f | ≥ (cid:90) S Kf as desired.The result for f ∈ s is proved in the same way.If f ∈ s we write, according to Lemma 2.1.31, |∇ / f | = | e θ ( e θ f ) | + | e θ Φ e θ f | = | e θ h | + | e θ Φ | | e θ f | = ( e θ e θ f − e θ (Φ) e θ f ) + e θ [( e θ f ) ] e θ (Φ) . Integrating by parts as before, (cid:90) S e θ [( e θ f ) ] e θ (Φ) da S = − (cid:90) π (cid:90) π ( e θ f ) e θ ( e Φ e θ Φ) √ γdθdϕ = (cid:90) S K ( e θ f ) da S . Thus, (cid:90) S |∇ / f | ≥ (cid:90) S K ( e θ f ) . As a corollary we deduce the following,
Corollary 2.1.33.
The following hold true for reduced scalars,1. If f ∈ s , (cid:90) S | d/ f | ≥ (cid:90) S Kf . (2.1.31)
2. If f ∈ s , (cid:90) S | d/ f | ≥ (cid:90) S Kf . (2.1.32) Proof.
According to (2.1.22), (cid:90) S (cid:0) |∇ / f | + Kf (cid:1) = (cid:90) S | d/ f | . .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES (cid:90) S | d/ f | ≥ (cid:90) S Kf . According to (2.1.23) (cid:90) S (cid:0) |∇ / f | + 4 Kf (cid:1) = 2 (cid:90) S | d/ f | . Hence, 2 (cid:90) S | d/ f | ≥ (cid:90) S Kf as desired. Corollary 2.1.34.
Under the following mild assumptions on the Gauss curvature K = 1 r + O (cid:16) (cid:15)r (cid:17) , re θ ( K ) = O (cid:16) (cid:15)r (cid:17) , the following holds.1. If f ∈ s is orthogonal to the kernel of d (cid:63) / , i.e. (cid:82) S f = 0 , then, we have (cid:90) S | d (cid:63) / f | ≥ (cid:90) S (1 + O ( (cid:15) )) Kf . (2.1.33)
2. If f ∈ s is orthogonal to the kernel of d (cid:63) / , i.e (cid:82) S f e Φ = 0 , then, we have (cid:90) S | d/ f | ≥ (cid:90) S (1 + O ( (cid:15) )) Kf and (cid:90) S | d (cid:63) / f | ≥ (cid:90) S (1 + O ( (cid:15) )) Kf . (2.1.34) Proof.
We start with the first assertion. If f ∈ s satisfies (cid:82) S f = 0 then, f is orthogonalto 1 which generates the kernel of d (cid:63) / , and hence, f is in the image of d/ , i.e. there exists h ∈ s such that f = d/ h. We deduce (cid:90) S ( d (cid:63) / ( f )) = (cid:90) S ( d (cid:63) / d/ h ) = (cid:90) S ( d (cid:63) / d/ ) hh. CHAPTER 2. PRELIMINARIES
Now, the above Poincar´e inequality for d/ and the assumption on K implies a lower boundfor the spectrum of the selfadjoint operator d (cid:63) / d/ by 2 K (1 + O ( (cid:15) )), and hence (cid:90) S ( d (cid:63) / ( f )) ≥ (cid:90) S K (1 + O ( (cid:15) ))( d (cid:63) / d/ ) hh ≥ (cid:90) S (1 + O ( (cid:15) ))( d/ h ) ≥ (cid:90) S (1 + O ( (cid:15) )) f which yields the first assertion.Assume now that f ∈ s satisfies (cid:82) S f e Φ = 0 i.e. , f is orthogonal to e Φ which generatesthe kernel of d (cid:63) / , and hence, f is in the image of d/ , i.e. there exists h ∈ s such that f = d/ h. We deduce (cid:90) S ( d/ f ) = (cid:90) S ( d/ d/ h ) (cid:90) S d (cid:63) / d/ d/ h d/ h = (cid:90) S ( d/ d (cid:63) / + 2 K ) d/ h d/ h = (cid:90) S ( d (cid:63) / d/ ) hh + (cid:90) K ( d/ h ) . Now, the above Poincar´e inequality for d/ and the assumption on K implies a lower boundfor the spectrum of the selfadjoint operator d (cid:63) / d/ by 4 K (1 + O ( (cid:15) )), and hence (cid:90) S ( d/ f ) ≥ (cid:90) S K (1 + O ( (cid:15) ))( d (cid:63) / d/ ) hh + (cid:90) K ( d/ h ) ≥ (cid:90) S (1 + O ( (cid:15) ))( d/ h ) ≥ (cid:90) S (1 + O ( (cid:15) )) f . Together with the fact that (cid:90) S ( d (cid:63) / f ) = (cid:90) S d/ d (cid:63) / f f = (cid:90) S ( d (cid:63) / d/ − K ) f f = (cid:90) S ( d/ f ) − (cid:90) S Kf , this yields the second assertion and concludes the proof of the corollary. Lemma 2.1.35.
Assume that K = 1 r + O (cid:16) (cid:15)r (cid:17) , re θ ( K ) = O (cid:16) (cid:15)r (cid:17) , (cid:90) S e = r (cid:18) π O ( (cid:15) ) (cid:19) . .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES Then, If f ∈ s , we have the estimate, (cid:90) S (cid:12)(cid:12) f | (cid:46) r (cid:90) S | d (cid:63) / f | + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e Φ f (cid:12)(cid:12)(cid:12)(cid:12) . (2.1.35) More precisely, f = (cid:82) S f e Φ (cid:82) S e e Φ + f ⊥ (2.1.36) with (cid:90) S (cid:12)(cid:12) f ⊥ | (cid:46) r (cid:90) S | d (cid:63) / f | . Proof.
According to Corollary 2.1.34, see 2.1.34, if f ∈ s is orthogonal to the kernel of d (cid:63) / , i.e (cid:82) S f e Φ = 0, then, we have (cid:90) S | d (cid:63) / f | ≥ (cid:90) S (1 + O ( (cid:15) )) Kf . As a consequence f ⊥ = f − (cid:16) (cid:82) S fe Φ (cid:82) S e (cid:17) e Φ verifies, r − (cid:90) S | f ⊥ | (cid:46) (cid:90) S | d (cid:63) / ( f ⊥ ) | = (cid:90) S | d (cid:63) / f | from which we derive, (cid:90) S (cid:12)(cid:12) f − (cid:18) (cid:82) S f e Φ (cid:82) S e (cid:19) e Φ (cid:12)(cid:12) (cid:46) r (cid:90) S | d (cid:63) / f | or, (cid:90) S (cid:12)(cid:12) f | = r (cid:90) S | d (cid:63) / f | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e Φ f (cid:12)(cid:12)(cid:12)(cid:12) (cid:82) S e ≤ r (cid:90) S | d (cid:63) / f | + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e Φ f (cid:12)(cid:12)(cid:12)(cid:12) as desired.4 CHAPTER 2. PRELIMINARIES
Higher derivative operators and spacesDefinition 2.1.36.
Given f a k -reduced scalar and s a positive integer we define, d / s f = (cid:40) r p (cid:52) / pk , if s = 2 p,r p +1 d/ k (cid:52) / pk , if s = 2 p + 1 . (2.1.37) We also define the norms, (cid:107) f (cid:107) h s ( S ) : = s (cid:88) i =0 (cid:107) d / i f (cid:107) L ( S ) . (2.1.38) Lemma 2.1.37.
Assume the Gauss curvature K of S verifies the condition, K = 1 r + O ( (cid:15) ) , | r i ∇ / i K | = O ( (cid:15) ) , ≤ i ≤ [ s/
2] + 1 . Then, the following holds.1. If f is a k -scalar, reduced from (1+3) f , we have, (cid:107) f (cid:107) h s ( S ) ∼ s (cid:88) j =0 r j (cid:107)∇ / j f (cid:107) L ( S ) (2.1.39) where ∇ / denotes the usual covariant derivative operator on S .2. Equivalently, the norm r − s (cid:107) f (cid:107) h s ( S ) of a reduced scalar f ∈ s s ( S ) can be defined asthe sum of L norms of any allowable sequence of Hodge operators d/ a , d (cid:63) / a appliedto f .Proof. For s = 1 , s the proof follows, step by step, by a simple commutation argument betweencovariant derivatives and (cid:52) / k and applications of Proposition 2.1.30. The proof of thesecond part follows from our reduced elliptic estimates and definition of the reducedHodge operators.As a consequence of the lemma we can derive the reduce form of the standard Sobolevand product Sobolev inequalities. Before stating the result we pause to define the productof two reduced scalars. .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES Definition 2.1.38.
Let f ∈ s a be reduced from an S a tensor and g ∈ s b reduced froman S b tensor. We define the product f · g to be the reduction of any product betweenthe corresponding tensors on S , i.e. any contraction of the tensor product between them.Thus f · g ∈ s a + b − c where c denotes the number of indices affected by the contraction. Examples.
Here are the most relevant examples for us. • f ∈ s , g ∈ s k in which case f · g ∈ s k and equals f g . • f ∈ s , g ∈ s k in which case f · g ∈ s k − or f · g ∈ s k +1 and in both cases f · g = f g as simple product of the reduced scalars. • f ∈ s , g ∈ s k in which case f · g ∈ s k − or f · g ∈ s k or f · g ∈ s k +2 . In the first case f · g = 2 f g . In the second case and third cases f · g = f g as simple product of thereduced scalars. Lemma 2.1.39.
Let f ∈ s a ( S ) , g ∈ s b ( S ) , a ≥ b , a > , and f · g ∈ s a + b − c where ≤ c ≤ ( a − b ) denotes the order of contraction. Then, d/ a + b − c ( f g ) = f d/ b g + g (cid:16) (1 − ca ) d/ a f − ca d (cid:63) / a +1 f (cid:17) ,d (cid:63) / a + b − c +1 ( f g ) = f d (cid:63) / b +1 g + (cid:16) ( − ca ) d/ a f + ca d (cid:63) / a +1 f (cid:17) . (2.1.40) Proof.
Assume a ≥ b and c ≤ a − b . We write, d/ a + b − c ( f g ) = f d/ b g + g ( e θ f + ( a − c ) e θ (Φ) f ) . We look for reals
A, B wit A + B = 1 such that e θ f + ( a − c ) e θ (Φ) f = A d/ a f − B d (cid:63) / a +1 f = e θ f + a ( A − B ) e θ Φ f. Therefore, a (1 − B ) = a − c i.e. B = ca , A = 1 − ca and we derive, d/ a + b − c ( f g ) = f d/ b g + g (cid:16) (1 − ca ) d/ a f − ca d (cid:63) / a +1 f (cid:17) . Also, d (cid:63) / a + b − c +1 ( f g ) = f d (cid:63) / b +1 g + g ( − e θ ( f ) + ( a − c ) e θ (Φ) f ) . CHAPTER 2. PRELIMINARIES
As before we write, with A + B = − − e θ ( f ) + ( a − c ) e θ (Φ) f = A d/ a f − B d (cid:63) / a +1 f = − e θ f + a ( A − B ) e θ Φ f. Hence, − a ( − − B ) = ( a − c )i.e. B = − ca , A = − ca . Hence, d (cid:63) / a + b − c +1 ( f g ) = f d (cid:63) / b +1 g + (cid:16) ( − ca ) d/ a f + ca d (cid:63) / a +1 f (cid:17) as desired. Proposition 2.1.40.
The following results hold true for k -reduced scalars on S ,1. If f ∈ s k we have, (cid:107) f (cid:107) L ∞ ( S ) (cid:46) r − (cid:107) f (cid:107) h ( S ) .
2. Given two reduced scalars f, g we have, (cid:107) f · g (cid:107) h s ( S ) (cid:46) r − (cid:16) (cid:107) f (cid:107) h [ s/ ( S ) (cid:107) g (cid:107) h s ( S ) + (cid:107) g (cid:107) h [ s/ ( S ) (cid:107) f (cid:107) h s ( S ) (cid:17) where [ s/ denotes the largest integer smaller than s/ .Proof. Both statements are classical for S k ( S ) tensors with respect to the norm on theright hand side of (2.1.39). A direct proof can also be derived using Lemma 2.1.39 andthe equivalence definition of the h s ( S ) norms. S -averagesDefinition 2.1.41. Given any f ∈ s we denote its average by, ¯ f : = 1 | S | (cid:90) S f, ˇ f := f − ¯ f. The following follows immediately from the definition. .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES Lemma 2.1.42.
For any two scalar reduced scalars f and g in s we have f g = f g + ˇ f ˇ g, and, f g − f g = ˇ f g + f ˇ g + (cid:0) ˇ f ˇ g − ˇ f ˇ g (cid:1) . Remark 2.1.43.
In view of the notations above, we may rewrite the Poincar´e inequal-ity for d (cid:63) / as follows. Under mild assumptions on the Gauss curvature ( K = r − + O ( (cid:15)r − ) , re θ ( K ) = O ( (cid:15)r − ) ), we have for any f ∈ s (cid:90) S | d (cid:63) / f | ≥ (cid:90) S (1 + O ( (cid:15) )) K ( ˇ f ) . S -foliations In this section we record the main equations associated to general, Z -invariant Einsteinvacuum spacetimes ( M , g ). We start by recalling the spacetime framework of [17] andthen we show how the null structure and Bianchi identities simplify in the reduced picture.Throughout this section we consider given an invariant S -foliation and a fixed adaptednull pair e , e , i.e. future directed Z - invariant, polarized, null vectors orthogonal to theleaves S of the foliation such as g ( e , e ) = − Definition 2.1.44.
We denote by S k ( M ) the set of k -covariant polarized tensors on M tangent to the S -foliation and which restrict to S k ( S ) on any S -surface of the foliationand by s k ( M ) their corresponding reductions. Spacetime null decompositions
Following [17] we define the spacetime Ricci coefficients, (1+3) χ AB : = g ( D A e , e B ) , (1+3) ξ A := 12 g ( D e , e A ) , (1+3) η A := 12 g ( D e , e A ) , (1+3) ζ A : = 12 g ( D A e , e ) , (1+3) ω := 14 g ( D e , e ) , (2.1.41)and interchanging e , e , (1+3) χ AB : = g ( D A e , e B ) , (1+3) ξ A := 12 g ( D e , e A ) , (1+3) η A := 12 g ( D e , e A ) , (1+3) ζ A : = − g ( D A e , e ) , (1+3) ω := 14 g ( D e , e ) . (2.1.42) From now on, an invariant S foliation is automatically assumed to be a Z invariant polarized foliation. CHAPTER 2. PRELIMINARIES
We also define the spacetime null curvature components, (1+3) α AB : = R A B , (1+3) β A := 12 R A , (1+3) ρ := 14 R , (1+3) α AB : = R A B , (1+3) β A := 12 R A , (1+3) (cid:63) ρ := 14 (cid:63) R . (2.1.43) Reduced null decompositions
We define the spacetime Ricci coefficients as follows
Definition 2.1.45 ( Ricci coefficients ) . Let e , e , e θ be a reduced null frame. The followingscalars χ = g ( D θ e , e θ ) , χ = g ( D θ e , e θ ) ,η = 12 g ( D e , e θ ) , η = 12 g ( D e , e θ ) ,ξ = 12 g ( D e , e θ ) , ξ = 12 g ( D e , e θ ) ,ω = 14 g ( D e , e ) , ω = 14 g ( D e , e ) ,ζ = 12 g ( D θ e , e ) , (2.1.44) are called the Ricci coefficients associated to our canonical null pair. Lemma 2.1.46.
The following lemma follows easily from the definitions, D e = − ωe + 2 ξe θ , D e = − ωe + 2 ξe θ ,D e = 2 ωe + 2 ηe θ , D e = 2 ωe + 2 ηe θ ,D e θ = ηe + ξe , D e θ = ξe + ηe ,D θ e = − ζe + χe θ , D θ e = ζe + χe θ ,D θ e θ = 12 χe + 12 χe . (2.1.45) Definition 2.1.47.
The null components of the Ricci curvature tensor of the metric g are denoted by R = α, R = α, R θ = β, R θ = β, R θθ = R = ρ, R = ρ. Recall that the scalar curvature of the reduced metric g vanishes, R = 0, and hence R = R θθ . .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES Comparison to the space-time frame
Let e , e , e θ be a null frame for the reduced metric g and e , e , e θ , e ϕ = X − / ∂ ϕ theaugmented adapted 3 + 1 frame for g . Recall that we have denoted, (1+3) χ, (1+3) ξ, (1+3) η, (1+3) η, (1+3) ζ, (1+3) ω, (1+3) χ, (1+3) ξ, (1+3) ω, the standard (as defined in [17] ) space-time Ricci coefficients and by (1+3) α, (1+3) β, (1+3) ρ, (1+3) (cid:63) ρ, (1+3) β, (1+3) α, the null decomposition of the curvature tensor R . Proposition 2.1.48.
The following relations between the spacetime and reduced Ricciand curvature null components hold true, • We have, (1+3) χ θθ = χ, (1+3) χ θϕ = 0 , (1+3) χ ϕϕ = e (Φ) , (1+3) χ θθ = χ, (1+3) χ θϕ = 0 , (1+3) χ ϕϕ = e (Φ) , (1+3) α θθ = α, (1+3) α θϕ = 0 , (1+3) α ϕϕ = − α, (1+3) α θθ = α, (1+3) α θϕ = 0 , (1+3) α ϕϕ = − α. • All e ϕ components of (1+3) η, (1+3) η, (1+3) ζ, (1+3) ξ, (1+3) ξ, (1+3) β, (1+3) β vanish and, (1+3) η θ = η, (1+3) η θ = η, (1+3) ζ θ = ζ, (1+3) ξ θ = ξ, (1+3) ξ θ = ξ, and
10 (1+3) β θ = β, (1+3) β θ = − β. Also, (1+3) ω = ω, (1+3) ω = ω, (1+3) ρ = ρ, (1+3) (cid:63) ρ = 0 . • We have, (1+3) tr χ = χ + e (Φ) , (1+3) tr χ = χ + e (Φ) . Recalling, see definition in [17], (1+3) (cid:98) χ AB = (1+3) χ AB −
12 ( (1+3) tr χ ) g/ AB , (1+3) (cid:98) χ AB = (1+3) χ AB −
12 ( (1+3) tr χ ) g/ AB , we have (1+3) (cid:98) χ θθ = 12 ( χ − e (Φ)) , (1+3) (cid:98) χ θθ = 12 (cid:0) χ − e (Φ) (cid:1) . Note the change of sign for the β component. CHAPTER 2. PRELIMINARIES
Proof.
We check only the less obvious relations such as those involving the null compo-nents of curvature. Using (2.1.5) and (2.1.6) we deduce, (1+3) α θθ = R θ θ = R θ θ = g θθ R = α, (1+3) α θϕ = R θ ϕ = 0 , (1+3) α ϕϕ = R ϕ ϕ = − R = − α, (1+3) β θ = R θ = R θ = − g R θ = 2 β, (1+3) β ϕ = R ϕ = 0 , (1+3) ρ = R = R = − g R = 4 ρ, (1+3) (cid:63) ρ = (cid:63) R = 0 , (1+3) β θ = R θ = R θ = g R θ = − β. Definition 2.1.49.
We introduce the notation, ϑ : = χ − e (Φ) , κ := (1+3) tr χ = χ + e (Φ) ,ϑ : = χ − e (Φ) , κ := (1+3) tr χ = χ + e (Φ) . Thus, (1+3) (cid:98) χ θθ = (1+3) (cid:98) χ ϕϕ = 12 ϑ, (1+3) (cid:98) χ θθ = (1+3) (cid:98) χ ϕϕ = 12 ϑ. In particular, χ = ( ϑ + κ ) and χ = ( ϑ + κ ) . Remark 2.1.50.
In view of Proposition 2.1.48 we have,1. The quantities κ, κ, ω, ω, ρ are reduced scalars in s .2. The quantities η, η, ζ, ξ, ξ, β, β are reduced scalars in s .3. The quantities ϑ, ϑ, α, α are reduced scalars in s . Commutation identities
We record first the commutation relations between the elements of the frame,[ e θ , e ] = 12 ( κ + ϑ ) e θ + ( ζ − η ) e − ξe , [ e θ , e ] = 12 ( κ + ϑ ) e θ − ( ζ + η ) e − ξe , [ e , e ] = 2 ωe − ωe + 2( η − η ) e θ . .1. AXIALLY SYMMETRIC POLARIZED SPACETIMES Lemma 2.1.51.
The following commutation formulae hold true for reduced scalars.1. If f ∈ s k , [ d/ k , e ] f = 12 κ d/ k f + Com k ( f ) ,Com k ( f ) = − ϑ d (cid:63) / k +1 f + ( ζ − η ) e f − kηe Φ f − ξ ( e f + ke (Φ) f ) − kβf, [ d/ k , e ] = 12 κ d/ k f + Com k ( f ) , Com k ( f ) = − ϑ d (cid:63) / k +1 f − ( ζ + η ) e f − kηe Φ f − ξ ( e f + ke (Φ) f ) − kβf. (2.1.46)
2. If f ∈ s k − [ d (cid:63) / k , e ] f = 12 κ d (cid:63) / k f + Com ∗ k ( f ) ,Com ∗ k ( f ) = − ϑ d/ k − f − ( ζ − η ) e f − ( k − ηe Φ f + ξ ( e f − ( k − e (Φ) f ) − ( k − βf, [ d (cid:63) / k , e ] f = 12 κ d (cid:63) / k f + Com ∗ k ( f ) , Com ∗ k ( f ) = − ϑ d/ k − f + ( ζ + η ) e f − ( k − ηe Φ f + ξ ( e f − ( k − e (Φ) f ) − ( k − βf. (2.1.47) Proof.
We write, [ e θ + ke θ (Φ) , e ] f = [ e θ , e ] f − k ( e e θ Φ) f. Recall that (see (2.1.4)), D a D b Φ = R ab − D a Φ D b Φ. Hence, e e θ Φ − ξe Φ − ηe Φ = D D θ Φ = R θ − D Φ D θ Φ = β − e Φ e θ Φ . Thus, e e θ Φ = β − e Φ e θ Φ + ηe (Φ) + ξe Φ . CHAPTER 2. PRELIMINARIES
We deduce, since e Φ = ( κ − ϑ ),[ e θ + ke θ (Φ) , e ] f = [ e θ , e ] f − k (cid:0) β − e Φ e θ Φ + ηe (Φ) + ξe Φ (cid:1) f = 12 ( κ + ϑ ) e θ f + ( ζ − η ) e f − ξe f − k (cid:0) β − e Φ e θ Φ + ηe (Φ) + ξe Φ (cid:1) f = 12 ( κ + ϑ ) e θ f + k
12 ( κ − ϑ ) e θ Φ f + ( ζ − η ) e f − kηe Φ f − ξ ( e f + ke (Φ) f ) − kβf = 12 κ d/ k f + 12 ϑ ( e θ f − ke θ Φ f )+ ( ζ − η ) e f − kηe Φ f − ξ ( e f + ke (Φ) f ) − kβf i.e., recalling the definition of d (cid:63) / k +1 ,[ e θ + ke θ (Φ) , e ] f = 12 κ d/ k f + Com k ( f ) ,Com ( f ) k = − ϑ d (cid:63) / k +1 f + ( ζ − η ) e f − kηe Φ f − ξ ( e f + ke (Φ) f ) − kβf. The other commutation formulae are proved in the same manner.
In standard coordinates the Schwarzschild metric has the form, ds = − Υ dt + Υ − dr + r dθ + X dϕ , (2.1.48)where, Υ := 1 − mr , X = r sin θ. We denote by T the stationary Killing vectorfield T = ∂ t and by Z = ∂ ϕ the axialsymmetric one. Recall the regular, Z -invariant optical functions in the exterior region r ≥ m of Schwarzschild u = t − r ∗ , u = t + r ∗ , dr ∗ dr = Υ − (2.1.49)with r ∗ = r + 2 m log( r m − L := − g ab ∂ a v∂ b = Υ − ∂ t − ∂ r , L := − g ab ∂ a u∂ b = Υ − ∂ t + ∂ r . (2.1.50)Clearly, g ( L, L ) = g ( L, L ) = 0 , g ( L, L ) = − − , D L L = D L L = 0 . .2. MAIN EQUATIONS Definition 2.1.52.
We can use the null geodesic generators
L, L to define the followingcanonical null pairs. In all cases all curvature components vanish identically except, (1+3) ρ = − mr . (2.1.51)
1. The null frame ( e , e ) for which e is geodesic (which is regular towards the futurefor all r > ) is given by e = L = Υ − ∂ t − ∂ r , e = Υ L = ∂ t + Υ ∂ r , Υ = 1 − mr . (2.1.52) All Ricci coefficients vanish except, χ = Υ r , χ = − r , ω = − mr , ω = 0 .
2. The null frame ( e , e ) for which e is geodesic. e = L = Υ − ∂ t + ∂ r , e = Υ L = ∂ t − Υ ∂ r . All Ricci coefficients vanish except, χ = 1 r , χ = − Υ r , ω = 0 , ω = mr . Note that the null pair (2.1.52) is regular along the future event horizon as can be easilyseen by studying the behavior . of future directed ingoing null geodesics near r = 2 m . In this section we translate the null structure and null Bianchi identities associated to an S -foliation in the reduced picture. We start with general, Z -invariant, S foliation . Wethen consider the special case of geodesic foliations. S -foliations We consider a fixed Z -invariant S -foliation with a fixed Z -invariant null frame e , e . i.e. the null geodesics in the direction of L reach the horizon in finite proper time. Note that, on theother hand, the past null geodesics in the direction of L still meet the horizon in infinite proper time. CHAPTER 2. PRELIMINARIES
Null structure equations
We simply translate the well known spacetime null structure equations (see proposition7.4.1 in [17]) in the reduced picture. Thus the spacetime equation , ∇ / (cid:98) χ + tr χ (cid:98) χ = ∇ / (cid:98) ⊗ ξ − ω (cid:98) χ + ( η + η − ζ ) (cid:98) ⊗ ξ − α becomes , e ( ϑ ) + κ ϑ = 2( e θ ( ξ ) − e θ (Φ) ξ ) − ω ϑ + 2( η + η − ζ ) ξ − α. (2.2.1)The spacetime equation, e (tr χ ) + 12 tr χ = 2div / ξ − ω tr χ + 2 ξ · ( η + η − ζ ) − (cid:98) χ · (cid:98) χ becomes, e ( κ ) + 12 κ + 2 ω κ = 2( e θ ξ + e θ (Φ) ξ ) + 2( η + η − ζ ) ξ − ϑ ϑ. (2.2.2)The spacetime equation, ∇ / (cid:98) χ + 12 tr χ (cid:98) χ = ∇ / (cid:98) ⊗ η + 2 ω (cid:98) χ −
12 tr χ (cid:98) χ + ξ (cid:98) ⊗ ξ + η (cid:98) ⊗ η becomes, e ϑ + 12 κ ϑ − ωϑ = 2( e θ η − e θ (Φ) η ) −
12 tr χ ϑ + 2( ξ ξ + η ) . The spacetime equation, ∇ / tr χ + 12 tr χ tr χ = 2div / η + 2 ρ + 2 ω tr χ − (cid:98) χ · (cid:98) χ + 2( ξ · ξ + η · η )becomes, e ( κ ) + 12 κ κ − ωκ = 2( e θ η + e θ (Φ) η ) + 2 ρ − ϑ ϑ + 2( ξ ξ + η η ) . The spacetime equation, ∇ / ζ = − β − ∇ / ω − (cid:98) χ · ( ζ + η ) −
12 tr χ ( ζ + η ) + 2 ω ( ζ − η ) + ( (cid:98) χ + 12 tr χ ) ξ + 2 ωξ Note however that the notation in [17] are different, see section 7.3 for the definitions. For convenience we drop the (1+3) labels in what follows. recall that (1+3) (cid:98) χ θθ = ϑ .2. MAIN EQUATIONS (1+3) β = − β !), e ζ + 12 κ ( ζ + η ) − ω ( ζ − η ) = β − e θ ( ω ) + 2 ωξ + 12 κ ξ − ϑ ( ζ + η ) + 12 ϑ ξ. The spacetime equation, ∇ / ξ − ∇ / η = − β + 4 ωξ + (cid:98) χ · ( η − η ) + 12 tr χ ( η − η )becomes , e ( ξ ) − e ( η ) = β + 4 ωξ + 12 κ ( η − η ) + 12 ϑ ( η − η ) . The spacetime equation, ∇ / ω + ∇ / ω = ρ + 4 ωω + ξ · ξ + ζ · ( η − η ) − η · η becomes e ω + e ω = ρ + 4 ωω + ξ ξ + ζ ( η − η ) − η η. The spacetime Codazzi equation, (1+3) div / (1+3) (cid:98) χ = (1+3) β + 12 ( (1+3) ∇ / (1+3) tr χ − (1+3) tr χ (1+3) ζ ) + (1+3) (cid:98) χ · (1+3) ζ becomes , 12 ( e θ ( ϑ ) + 2 e θ (Φ) ϑ ) = − β + 12 ( e θ ( κ ) − κζ ) + 12 ϑζ. The Gauss equation, K = − (1+3) tr χ (1+3) tr χ + 12 (1+3) (cid:98) χ (1+3) (cid:98) χ − (1+3) ρ becomes, K = − κκ + 14 ϑϑ − ρ. We summarize the results in the following proposition. Note that (1+3) β = − β and d (cid:63) / f = − e θ ( f ). Note that (1+3) β θ = − β , (1+3) (cid:98) χ = ϑ . CHAPTER 2. PRELIMINARIES
Proposition 2.2.1. e ( ϑ ) + κ ϑ + 2 ω ϑ = − α − d (cid:63) / ξ + 2( η + η − ζ ) ξ,e ( κ ) + 12 κ + 2 ω κ = 2 d/ ξ + 2( η + η − ζ ) ξ − ϑ ,e ϑ + 12 κ ϑ − ωϑ = − d (cid:63) / η − κ ϑ + 2( ξ ξ + η ) ,e ( κ ) + 12 κ κ − ωκ = 2 d/ η + 2 ρ − ϑ ϑ + 2( ξ ξ + η η ) ,e ζ + 12 κ ( ζ + η ) − ω ( ζ − η ) = β + 2 d (cid:63) / ω + 2 ωξ + 12 κ ξ − ϑ ( ζ + η ) + 12 ϑ ξ,e ( ξ ) − ωξ − e ( η ) = β + 12 κ ( η − η ) + 12 ϑ ( η − η ) ,e ω + e ω = ρ + 4 ωω + ξ ξ + ζ ( η − η ) − η η. (2.2.3) In view of the symmetry e − e , we also derive, e ( ϑ ) + κ ϑ + 2 ωϑ = − α − d (cid:63) / ξ + 2( η + η + 2 ζ ) ξ,e ( κ ) + 12 κ + 2 ω κ = 2 d/ ξ + 2( η + η + 2 ζ ) ξ − ϑ ,e ϑ + 12 κ ϑ − ωϑ = − d (cid:63) / η − κ ϑ + 2( ξ ξ + η ) ,e ( κ ) + 12 κ κ − ωκ = 2 d/ η + 2 ρ − ϑ ϑ + 2( ξ ξ + η η ) , − e ζ + 12 κ ( − ζ + η ) + 2 ω ( ζ + η ) = β + 2 d (cid:63) / ω + 2 ωξ + 12 κ ξ − ϑ ( − ζ + η ) − ϑ ξ,e ( ξ ) − e ( η ) = β + 4 ωξ + 12 κ ( η − η ) + 12 ϑ ( η − η ) ,e ω + e ω = ρ + 4 ωω + ξ ξ + ζ ( η − η ) − η η. (2.2.4) We also have the Codazzi equations, d/ ϑ = − β − d (cid:63) / κ − ζκ + ϑ ζ,d/ ϑ = − β − d (cid:63) / κ + ζκ − ϑ ζ, and the Gauss equation, K = − ρ − κ κ + 14 ϑ ϑ. .2. MAIN EQUATIONS We now translate the spacetime null Bianchi identities of [17] (see proposition 7.3.2.) inthe reduced picture. The spacetime equation (note that D (cid:63) / β := −
12 (1+3) ∇ / ⊗ β ), ∇ / α + 12 tr χ α = − D (cid:63) / β + 4 ωα − (cid:98) χρ + (cid:63) (cid:98) χ (cid:63) ρ ) + ( ζ + 4 η ) ⊗ β becomes (note that (cid:63) ρ = 0), e α + 12 κα = ( e θ ( β ) − ( e θ Φ) β ) + 4 ωα − ϑρ + ( ζ + 4 η ) β. (2.2.5)The spacetime equation, ∇ / β + 2tr χβ = div / α − ωβ + (2 ζ + η ) · α + 3( ξρ + (cid:63) ξ (cid:63) ρ )becomes, e β + 2 κβ = ( e θ α + 2( e θ Φ) α ) − ωβ + (2 ζ + η ) α + 3 ξρ. (2.2.6)The spacetime equation, ∇ / β + tr χβ = D (cid:63) / ( − ρ, (cid:63) ρ ) + 2 (cid:98) χ · β + 2 ω β + ξ · α + 3( ηρ + (cid:63) η (cid:63) ρ )becomes (recall (1+3) β θ = − β ), e β + κβ = e θ ( ρ ) + 2 ωβ + 3 ηρ − ϑβ + ξα. (2.2.7)The spacetime equation, e ρ + 32 tr χρ = div / β − (cid:98) χ · α + ζ · β + 2( η · β − ξ · β )becomes, e ρ + 32 κρ = ( e θ ( β ) + ( e θ Φ) β ) − ϑ α + ζ β + 2( η β + ξ β ) . (2.2.8)Indeed note that, (1+3) (cid:98) χ · (1+3) α = 2 (1+3) (cid:98) χ θθ (1+3) α θθ = ϑα. All other equations in the proposition below are derived using the e − e symmetry. Wesummarize the results in the following proposition.8 CHAPTER 2. PRELIMINARIES
Proposition 2.2.2. e α + 12 κα = − d (cid:63) / β + 4 ωα − ϑρ + ( ζ + 4 η ) β,e β + 2 κβ = d/ α − ωβ + (2 ζ + η ) α + 3 ξρ,e β + κβ = − d (cid:63) / ρ + 2 ωβ + 3 ηρ − ϑβ + ξα,e ρ + 32 κρ = d/ β − ϑ α + ζ β + 2( η β + ξ β ) ,e ρ + 32 κρ = d/ β − ϑ α − ζ β + 2( η β + ξ β ) ,e β + κβ = − d (cid:63) / ρ + 2 ωβ + 3 ηρ − ϑβ + ξα,e β + 2 κ β = d/ α − ω β + ( − ζ + η ) α + 3 ξρ,e α + 12 κ α = − d (cid:63) / β + 4 ωα − ϑρ + ( − ζ + 4 η ) β. (2.2.9) Mass aspect functions
We define the mass aspect functions, µ : = − d/ ζ − ρ + 14 ϑϑ,µ : = d/ ζ − ρ + 14 ϑϑ. (2.2.10)One can derive useful propagation equations, in the e direction for µ and in the e direction for µ by using the null structure and null Bianchi equations, see [17] and [37].In the next section we will do this in the context of null-geodesic foliations. Definition 2.2.3.
The Hawking mass m = m ( S ) of S is defined by the formula, mr = 1 + 116 π (cid:90) S κκ. (2.2.11) Proposition 2.2.4.
The following identities hold true. .2. MAIN EQUATIONS
1. The average of ρ is given by the formulas, ρ = − mr + 116 πr (cid:90) S ϑϑ. (2.2.12)
2. The average of the mass aspect function is, µ = µ = 2 mr . (2.2.13)
3. The average of κ and κ are related by, κ κ = − r − ˇ κ ˇ κ (2.2.14) where Υ = 1 − mr .Proof. We have from the Gauss equation K = − κκ + 14 ϑϑ − ρ. Integrating on S and using the Gauss Bonnet formula, we infer4 π = − (cid:90) S κκ + 14 (cid:90) S ϑϑ − (cid:90) S ρ. Together with the definition of the Hawking mass, we infer (cid:90) S ρ = − π (cid:18) π (cid:90) S κκ (cid:19) + 14 (cid:90) S ϑϑ = − πmr + 14 (cid:90) S ϑϑ and hence ρ = − mr + 116 πr (cid:90) S ϑϑ. which proves our first identity. The second identity follows easily from the definition of µ, µ and the first formula. Thus, for example, µ = 1 | S | (cid:90) S µ = 1 | S | (cid:90) S (cid:18) − d/ ζ − ρ + 14 ϑϑ (cid:19) = − ρ + 14 | S | (cid:90) S ϑϑ = 2 mr . CHAPTER 2. PRELIMINARIES
To prove the last identity we remark that, in view of the definition of the Hawking mass, − Υ = 2 mr − π (cid:90) S κκ = 116 π (cid:18) | S | κ κ + (cid:90) S ˇ κ ˇ κ (cid:19) and hence κ κ = − π Υ | S | − | S | (cid:90) S ˇ κ ˇ κ = − r − ˇ κ ˇ κ. This concludes the proof of the proposition.
We restrict our attention to geodesic foliations, i.e. geodesic foliations by Z invariantoptical functions. Basic definitions
Assume given an outgoing optical function u , i.e. Z -invariant solution of the equation, g αβ ∂ α u∂ β u = g ab ∂ a u∂ b u = 0and L = − g ab ∂ b u∂ a its null geodesic generator. We choose e such that, e = ςL, L ( ς ) = 0 . (2.2.15) Remark 2.2.5.
In our definition of a GCM admissible spacetime, see section 3.1, weinitialize ς on the spacelike hypersurface Σ ∗ . We then choose s such that e ( s ) = 1 . (2.2.16)The functions u, s generate what is called an outgoing geodesic foliation. Let S u,s be the2-surfaces of intersection between the level surfaces of u and s . We choose e the unique Z -invariant null vectorfield orthogonal to S u,s and such that g ( e , e ) = −
2. We then let e θ to be unit tangent to S u,s , Z -invariant and orthogonal to Z . We also introduceΩ := e ( s ) . (2.2.17) .2. MAIN EQUATIONS Lemma 2.2.6.
We have ω = ξ = 0 , η = − ζ, (2.2.18) ς = 2 e ( u ) ,e ( ς ) = 0 ,e θ (log ς ) = η − ζ,e θ (Ω) = − ξ − ( η − ζ )Ω ,e (Ω) = − ω. (2.2.19) Proof.
Recall that L is geodesic, e = ςL and L ( ς ) = 0. This immediately implies that e is geodesic, and hence we have ω = ξ = 0 . Applying the vectorfield [ e , e θ ] = ( η + ζ ) e + ξe − χe θ to s , and since e ( s ) = 1 and e θ ( s ) = 0, we derive, η + ζ = 0 . Next, note that e ( u ) = g ( e , − L ) = − ς − g ( e , e ) = 2 ς and hence ς = 2 e ( u ) . Applying the vectorfield [ e , e θ ] = ξe + ( η − ζ ) e − χe θ to u and making use of the relation e ( u ) = e θ ( u ) = 0 we deduce,( η − ζ ) e ( u ) = e ( e θ u ) − e θ e ( u ) = − e θ e ( u )2 CHAPTER 2. PRELIMINARIES which together with the identity ς = 2 /e ( u ) yields η − ζ = − e θ log( e u ) = − e θ log (cid:18) ς (cid:19) = e θ (log ς )and hence e θ (log ς ) = η − ζ. Applying the vectorfield [ e , e θ ] = ξe + ( η − ζ ) e − χe θ to s we deduce, since e ( s ) = 1, e θ ( s ) = 0 and e ( s ) = Ω, e θ (Ω) = − ξ − ( η − ζ )Ω . Finally applying [ e , e ] = − ωe − η − η ) e θ + 2 ωe to s , and using e ( s ) = 1 and e θ ( s ) = 0, we infer e ( e ( s )) = − ω , i.e. e (Ω) = − ω asdesired. Remark 2.2.7.
In the particular case when ς is constant we have η = ζ = − η . InSchwarzschild, relative to the standard outgoing geodesic frame, we have ς = 1 , Ω = − Υ = − (cid:18) − mr (cid:19) . Basic equationsProposition 2.2.8.
Relative to an outgoing geodesic foliation we have .2. MAIN EQUATIONS
1. The reduced null structure equations take the form, e ( ϑ ) + κ ϑ = − α,e ( κ ) + 12 κ = − ϑ ,e ζ + κζ = − β − ϑζ,e ( η − ζ ) + 12 κ ( η − ζ ) = − ϑ ( η − ζ ) ,e ϑ + 12 κ ϑ = 2 d (cid:63) / ζ − κ ϑ + 2 ζ ,e ( κ ) + 12 κ κ = − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ ,e ω = ρ + ζ (2 η + ζ ) ,e ( ξ ) = − e ( ζ ) + β − κ ( ζ + η ) − ϑ ( ζ + η ) , and e ( ϑ ) + κ ϑ + 2 ω ϑ = − α − d (cid:63) / ξ + 2( η − ζ ) ξ,e ( κ ) + 12 κ + 2 ω κ = 2 d/ ξ + 2( η − ζ ) ξ − ϑ ,e ζ + 12 κ ( ζ + η ) − ω ( ζ − η ) = β + 2 d (cid:63) / ω + 12 κ ξ − ϑ ( ζ + η ) + 12 ϑ ξ,e ϑ + 12 κ ϑ − ωϑ = − d (cid:63) / η − κ ϑ + 2 η ,e ( κ ) + 12 κ κ − ωκ = 2 d/ η + 2 ρ − ϑ ϑ + 2 η , and d/ ϑ = − β − d (cid:63) / κ − ζκ + ϑ ζ,d/ ϑ = − β − d (cid:63) / κ + ζκ − ϑ ζ,K = − ρ − κ κ + 14 ϑ ϑ.
2. The null Bianchi identities are given in this case by e α + 12 κα = − d (cid:63) / β + 4 ωα − ϑρ + ( ζ + 4 η ) β,e β + 2 κβ = d/ α + ζα,e β + κβ = − d (cid:63) / ρ + 2 ωβ + 3 ηρ − ϑβ + ξα,e ρ + 32 κρ = d/ β − ϑ α − ζβ, CHAPTER 2. PRELIMINARIES e ρ + 32 κρ = d/ β − ϑ α − ζ β + 2( η β + ξ β ) ,e β + κβ = − d (cid:63) / ρ − ζρ − ϑβ,e β + 2 κ β = d/ α − ω β + ( − ζ + η ) α + 3 ξρ,e α + 12 κ α = − d (cid:63) / β − ϑρ − ζβ.
3. The mass aspect function µ = − d/ ζ − ρ + ϑϑ , defined in (2.2.10) verifies thetransport equation, e ( µ ) + 32 κµ = Err [ e µ ] , Err [ e µ ] : = 12 κζ + e θ ( κ ) ζ + d/ ( ϑζ ) − κϑ . Proof.
Concerning the null structure equations we only need to derive the equation for η − ζ . According to Proposition (2.2.1) we have, e ( ξ ) − e ( η ) = β + 4 ωξ + κ ( η − η ) + ϑ ( η − η )which becomes e η = − β − κ ( η − η ) − ϑ ( η − η )and, − e ζ + 12 κ ( − ζ + η ) + 2 ω ( ζ + η ) = β + 2 d (cid:63) / ω + 2 ωξ + κ ξ − ϑ ( − ζ + η ) − ϑ ξ which becomes, e ζ = − κζ − β − ϑζ. Hence, e ( ζ − η ) = − κζ − ϑζ + 12 κ ( η − η ) + 12 ϑ ( η − η )= κ (cid:18) − ζ + 12 ( η − η ) (cid:19) + ϑ (cid:18) − ζ + 12 ( η − η ) (cid:19) . Since ζ = − η we deduce − ζ + ( η − η ) = ( − ζ + η ) and thus, e ( ζ − η ) = − κ ( ζ − η ) − ϑ ( ζ − η ) .2. MAIN EQUATIONS µ . We have e ( µ ) = − [ e , d/ ] ζ − d/ e ( ζ ) − e ( ρ ) + 14 e ( ϑϑ )= 12 κ d/ ζ − ϑ d (cid:63) / ζ + e (Φ) ζ − βζ − d/ ( − κζ − β − ϑζ ) − (cid:18) − κρ + d/ β − ϑ α − ζβ (cid:19) + 14 ϑ (cid:18) − κ ϑ + 2 d (cid:63) / ζ − κ ϑ + 2 ζ (cid:19) + 14 ϑ ( − κ ϑ − α )= 32 κ (cid:18) d/ ζ + ρ − ϑϑ (cid:19) − ϑ d (cid:63) / ζ + 12 ( κ − ϑ ) ζ + e θ ( κ ) ζ + d/ ( ϑζ )+ 14 ϑ (cid:18) d (cid:63) / ζ − κϑ + 2 ζ (cid:19) and hence e ( µ ) + 32 κµ = 12 κζ + e θ ( κ ) ζ + d/ ( ϑζ ) − κϑ as desired. This concludes the proof of the proposition. Transport equations for S -averagesProposition 2.2.9. For any scalar function f , we have, e (cid:18)(cid:90) S f (cid:19) = (cid:90) S ( e ( f ) + κf ) ,e (cid:18)(cid:90) S f (cid:19) = (cid:90) S ( e ( f ) + κf ) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) , (2.2.20) where the error term is given by the formulaErr (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) : = − ς − ˇ ς (cid:90) S ( e ( f ) + κf ) + ς − (cid:90) S ˇ ς ( e ( f ) + κf )+ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:90) S ( e f + κf ) − ς − Ω (cid:90) S ˇ ς ( e f + κf ) − ς − (cid:90) S ˇΩ ς ( e f + κf ) . CHAPTER 2. PRELIMINARIES
In particular, we have e ( r ) = r κ, e ( r ) = r κ + A ) (2.2.21) where A : = − ς − κ ˇ ς + κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) + ς − ˇ ς ˇ κ − ς − Ω ˇ ς ˇ κ − ς − ˇΩ ςκ. (2.2.22) Proof.
See section A.1.
Corollary 2.2.10.
For a reduced scalar f , we have e (cid:18)(cid:90) S f e Φ (cid:19) = (cid:90) S (cid:18) e ( f ) + (cid:18) κ − ϑ (cid:19) f (cid:19) e Φ and e (cid:18)(cid:90) S f e Φ (cid:19) = (cid:90) S (cid:18) e ( f ) + (cid:18) κ − ϑ (cid:19) f (cid:19) e Φ + Err (cid:20) e (cid:18)(cid:90) S f e Φ (cid:19)(cid:21) . Proof.
In view of Proposition 2.2.9, we have e (cid:18)(cid:90) S f e Φ (cid:19) = (cid:90) S (cid:16) e ( f e Φ ) + κf e Φ (cid:17) = (cid:90) S (cid:16) e ( f ) + ( κ + e Φ) f (cid:17) e Φ = (cid:90) S (cid:18) e ( f ) + (cid:18) κ − ϑ (cid:19) f (cid:19) e Φ as desired.Also, using again Proposition 2.2.9, we have e (cid:18)(cid:90) S f e Φ (cid:19) = (cid:90) S (cid:16) e ( f e Φ ) + κf e Φ (cid:17) + Err (cid:20) e (cid:18)(cid:90) S f e Φ (cid:19)(cid:21) = (cid:90) S (cid:16) e ( f ) + ( κ + e Φ) f (cid:17) e Φ + Err (cid:20) e (cid:18)(cid:90) S f e Φ (cid:19)(cid:21) = (cid:90) S (cid:18) e ( f ) + (cid:18) κ − ϑ (cid:19) f (cid:19) e Φ + Err (cid:20) e (cid:18)(cid:90) S f e Φ (cid:19)(cid:21) as desired. .2. MAIN EQUATIONS Corollary 2.2.11.
Given a scalar function f we have, e ( f ) = e ( f ) + ˇ κ ˇ f ,e ( ˇ f ) = e ( f ) − e ( f ) − ˇ κ ˇ f , (2.2.23) and e (cid:0) f (cid:1) = e ( f ) + Err [ e f ] ,e ( ˇ f ) = e ( f ) − e ( f ) − Err [ e ( f )] , (2.2.24) where,Err [ e ( f )] = − ς − ˇ ς (cid:0) e f + κf − κf (cid:1) + ς − (cid:16) ˇ ς ( e f + κf ) − ˇ ς ˇ κ f (cid:17) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:16) e f + κf ) − κ f (cid:17) − ς − Ω (cid:16) ˇ ς ( e f + κf ) − ˇ ς ˇ κ f (cid:17) − ς − (cid:16) ˇΩ ς ( e f + κf ) − ˇΩ ς κf (cid:17) + ˇ κ ˇ f . (2.2.25) Proof.
We have, recalling Lemma 2.1.42 and | S | = 4 πr , e ( f ) = e (cid:18) (cid:82) S f | S | (cid:19) = 1 | S | (cid:90) S ( e ( f ) + κf ) − e ( | S | ) | S | f = e ( f ) + κf − e rr f = e ( f ) + κ f − κ f = e ( f ) + ˇ κ ˇ f . This also yields e ( ˇ f ) = e ( f ) − e ( f ) = e ( f ) − e ( f ) − ˇ κ ˇ f as desired.Similarly, e ( f ) = e (cid:18) (cid:82) S f | S | (cid:19) = 1 | S | e (cid:18)(cid:90) S f (cid:19) − e ( r ) r f = 1 | S | (cid:90) S ( e f + κf ) + 1 | S | Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) − ( κ + A ) f = e ( f ) + κf − κf + 1 | S | Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) − Af = e ( f ) + ˇ κ ˇ f + 1 | S | Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) − Af . CHAPTER 2. PRELIMINARIES
We deduce, e ( f ) = e ( f ) + Err[ e ( f )]where, recalling the definitions of the error terms Err (cid:2) e (cid:0)(cid:82) S f (cid:1)(cid:3) and A ,Err[ e ( f )] = ˇ κ ˇ f + 1 | S | Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) − A f = ˇ κ ˇ f − ς − ˇ ς e f + κf + ς − ˇ ς ( e f + κf ) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) e f + κf − ς − Ω ˇ ς ( e f + κf ) − ς − ˇΩ ς ( e f + κf ) − f (cid:16) − ς − κ ˇ ς + ς − ˇ ς ˇ κ + κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) − ς − Ω ˇ ς ˇ κ − ς − ˇΩ ςκ (cid:17) , i.e., Err[ e ( f )] = ˇ κ ˇ f − ς − ˇ ς (cid:0) e f + κf − κf (cid:1) + ς − (cid:16) ˇ ς ( e f + κf ) − ˇ ς ˇ κ f (cid:17) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:16) e f + κf ) − κ f (cid:17) − ς − Ω (cid:16) ˇ ς ( e f + κf ) − ˇ ς ˇ κ f (cid:17) − ς − (cid:16) ˇΩ ς ( e f + κf ) − ˇΩ ςκf (cid:17) as stated. Finally e ( ˇ f ) = e f − e ( f ) = e f − e ( f ) − Err[ e f ]which ends the proof of the corollary.The following is also an immediate application of Proposition 2.2.9. Corollary 2.2.12. If f verifies the scalar equation e ( f ) + p κf = F, then, e ( r p f ) = r p F. Commutation identities revisited
We revisit the general commutation identities of Lemma 2.1.51 in an outgoing geodesicfoliation. .2. MAIN EQUATIONS Lemma 2.2.13.
The following commutation formulae holds true,1. If f ∈ s k , [ r d/ k , e ] f = r (cid:20) Com k ( f ) + 12 ˇ κ d/ k f (cid:21) , [ r d/ k , e ] f = r (cid:20) Com k ( f ) + 12 ( − A + ˇ κ ) d/ k f (cid:21) . (2.2.26)
2. If f ∈ s k − [ r d (cid:63) / k , e ] f = r (cid:20) Com ∗ k ( f ) + 12 ˇ κ d (cid:63) / k f (cid:21) , [ r d (cid:63) / k , e ] f = r (cid:20) Com ∗ k ( f ) + 12 ( − A + ˇ κ ) d (cid:63) / k f (cid:21) . (2.2.27) Also, we have
Com k ( f ) = − ϑ d (cid:63) / k +1 f + ( ζ − η ) e f − kηe Φ f − ξ ( e f + ke (Φ) f ) − kβf, Com k ( f ) = − ϑ d (cid:63) / k +1 f + kζe Φ f − kβf,Com ∗ k ( f ) = − ϑ d/ k − f − ( ζ − η ) e f − ( k − ηe Φ f + ξ ( e f − ( k − e (Φ) f ) − ( k − βf, Com ∗ k ( f ) = − ϑ d/ k − f + ( k − ζe Φ f − ( k − βf. Proof.
We make use of the commutation Lemma 2.1.51 and the definition of A , see Propo-sition 2.2.9, to write, for f ∈ s k ,[ r d/ k , e ] f = r [ d/ k , e ] f − e ( r ) d/ k f = 12 rκ d/ k f + r Com k ( f ) − r κ d/ k f = r (cid:20) Com k ( f ) + 12 ˇ κ d/ k f (cid:21) [ r d/ k , e ] f = r [ d/ k , e ] f − e ( r ) d/ k f = 12 rκ d/ k f + rCom k ( f ) − r A + κ ) d/ k f = r (cid:20) Com k ( f ) + 12 ( − A + ˇ κ ) d/ k f (cid:21) . CHAPTER 2. PRELIMINARIES
The remaining formulae are proved in the same manner. Also, the form of
Com k ( f ),Com k ( f ), Com ∗ k ( f ) and Com ∗ k ( f ) follows from Lemma 2.1.51 and the fact that we have ξ = η + ζ = 0 in an outgoing geodesic foliation.We also record here for future use the following lemma. Lemma 2.2.14.
Let T = ( e + Υ e ) , with Υ = 1 − mr . We have, [ T , e ] = (cid:18)(cid:16) ω − mr (cid:17) − m r (cid:18) κ − r (cid:19) + e ( m ) r (cid:19) e + ( η + ζ ) e θ , [ T , e ] = (cid:18) − Υ (cid:16) ω − mr (cid:17) − m r (cid:18) κ + 2Υ r (cid:19) − m r A + e ( m ) r (cid:19) e − ( η + ζ )Υ e θ . (2.2.28) Proof.
Recall that [ e , e ] = 2 ωe + 2( η + ζ ) e θ . Thus,[ T , e ] = 12 [ e + Υ e , e ] = 12 (cid:18) ωe + 2( η + ζ ) e θ − e (cid:18) − mr (cid:19) e (cid:19) = (cid:18) ω − mr e ( r ) + e ( m ) r (cid:19) e + ( η + ζ ) e θ = (cid:18)(cid:16) ω − mr (cid:17) − mr ( e ( r ) −
1) + e ( m ) r (cid:19) e + ( η + ζ ) e θ = (cid:18)(cid:16) ω − mr (cid:17) − mr (cid:16) r κ − (cid:17) + e ( m ) r (cid:19) e + ( η + ζ ) e θ = (cid:18)(cid:16) ω − mr (cid:17) − m r (cid:18) κ − r (cid:19) + e ( m ) r (cid:19) e + ( η + ζ ) e θ and, [ T , e ] = 12 [ e + Υ e , e ] = 12 (cid:18) Υ ( − ωe − η + ζ ) e θ ) − e (cid:18) − mr (cid:19) e (cid:19) = (cid:18) − Υ ω − mr e ( r ) + e ( m ) r (cid:19) e − Υ( η + ζ ) e θ = (cid:18) − Υ (cid:16) ω − mr (cid:17) − Υ mr − mr r κ + A ) + e ( m ) r (cid:19) e − Υ( η + ζ ) e θ = (cid:18) − Υ (cid:16) ω − mr (cid:17) − m r (cid:18) κ + 2Υ r (cid:19) − m r A + e ( m ) r (cid:19) e − Υ( η + ζ ) e θ which concludes the proof of the lemma. .2. MAIN EQUATIONS Remark 2.2.15.
When applying the formulas of Lemma 2.2.14 to a k reduced scalar f ∈ s k , the term ( η + ζ ) e θ ( f ) should correspond to a reduced scalar. In fact, recallingRemark 2.1.23, we can write, ζe θ ( f ) = 12 ζ ( d/ k f − d (cid:63) / k +1 f ) which can indeed be shown to be a k -reduced scalar in s k . Derivatives of the Hawking massProposition 2.2.16 (Derivatives of the Hawking mass) . We have the following identitiesfor the Hawking mass, e ( m ) = r π (cid:90) S Err , (2.2.29) and e ( m ) = (cid:0) − ς − ˇ ς (cid:1) r π (cid:90) S Err + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) r π (cid:90) S Err + ς − r π (cid:90) S ˇ ς (cid:0) ρ ˇ κ + 2 ˇ ρκ + 2 κ d/ η + 2 κ d/ ξ + Err (cid:1) − ς − r π (cid:90) S (Ωˇ ς + ˇΩ ς ) (2 ρ ˇ κ + 2 ˇ ρκ − κ d/ ζ + Err ) − mr ς − (cid:104) − ˇ ς ˇ κ + Ω ˇ ς ˇ κ + ˇΩ ςκ (cid:105) , (2.2.30) where Err := 2ˇ κ ˇ ρ + 2 e θ ( κ ) ζ − κϑ −
12 ˇ κϑϑ + 2 κζ , Err := 2 ˇ ρ ˇ κ − e θ ( κ ) η − e θ ( κ ) ξ −
12 ˇ κϑϑ + 2 κη + 2 κ (cid:0) η − ζ (cid:1) ξ − κϑ , Err := 2 ˇ ρ ˇ κ − κϑ − κϑϑ + 2 κζ , Err := 2 ˇ ρ ˇ κ + κ (cid:18) η − ϑϑ (cid:19) + 2 κ (cid:0) η − ζ (cid:1) ξ − κϑ . Proof.
The proof relies on the definition of the Hawking mass m given by the formula mr = 1+ π (cid:82) S κκ , Proposition 2.2.9, and the the null structure equations for e ( κ ), e ( κ ), e ( κ ) and e ( κ ) provided by Proposition 2.2.8. We refer to section A.2 for the details.2 CHAPTER 2. PRELIMINARIES
Transport equations for main averaged quantitiesLemma 2.2.17.
The following equations hold true e (cid:18) κ − r (cid:19) + 12 κ (cid:18) κ − r (cid:19) = − ϑ + 12 ˇ κ ,e (cid:16) ω − mr (cid:17) = ρ + 2 mr + mr (cid:18) κ − r (cid:19) − e ( m ) r + 3 ζ (2 η + ζ ) + ˇ κ ˇ ω, (2.2.31) and e (cid:18) κ − r (cid:19) + 12 κ (cid:18) κ − r (cid:19) = 2 ω (cid:18) κ − r (cid:19) + 4 r (cid:16) ω − mr (cid:17) + 2 (cid:18) ρ + 2 mr (cid:19) − ς − (cid:18) − κ κ + 2 ω κ + 2 ρ (cid:19) ˇ ς − κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) − r ς − κ ˇ ς + 1 r κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) + Err (cid:20) e (cid:18) κ − r (cid:19)(cid:21) , (2.2.32) where,Err (cid:20) e (cid:18) κ − r (cid:19)(cid:21) := 2 η + 2ˇ ω ˇ κ −
12 ˇ κ ˇ κ − ϑϑ + 1 r ς − ˇ ς ˇ κ − r ς − Ω ˇ ς ˇ κ − r ς − ˇΩ ςκ − ς − ˇ ς (cid:18)
12 ˇ κ ˇ κ + 2ˇ ω ˇ κ − ϑϑ + 2 η (cid:19) + ς − (cid:32) ˇ ς (cid:18) κκ + 2 ωκ + 2 ˇ ρ + 2 d/ η − ϑϑ + 2 η (cid:19) − ˇ ς ˇ κ κ (cid:33) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18)
12 ˇ κ − ϑ (cid:19) − ς − Ω (cid:32) ˇ ς (cid:18) κ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) − ς − (cid:32) ˇΩ ς (cid:18) κ − ϑ (cid:19) − ˇΩ ς κ κ (cid:33) + ˇ κ ˇ κ. (2.2.33) Proof.
The proof relies on Corollary 2.2.11 and the null structure equations for e ( κ ) and e ( κ ) provided by Proposition 2.2.8. We refer to section A.3 for the details. .2. MAIN EQUATIONS Transport equations for main checked quantitiesProposition 2.2.18 (Transport equations for checked quantities) . We have the followingtransport equations in the e direction, e ˇ κ + κ ˇ κ = Err [ e ˇ κ ] , Err [ e ˇ κ ] : = −
12 ˇ κ −
12 ˇ κ −
12 ( ϑ − ϑ ) ,e ˇ κ + 12 κ ˇ κ + 12 ˇ κκ = − d/ ζ + 2 ˇ ρ + Err [ e ˇ κ ] , Err [ e ˇ κ ] : = −
12 ˇ κ ˇ κ −
12 ˇ κ ˇ κ + (cid:18) − ϑϑ + 2 ζ (cid:19) − (cid:18) − ϑϑ + 2 ζ (cid:19) ,e ˇ ω = ˇ ρ + Err [ e ˇ ω ] , Err [ e ˇ ω ] : = − ˇ κ ˇ ω + ( ζ (2 η + ζ ) − ζ (2 η + ζ )) , (2.2.34) e ˇ ρ + 32 κ ˇ ρ + 32 ρ ˇ κ = d/ β + Err [ e ˇ ρ ] , Err [ e ˇ ρ ] : = −
32 ˇ κ ˇ ρ + 12 ˇ κ ˇ ρ − (cid:18) ϑα + ζβ (cid:19) + (cid:18) ϑα + ζβ (cid:19) ,e ˇ µ + 32 κ ˇ µ + 32 µ ˇ κ = Err [ e ˇ µ ] , Err [ e ˇ µ ] : = −
32 ˇ κ ˇ µ + 12 ˇ κ ˇ µ + Err [ e µ ] − Err [ e µ ] ,e ( ˇΩ) = − ω + ˇ κ ˇΩ . (2.2.35) Also in the e direction, e (ˇ κ ) = 2 d/ η + 2 ˇ ρ −
12 ( κ ˇ κ + κ ˇ κ ) + 2 ( ω ˇ κ + κ ˇ ω )+ ς − (cid:18) − κ κ + 2 ω κ + 2 ρ (cid:19) ˇ ς + 12 κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) + Err [ e ˇ κ ] ,e (ˇ κ ) + κ ˇ κ = 2 d/ ξ − ω κ + ω ˇ κ ) + ς − ˇ ς (cid:18) − κ − ω κ (cid:19) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) − κ κ + 2 ρ (cid:19) + Err [ e (ˇ κ )] ,e ˇ ρ + 32 κ ˇ ρ = − ρ ˇ κ + d/ β − κ ρς − ˇ ς + 32 κ ρ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) + Err [ e ˇ ρ ] , (2.2.36)4 CHAPTER 2. PRELIMINARIES with error terms given by,Err [ e ˇ κ ] := 2 (cid:16) η − η (cid:17) −
12 ˇ κ ˇ κ + 2ˇ ω ˇ κ − (cid:0) ϑϑ − ϑϑ (cid:1) + ς − ˇ ς (cid:18)
12 ˇ κ ˇ κ + 2ˇ ω ˇ κ − ϑϑ + 2 η (cid:19) − ς − (cid:32) ˇ ς (cid:18) κκ + 2 ωκ + 2 ˇ ρ + 2 d/ η − ϑϑ + 2 η (cid:19) − ˇ ς ˇ κ κ (cid:33) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18)
12 ˇ κ − ϑ (cid:19) + ς − Ω (cid:32) ˇ ς (cid:18) κ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) + ς − (cid:32) ˇΩ ς (cid:18) κ − ϑ (cid:19) − ˇΩ ς κ κ (cid:33) − ˇ κ ˇ κ, (2.2.37) Err [ e (ˇ κ )] := −
12 ˇ κ − ω ˇ κ + 2( η − ζ ) ξ − η − ζ ) ξ − (cid:16) ϑ − ϑ (cid:17) − ς − (cid:32) ˇ ς (cid:18) κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) + ς − Ω (cid:32) ˇ ς (cid:18) κ κ − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ (cid:19) − ˇ ς ˇ κ κ (cid:33) + ς − (cid:32) ˇΩ ς (cid:18) κ κ − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ (cid:19) − ˇΩ ς κ κ (cid:33) − ˇ κ , (2.2.38) and Err [ e ˇ ρ ] := − (cid:18) ϑα + ζβ − ηβ − ξβ (cid:19) + (cid:18) ϑα + ζβ − ηβ − ξβ (cid:19) −
32 ˇ κ ˇ ρ + ς − ˇ ς (cid:18) −
12 ˇ κ ˇ ρ − ϑ α − ζ β + 2( η β + ξ β ) (cid:19) − ς − (cid:32) ˇ ς (cid:18) − κρ + d/ β − ϑ α − ζ β + 2( η β + ξ β ) (cid:19) − ˇ ς ˇ κ ρ (cid:33) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) −
12 ˇ κ ˇ ρ − ϑ α − ζβ (cid:19) + ς − Ω (cid:32) ˇ ς (cid:18) − κρ + d/ β − ϑ α − ζβ (cid:19) − ˇ ς ˇ κ ρ (cid:33) + ς − (cid:32) ˇΩ ς (cid:18) − κρ + d/ β − ϑ α − ζβ (cid:19) − ˇΩ ς κ (cid:33) − ˇ κ ˇ ρ. (2.2.39) .2. MAIN EQUATIONS Proof.
The proof relies on Corollary 2.2.11 and the null structure equations of Proposition2.2.8. We refer to section A.4 for the details.
We derive below additional equations for ω, η, ξ . Proposition 2.2.19.
The following identities hold true for a general forward geodesicfoliation. • The scalar ω verifies d (cid:63) / ω = − κξ + (cid:18) κ + 2 ω + 12 ϑ (cid:19) η + e ( ζ ) − β + 12 κζ − ωζ + 12 ϑζ − ϑξ. • The reduced -form η verifies d/ d (cid:63) / η = κ (cid:0) − e ( ζ ) + β (cid:1) − e ( e θ ( κ )) − κ (cid:18) κζ − ωζ (cid:19) + 6 ρη − κe θ κ − κe θ ( κ ) + 2 ωe θ ( κ ) + 2 e θ ( ρ ) + Err [ d/ d (cid:63) / η ] , Err [ d/ d (cid:63) / η ] = (cid:18) d/ η − κϑ + 2 η (cid:19) η + 2 e θ ( η ) − κ (cid:18) ϑζ − ϑξ (cid:19) − ϑe θ ( κ ) − (cid:18) d/ η − ϑϑ + 2 η (cid:19) ζ − e θ ( ϑ ϑ ) − ϑ ξ − ϑϑη. • The reduced -form ξ verifies d/ d (cid:63) / ξ = − e ( e θ ( κ )) + κ (cid:0) e ( ζ ) − β (cid:1) + κ ζ − κe θ κ + 6 ρξ − ωe θ ( κ )+ Err [ d/ d (cid:63) / ξ ] , Err [ d/ d (cid:63) / ξ ] = (cid:18) d/ ξ + 12 κ ϑ + 2 ηξ − ϑ (cid:19) η + 2 e θ ( ηξ ) − e θ ( ϑ )+ κ (cid:18) ϑζ − ϑξ (cid:19) − ϑe θ κ − ϑϑξ − ζ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) + ξ (cid:16) − ϑϑ − d/ ζ + 2 ζ (cid:17) − ηζξ − e θ ( ζξ ) . Proof.
The proof relies on the null structure equations of Proposition 2.2.8, in particularthe ones for e ( ζ ), e ( κ ) and e ( κ ). We refer to section A.5 for the details.6 CHAPTER 2. PRELIMINARIES
All the equations of section 2.2.4 for outgoing geodesic foliations have a counterpart foringoing geodesic foliations. The corresponding equations can be easily deduced from theones in section 2.2.4 by performing the following substitutions u → u, s → s, C u → C u , S u,s → S u,s , r → r, m → m,e → e , e → e , e θ → e θ , e ( s ) = 1 → e ( s ) = − ,α → α, β → , β, ρ → ρ, µ → µ, β → β, α → α,ξ → ξ, ω → ω, κ → κ, ϑ → ϑ, η → η, η → η, ζ → − ζ, κ → κ,ϑ → ϑ, ω → ω, ξ → ξ, Ω = e ( s ) → Ω = e ( s ) , ς = 2 e ( u ) → ς = 2 e ( u ) ,κ − r → κ + 2 r , κ + 2Υ r → κ − r , ω − mr → ω + mr , ρ + 2 mr → ρ + 2 mr ,µ − mr → µ − mr , Ω + Υ → Ω − Υ , ς − → ς − ,A = 2 r e ( r ) − κ → A = 2 r e ( r ) − κ. ( u, s, θ, ϕ ) coordinatesProposition 2.2.20. Consider, in addition to the functions u, s, ϕ an additional Z in-variant function θ . Then, relative to the coordinates system ( u, s, θ, ϕ ) , the following holdtrue,1. The spacetime metric takes the form, g = − ςduds + ς Ω du + γ (cid:18) dθ − ς ( b − Ω b ) du − bds (cid:19) + e ( dϕ ) (2.2.40) where, Ω = e ( s ) , b = e ( θ ) , b = e ( θ ) , γ − = e θ ( θ ) . (2.2.41)
2. In these coordinates the reduced frame takes the form, ∂ s = e − b √ γe θ , ∂ u = ς (cid:18) e −
12 Ω e − √ γ ( b − b Ω) e θ (cid:19) , ∂ θ = √ γe θ . (2.2.42) .2. MAIN EQUATIONS
3. In the particular case when b = e ( θ ) = 0 we have, e ( γ ) = 2 χγ, e ( b ) = − ζ + η ) γ − / . (2.2.43) Proof.
First, from the fact that ( e , e , e θ ) forms a null frame, we easily verify that (2.2.42)holds. Then, (2.2.40) immediately follows from (2.2.42) and the fact that ( e , e , e θ ) formsa null frame.To prove the last statement, when b = e ( θ ) = 0, we start with,[ e , e ] = 2 ωe − ωe + 2( η − η ) e θ = − ζ + η ) e θ − ωe . Applying this to θ we derive,[ e , e ]( θ ) = ( − ζ + η ) e θ − ωe )( θ ) = − ζ + η ) e θ ( θ ) = − ζ + η ) γ − / . We deduce, e ( b ) = e ( e ( θ )) = − ζ + η ) γ − / . To prove the equation for γ we make use of,[ e , e θ ] = ( η + ζ ) e + ξe − χe θ = − χe θ so that e e θ ( θ ) = [ e , e θ ]( θ ) = − χe θ ( θ ) = − χγ − / . Thus e ( γ − / ) = − χγ − / from which e ( γ ) = 2 χγ. This concludes the proof of the lemma.
Remark 2.2.21.
In Schwarzschild, relative to the above coordinate system, we have ς = 1 , Ω = − Υ , b = b = 0 , γ = r , e Φ = r sin θ, so that we obtain outgoing Eddington-Finkelstein coordinates. Remark 2.2.22.
The ( u, s, θ, ϕ ) coordinates system, with the choice b = 0 (i.e. θ istransported by e ( θ ) = 0 ), will be used in section 3.7 and Chapter 9 in connection withour GCM procedure. CHAPTER 2. PRELIMINARIES ( u, r, θ, ϕ ) coordinatesProposition 2.2.23. Consider, in addition to the functions u, r, ϕ an additional Z in-variant function θ . Relative to the coordinates ( u, r, θ, ϕ ) the following hold true,1. The spacetime metric takes the form, g = − ςrκ dudr + ς ( κ + A ) κ du + γ (cid:18) dθ − ςbdu − b (cid:19) (2.2.44) where, b = e ( θ ) , b = e ( θ ) , γ = 1( e θ ( θ )) (2.2.45) and, Θ := 4 rκ dr − ς (cid:18) κ + Aκ (cid:19) du.
2. The reduced coordinates derivatives take the form, ∂ r = 2 rκ e − √ γrκ be θ ,∂ θ = √ γe θ ,∂ u = ς (cid:20) e − κ + Aκ e − √ γ (cid:18) b − (cid:18) κ + Aκ (cid:19) b (cid:19) e θ (cid:21) . (2.2.46)
3. To control e Φ , we will rely on the following transport equation e (cid:18) e Φ r sin θ − (cid:19) = e Φ r sin θ (ˇ κ − ϑ ) . (2.2.47) Proof.
First, from the fact that ( e , e , e θ ) forms a null frame, we easily verify that (2.2.46)holds. Then, (2.2.44) immediately follows from (2.2.46) and the fact that ( e , e , e θ ) formsa null frame.It remains to prove (2.2.47). It follows from e (cid:18) e Φ r sin θ − (cid:19) = e Φ r sin θ (cid:18) e (Φ) − e ( r ) r (cid:19) = e Φ r sin θ (cid:18)
12 ( κ − ϑ ) − κ (cid:19) = e Φ r sin θ (ˇ κ − ϑ )which concludes the proof of the lemma. .2. MAIN EQUATIONS Remark 2.2.24.
In Schwarzschild, relative to the above coordinate system, we have κ = 2 r , κ = − r , ς = 1 , A = 0 , b = b = 0 , γ = r , e Φ = r sin θ, so that we obtain outgoing Eddington-Finkelstein coordinates. Remark 2.2.25.
The ( u, r, θ, ϕ ) coordinates system, with the choice (2.2.52) for θ intro-duced below, will be used in Proposition 3.4.3 to prove the convergence to the outgoingEddington-Finkelstein coordinates of Schwarzschild. ( u, r, θ, ϕ ) coordinates We easily deduce an analog statement relative to ( u, r, θ, ϕ ) coordinates.
Proposition 2.2.26.
Consider, in addition to the functions u, r, ϕ an additional Z in-variant function θ . Relative to the coordinates ( u, r, θ, ϕ ) the following hold true,1. The spacetime metric takes the form, g = − ςrκ dudr + ς ( κ + A ) κ du + γ (cid:18) dθ − ςbdu − b (cid:19) (2.2.48) where, b = e ( θ ) , b = e ( θ ) , γ = 1( e θ ( θ )) (2.2.49) and, Θ := 4 rκ dr − ς (cid:18) κ + Aκ (cid:19) du.
2. The reduced coordinates derivatives take the form, ∂ r = 2 rκ e − √ γrκ be θ ,∂ θ = √ γe θ ,∂ u = ς (cid:20) e − κ + Aκ e − √ γ (cid:18) b − (cid:18) κ + Aκ (cid:19) b (cid:19) e θ (cid:21) . (2.2.50)0 CHAPTER 2. PRELIMINARIES
3. To control e Φ , we will rely on the following transport equation e (cid:18) e Φ r sin θ − (cid:19) = e Φ r sin θ (ˇ κ − ϑ ) . (2.2.51) Remark 2.2.27.
In Schwarzschild, relative to the above coordinate system, we have κ = 2 r , κ = − r , ς = 1 , A = 0 , b = b = 0 , γ = r , e Φ = r sin θ, so that we obtain ingoing Eddington-Finkelstein coordinates. Remark 2.2.28.
The ( u, r, θ, ϕ ) coordinates system, with the choice (2.2.52) for θ in-troduced below, will be used in Proposition 3.4.4 to prove the convergence to the ingoingEddington-Finkelstein coordinates of Schwarzschild. Initialization of θ We now introduce the coordinate function θ that will be used for the ( u, r, θ, ϕ ) coordinatessystem and for the ( u, r, θ, ϕ ) coordinates system, see Remarks 2.2.25 and 2.2.28. Lemma 2.2.29.
Let θ ∈ [0 , π ] be the Z -invariant scalar on M defined by, θ := cot − ( re θ (Φ)) . (2.2.52) Then, e Φ r sin θ = √ a . (2.2.53) where, a := e r + ( e θ ( e Φ )) − . (2.2.54) Moreover, we have in an outgoing geodesic foliation re θ ( θ ) = 1 + r ( K − r )1 + ( re θ (Φ)) ,e ( θ ) = − rβ + r ( − ˇ κ + A + ϑ ) e θ (Φ) + rξe (Φ) + rηe (Φ)1 + ( re θ (Φ)) ,e ( θ ) = − rβ + r ( − ˇ κ + ϑ ) e θ (Φ) − rζe (Φ)1 + ( re θ (Φ)) , and analog identities hold for an ingoing geodesic foliation. .2. MAIN EQUATIONS Proof.
In view of the definition of θ , we have θ ∈ [0 , π ], sin θ ≥ θ = 1 √ θ = 1 (cid:112) re θ (Φ)) = e Φ (cid:112) e + ( re θ ( e Φ ))) = e Φ r (cid:113) e r + ( e θ ( e Φ )) = e Φ r √ a . Hence e Φ r sin θ = (cid:114) e r + ( e θ ( e Φ )) = √ a . Also, we compute re θ ( θ ) = − r e θ e θ (Φ)1 + ( re θ (Φ)) . Next, recall that we have e θ e θ (Φ) = − K − ( e θ (Φ)) . We infer re θ ( θ ) = r ( K + ( e θ (Φ)) )1 + ( re θ (Φ)) = 1 + r ( K − r )1 + ( re θ (Φ)) . as desired.Also, we have in an outgoing geodesic foliation e ( θ ) = − re e θ (Φ) + e ( r ) e θ (Φ)1 + ( re θ (Φ)) = − r ( D D θ Φ + D D e θ Φ) + e ( r ) e θ (Φ)1 + ( re θ (Φ)) = − rβ + r (cid:16) e ( r ) r − e (Φ) (cid:17) e θ (Φ) − rζe (Φ)1 + ( re θ (Φ)) = − rβ + r ( − ˇ κ + ϑ ) e θ (Φ) − rζe (Φ)1 + ( re θ (Φ)) . CHAPTER 2. PRELIMINARIES
Finally, we compute in an outgoing geodesic foliation e ( θ ) = − re e θ (Φ) + e ( r ) e θ (Φ)1 + ( re θ (Φ)) = − r ( D D θ Φ + D D e θ Φ) + e ( r ) e θ (Φ)1 + ( re θ (Φ)) = − rβ + r (cid:16) e ( r ) r − e (Φ) (cid:17) e θ (Φ) + rξe (Φ) + rηe (Φ)1 + ( re θ (Φ)) = − rβ + r ( − ˇ κ + A + ϑ ) e θ (Φ) + rξe (Φ) + rηe (Φ)1 + ( re θ (Φ)) . This concludes the proof of the lemma.In view of (2.2.53), we will need to control the quantity a defined in (2.2.54). To this end,we will need the following lemma. Lemma 2.2.30.
The quantity a defined in (2.2.54) vanishes on the axis of symmetry andverifies the following identities in an outgoing geodesic foliation, e ( a ) = (ˇ κ − ϑ ) e r + 2 e θ ( e Φ ) (cid:16) β − e (Φ) ζ (cid:17) e Φ ,e θ ( a ) = 2 e θ (Φ) e (cid:18)(cid:18) ρ + 2 mr (cid:19) + 14 (cid:18) κκ + 4Υ r (cid:19) − ϑϑ (cid:19) ,e ( a ) = (cid:16) ˇ κ − A − ϑ (cid:17) e r + 2 e θ ( e Φ ) (cid:16) β + e (Φ) η + ξe (Φ) (cid:17) e Φ , and analog identities hold in an outgoing geodesic foliation.Proof. The vanishing on the axis follow easily from the fact that both e and e θ ( e Φ )) − R ab = D a D b Φ + D a Φ D b Φ , and (see Definition 2.1.47) R θ = β, R θ = β, R θθ = R = ρ, R = ρ. .2. MAIN EQUATIONS a = e r + ( e θ ( e Φ )) −
1, we compute in an outgoing geodesicfoliation e ( a ) = 2 e (Φ) e r − e ( r ) e r + 2 e θ ( e Φ ) e ( e θ ( e Φ ))= ( κ − ϑ ) e r − κe r + 2 e θ ( e Φ ) (cid:16) e ( e θ (Φ)) + e θ (Φ) e (Φ) (cid:17) e Φ = (ˇ κ − ϑ ) e r + 2 e θ ( e Φ ) (cid:16) β − e (Φ) ζ (cid:17) e Φ . Also e θ ( a ) = 2 e θ (Φ) e r + 2 e θ ( e Φ ) e θ ( e θ ( e Φ ))= 2 e θ (Φ) e r + 2 e θ ( e Φ ) (cid:16) e θ ( e θ (Φ)) + e θ (Φ) (cid:17) e Φ = 2 e θ (Φ) e r + 2 e θ ( e Φ ) (cid:16) ρ + D D θ e θ Φ (cid:17) e Φ = 2 e θ (Φ) e r + 2 e θ ( e Φ ) (cid:16) ρ + 12 χe Φ + 12 χe Φ (cid:17) e Φ = 2 e θ (Φ) e r + 2 e θ ( e Φ ) (cid:16) ρ + 14 κκ − ϑϑ (cid:17) e Φ = 2 e θ (Φ) e (cid:18)(cid:18) ρ + 2 mr (cid:19) + 14 (cid:18) κκ + 4Υ r (cid:19) − ϑϑ (cid:19) . Finally, we have in an outgoing geodesic foliation e ( a ) = 2 e (Φ) e r − e ( r ) e r + 2 e θ ( e Φ ) e ( e θ ( e Φ ))= ( κ − ϑ ) e r − (cid:16) κ + A (cid:17) e r + 2 e θ ( e Φ ) (cid:16) e ( e θ (Φ)) + e θ (Φ) e (Φ) (cid:17) e Φ = (cid:16) ˇ κ − A − ϑ (cid:17) e r + 2 e θ ( e Φ ) (cid:16) β + e (Φ) η + ξe (Φ) (cid:17) e Φ . This concludes the proof of the lemma.
Remark 2.2.31.
The function θ defined by (3.4.3) defines • together with the functions ( u, r, ϕ ) , a regular coordinates system with the axis ofsymmetry corresponding to θ = 0 , π , • together with the functions ( u, r, ϕ ) , a regular coordinates system with the axis ofsymmetry corresponding to θ = 0 , π . CHAPTER 2. PRELIMINARIES
Recall that in Schwarzschild all Ricci coefficients ξ, ξ, ϑ, ϑ, η, η, ζ and curvature compo-nents α, α, β, β vanish identically. In addition the check quantities ˇ κ, ˇ κ, ˇ ω, ˇ ω and ˇ ρ alsovanish. Thus, roughly, we expect that in perturbations of Schwarzschild these quantitiesstay small, i.e. of oder O ( (cid:15) ) for a sufficiently small (cid:15) . More precisely we say that a smooth,vacuum, Z -invariant, polarized spacetime is an O ( (cid:15) )-perturbation of Schwarzschild, orsimply O ( (cid:15) )-Schwarzschild, if the following are true relative to a Z -invariant null frame e , e , e θ , ξ, ξ, ϑ, ϑ, η, η, ζ, ˇ κ, ˇ κ, ˇ ω, ˇ ω α, α, β, β , ˇ ρ = O ( (cid:15) ) (2.3.1)Moreover, e ( r ) − r κ = O ( (cid:15) ) , e ( r ) − r κ = O ( (cid:15) ) , (2.3.2)where r is the area radius of the 2-spheres generated by e θ , e ϕ , see (2.1.12).In reality, of course, we expect that small perturbations of Schwarzschild, remain not onlyclose to the original Schwarzschild but also converge to a nearby Schwarzschild solutionbut for the discussion below this will suffice. Our definition of O ( (cid:15) )-Schwarzschild perturbations does not specify a particular frame. Inwhat follows we investigate how the main Ricci and curvature quantities change relative toframe transformations, i.e linear transformations which take null frames into null frames. Lemma 2.3.1.
A general null transformation can be written in the form, e (cid:48) = λ (cid:18) e + f e θ + 14 f e (cid:19) ,e (cid:48) θ = (cid:18) f f (cid:19) e θ + 12 f e + 12 f (cid:18) f f (cid:19) e ,e (cid:48) = λ − (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19) . (2.3.3) Proof.
It is straightforward to check that the transformation (2.3.3) takes null frames intonull frames. One can also check that it can be written in the form type(3) ◦ type(1) ◦ .3. PERTURBATIONS OF SCHWARZSCHILD AND INVARIANT QUANTITIES e , i.e. ( λ = 1 , f = 0), type 2 transformationsfix e , i.e. ( λ = 1 , f = 0) and type 3 transformations keep the directions of e , e i.e.( f = f = 0). Remark 2.3.2.
Note that f, f are reduced from spacetime forms while λ is reduced froma scalar. Remark 2.3.3.
A transformation consistent with O ( (cid:15) ) - Schwarzschild spacetimes musthave f, f = O ( (cid:15) ) and a := log λ = O ( (cid:15) ) . Proposition 2.3.4 (Transformation formulas) . Under a general transformation of type (2.3.3) , the Ricci coefficients and curvature components transform as follows: ξ (cid:48) = λ (cid:18) ξ + 12 λ − e (cid:48) ( f ) + ωf + 14 f κ (cid:19) + λ Err ( ξ, ξ (cid:48) ) , Err ( ξ, ξ (cid:48) ) = 14 f ϑ + l.o.t. ,ξ (cid:48) = λ − (cid:18) ξ + 12 λe (cid:48) ( f ) + ω f + 14 f κ (cid:19) + λ − Err ( ξ, ξ (cid:48) ) , Err ( ξ, ξ (cid:48) ) = − λf e (cid:48) ( f ) + 14 f ϑ + l.o.t. , (2.3.4) ζ (cid:48) = ζ − e (cid:48) θ (log( λ )) + 14 ( − f κ + f κ ) + f ω − f ω + Err ( ζ, ζ (cid:48) ) , Err ( ζ, ζ (cid:48) ) = 12 f e (cid:48) θ ( f ) + 14 ( − f ϑ + f ϑ ) + l.o.t. ,η (cid:48) = η + 12 λe (cid:48) ( f ) + 14 κf − f ω + Err ( η, η (cid:48) ) , Err ( η, η (cid:48) ) = 14 f ϑ + l.o.t. ,η (cid:48) = η + 12 λ − e (cid:48) ( f ) + 14 κf − f ω + Err ( η, η (cid:48) ) , Err ( η, η (cid:48) ) = − f λ − e (cid:48) ( f ) + 14 f ϑ + l.o.t. , (2.3.5) κ (cid:48) = λ ( κ + d/ (cid:48) ( f )) + λ Err ( κ, κ (cid:48) ) , Err ( κ, κ (cid:48) ) = f ( ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t. ,κ (cid:48) = λ − (cid:0) κ + d/ (cid:48) ( f ) (cid:1) + λ − Err ( κ, κ (cid:48) ) , Err ( κ, κ (cid:48) ) = − f e (cid:48) θ ( f ) + f ( − ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t. , (2.3.6)6 CHAPTER 2. PRELIMINARIES ϑ (cid:48) = λ ( ϑ − d (cid:63) / (cid:48) ( f )) + λ Err ( ϑ, ϑ (cid:48) ) , Err ( ϑ, ϑ (cid:48) ) = f ( ζ + η ) + f ξ + 14 f f κ + f f ω − f ω + l.o.t. ϑ (cid:48) = λ − (cid:0) ϑ − d (cid:63) / (cid:48) ( f ) (cid:1) + λ − Err ( ϑ, ϑ (cid:48) ) , Err ( ϑ, ϑ (cid:48) ) = − f e (cid:48) θ ( f ) + f ( − ζ + η ) + f ξ + 14 f f κ + f f ω − f ω + l.o.t. , (2.3.7) ω (cid:48) = λ (cid:18) ω − λ − e (cid:48) (log( λ )) (cid:19) + λ Err ( ω, ω (cid:48) ) , Err ( ω, ω (cid:48) ) = 14 f e (cid:48) ( f ) + 12 ωf f − f η + 12 f ξ + 12 f ζ − κf + 18 f f κ − ωf + l.o.t. ,ω (cid:48) = λ − (cid:18) ω + 12 λe (cid:48) (log( λ )) (cid:19) + λ − Err ( ω, ω (cid:48) ) , Err ( ω, ω (cid:48) ) = − f e (cid:48) ( f ) + ωf f − f η + 12 f ξ − f ζ − κf + 18 f f κ − ωf + l.o.t. (2.3.8) The lower order terms we denote by l.o.t. are linear with respect
Γ = { ξ, ξ, ϑ, κ, η, η, ζ, κ, ϑ } and quadratic or higher order in f, f , and do not contain derivatives of these latter.Also, α (cid:48) = λ α + λ Err ( α, α (cid:48) ) , Err ( α, α (cid:48) ) = 2 f β + 32 f ρ + l.o.t. ,β (cid:48) = λ (cid:18) β + 32 ρf (cid:19) + λ Err ( β, β (cid:48) ) , Err ( β, β (cid:48) ) = 12 f α + l.o.t. ,ρ (cid:48) = ρ + Err ( ρ, ρ (cid:48) ) , Err ( ρ, ρ (cid:48) ) = 32 ρf f + f β + f β + l.o.t. ,β (cid:48) = λ − (cid:18) β + 32 ρf (cid:19) + λ − Err ( β, β (cid:48) ) , Err ( β, β (cid:48) ) = 12 f α + l.o.t. ,α (cid:48) = λ − α + λ − Err ( α, α (cid:48) ) , Err ( α, α (cid:48) ) = 2 f β + 32 f ρ + l.o.t. (2.3.9) The lower order terms we denote by l.o.t. are linear with respect to the curvature quantities α, β, ρ, β, α and quadratic or higher order in f, f , and do not contain derivatives of theselatter. .3. PERTURBATIONS OF SCHWARZSCHILD AND INVARIANT QUANTITIES Proof.
See Appendix A.6.
Lemma 2.3.5.
In the particular case when λ = 1 , f = 0 , we have e (cid:48) = e + f e θ + 14 f e ,e (cid:48) θ = e θ + 12 f e ,e (cid:48) = e , and ξ (cid:48) = ξ + 12 e (cid:48) f + 14 κf + f ω + 14 f ϑ + 14 f η − f η + 12 f ζ − f κ − f ω − f ϑ − f ξ,ω (cid:48) = ω + 12 f ζ − ηf − f ω − f κ − f ϑ − f ξ,ζ (cid:48) = ζ − (cid:18) κ + ω (cid:19) f − f (cid:18) ϑ + 12 f ξ (cid:19) ,η (cid:48) = η + 12 e (cid:48) ( f ) − f ω − f ξ. Proof.
The proof follows from Proposition 2.3.4 by setting λ = 1 , f = 0. Since we needprecise formulas for the error terms, we provide a proof in section A.9. Lemma 2.3.6 (Transport equations for ( f, f , λ )) . Assume that we have in the new nullframe ( e (cid:48) , e (cid:48) , e (cid:48) θ ) of type (2.3.3) ξ (cid:48) = 0 , ω (cid:48) = 0 , ζ (cid:48) + η (cid:48) = 0 . Then, ( f , f, log( λ )) satisfy the following transport equations λ − e (cid:48) ( f ) + (cid:16) κ ω (cid:17) f = − ξ + E ( f, Γ) ,λ − e (cid:48) (log( λ )) = 2 ω + E ( f, Γ) ,λ − e (cid:48) ( f ) + κ f = − ζ + η ) + 2 e (cid:48) θ (log( λ )) + 2 f ω + E ( f, f , Γ) , where E , E and E are given by E ( f, Γ) = − ϑf + l.o.t. ,E ( f, Γ) = f ζ − f ω − ηf − f κ + l.o.t. ,E ( f, f , Γ) = − f e (cid:48) θ ( f ) − f ϑ + l.o.t. , CHAPTER 2. PRELIMINARIES
Here, l.o.t. denote terms which are cubic or higher order in f, f (or in f only in the caseof E and E ) and ˇΓ and do not contain derivatives of these quantities, where Γ and ˇΓ denotes the Ricci coefficients and renormalized Ricci coefficients w.r.t. the original nullframe ( e , e , e θ ) .Proof. See section A.7.To avoid a potential log loss for the third equation in Lemma 2.3.6, i.e. the transportequation for f , we state the following renormalized version of the lemma. Corollary 2.3.7.
Assume given a null frame ( e , e , e θ ) associated to an outgoing geodesicfoliation as in section 2.2.4, and let r denote the corresponding area radius. Assume thatwe have in the new null frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) of type (2.3.3) ξ (cid:48) = 0 , ω (cid:48) = 0 , ζ (cid:48) + η (cid:48) = 0 . Then, ( f , f, log( λ )) satisfy the following transport equations λ − e (cid:48) ( rf ) = E (cid:48) ( f, Γ) ,λ − e (cid:48) (log( λ )) = E (cid:48) ( f, Γ) ,λ − e (cid:48) (cid:16) rf − r e (cid:48) θ (log( λ )) + rf Ω (cid:17) = E (cid:48) ( f, f , λ, Γ) , where E (cid:48) ( f, Γ) = − r κf − r ϑf + l.o.t. ,E (cid:48) ( f, Γ) = f ζ − f ω − ηf − f κ + l.o.t. ,E (cid:48) ( f, f , λ, Γ) = − r κf + r (cid:18) ˇ κ − (cid:18) κ − r (cid:19)(cid:19) e (cid:48) θ (log( λ )) + r (cid:16) d/ (cid:48) ( f ) + λ − ϑ (cid:48) (cid:17) e (cid:48) θ (log( λ )) − r κ Ω f + rE ( f, f , Γ) − r e (cid:48) θ ( E ( f, Γ)) + r Ω E ( f, Γ) , and where E , E and E are given in Lemma 2.3.6.Proof. See section A.8. Γ g and Γ b Many of the identities which we present below, contain a huge number of O ( (cid:15) ) terms. Inwhat follows we introduce schematic notation meant to keep track of the most important .3. PERTURBATIONS OF SCHWARZSCHILD AND INVARIANT QUANTITIES g and Γ b is consistentwith our main bootstrap assumptions BA-E on energy and
BA-D on decay, see section3.4.1.
Definition 2.3.8.
We divide the small connection coefficient terms (relative to an arbi-trary null frame) into Γ (0) g = (cid:26) rξ, ϑ, ζ, η, r e ( r ) − κ, r e θ ( r ) (cid:27) , Γ (0) b = (cid:26) η, ϑ, ξ, r e ( r ) − κ (cid:27) . For higher derivatives we introduce, Γ (1) g = (cid:110) d Γ (0) g , r e θ ( ω ) , re θ ( κ ) , re θ ( κ ) (cid:111) , Γ (1) b = (cid:110) d Γ (0) b , re θ ( ω ) (cid:111) , and for s ≥ , Γ ( s ) g = d ≤ s Γ g , Γ ( s ) b = d ≤ s Γ b , where we have introduced the notations d = { e , re , d / } , with angular derivatives d / of reduced scalars in s k defined by (2.1.37) . Remark 2.3.9.
According to the main bootstrap assumptions
BA-E , BA-D (see section3.4.1), the terms Γ b behave worse in powers of r than the terms in Γ g . Thus, in thecalculations below, we replace the terms of the form Γ ( s ) g + Γ ( s ) b by Γ ( s ) b . Given the form ofthe bootstrap assumptions, we may also replace r − Γ ( s ) b by Γ ( s ) g . We will denote l.o.t. thecubic and higher error terms in ˇΓ , ˇ R . We also include in l.o.t. terms which decay fasterin powers of r than the main quadratic terms. q Note from the transformation formulas of Proposition 2.3.4 that the only quantities whichremain invariant up to quadratic or higher order error terms are α , α and ρ . Among theseonly α, α vanish in Schwarzschild. We call such quantities O ( (cid:15) ) invariant. In what followswe show that, in addition to these two invariants, there exist other important invariants. In the frames we are using, we have in fact ξ = 0 for r ≥ m so that it behaves in fact better thanthe other components of Γ (0) g . CHAPTER 2. PRELIMINARIES
Lemma 2.3.10.
The expression, e ( e ( α )) + (2 κ − ω ) e ( α ) + (cid:18) − e ( ω ) + 8 ω − ω κ + 12 κ (cid:19) α is an O ( (cid:15) ) invariant. It is also a conformal invariant, i.e. invariant under transforma-tions (2.3.3) with f = f = 0 .Proof. Clearly the quantity vanishes in Schwarzschild and is an O ( (cid:15) ) invariant. Fora conformal transformation, the result follows by a straightforward application of thetransformation properties of Proposition 2.3.4 in the particular case where f = f = 0. Remark 2.3.11.
Alternatively one can also define the corresponding quantity obtainedby interchanging e , e , i.e. e ( e ( α )) + (2 κ − ω ) e ( α ) + (cid:18) − e ( ω ) + 8 ω − ωκ + 12 κ (cid:19) α. Note that it differs by O ( (cid:15) ) from the previous one. Definition 2.3.12.
Given a general null frame ( e , e , e θ ) , and given a scalar function r satisfying the assumptions for section 2.3.2, i.e. r e ( r ) − κ ∈ Γ g , r e θ ( r ) ∈ Γ g , r e ( r ) − κ ∈ Γ b , we defined our main quantity q as q := r (cid:20) e ( e ( α )) + (2 κ − ω ) e ( α ) + (cid:18) − e ( ω ) + 8 ω − ω κ + 12 κ (cid:19) α (cid:21) . (2.3.10) q In this section, we state three identities involving the quantity q defined by (2.3.10). Allcalculations are made in a general frame. Proposition 2.3.13.
We have q = r (cid:18) d (cid:63) / d (cid:63) / ρ + 34 κρϑ + 34 κρϑ (cid:19) + Err [ q ] (2.3.11) with error term written schematically in the formErr [ q ] = r e η · β + r d ≤ (cid:0) Γ b · Γ g ) . (2.3.12) .4. INVARIANT WAVE EQUATIONS Proof.
See section A.10The following consequence of Proposition 2.3.13 will prove to be very useful in the sequel.
Proposition 2.3.14.
We have e ( r q ) = r (cid:26) d (cid:63) / d (cid:63) / d/ β − ρ d (cid:63) / d (cid:63) / κ − κρ d (cid:63) / ζ − κρα + 34 (2 ρ − κκρ ) ϑ (cid:27) + Err [ e ( r q )] , (2.3.13) where the error term Err [ e ( r q )] is given schematically byErr [ e ( r q )] = r Γ b q + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) . (2.3.14) Proof.
See section A.11.We deduce from Proposition 2.3.14 the following nonlinear version of the Teukolsky-Starobinski identity.
Proposition 2.3.15.
The following identity holds true in ( int ) M , e ( r e ( r q )) + 2 ωr e ( r q ) = r (cid:26) d (cid:63) / d (cid:63) / d/ d/ α + 32 ρ (cid:16) κe − κe (cid:17) α (cid:27) + Err [ T S ] , (2.3.15) where the error term Err [ T S ] is given schematically byErr [ T S ] = r (cid:0) d / Γ b + r Γ b · Γ b ) · α + r (cid:0) Γ b e ( r q ) + ( d ≤ Γ b ) r q (cid:17) + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) . Proof.
See section A.12.
In this section, we write wave equations for the invariant quantities α , α and q .02 CHAPTER 2. PRELIMINARIES
Lemma 2.4.1.
With respect to a general S -foliation we have, for a reduced scalar ψ ∈ s , (cid:3) g ψ = −
12 ( e e + e e ) ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ + (cid:18) ω − κ (cid:19) e ψ + ( η + η ) e θ ψ. (2.4.1) Also, (cid:3) g ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ, (cid:3) g ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ. Proof.
We calculate, in spacetime, (cid:3) g ψ = g D D ψ + g D D ψ + δ AB D A D B ψ = −
12 ( D D + D D ) ψ + g AB D A D B ψ. Now, δ AB D A D B ψ = (cid:52) / ψ − (1+3) tr χe ψ − (1+3) tr χe ψ, D D ψ = e e ψ − ωe ψ − ηe θ ψ, D D ψ = e e ψ − ωe ψ − ηe θ ψ. Hence, (cid:3) g ψ = −
12 ( e e + e e ) ψ + (cid:52) / ψ − (1+3) tr χe ψ − (1+3) tr χe ψ + ωe ψ + ηe θ ψ + ωe ψ + ηe θ ψ = −
12 ( e e + e e ) ψ + (cid:52) / ψ + (cid:18) ω − (1+3) tr χ (cid:19) e ψ + (cid:18) ω − (1+3) tr χ (cid:19) e ψ + ( η + η ) e θ ψ. Since, 12 e e ψ = 12 e e ψ + ωe ψ − ωe ψ + ( η − η ) e θ ψ we also have, (cid:3) g ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − (1+3) tr χ (cid:19) e ψ − (1+3) tr χe ψ + 2 ηe θ ψ. Since κ = (1+3) tr χ, κ = (1+3) tr χ , this concludes the proof of the lemma. .4. INVARIANT WAVE EQUATIONS Definition 2.4.2.
Given a reduced k -scalar ψ ∈ s k we define, (cid:3) k ψ = −
12 ( e e + e e ) ψ + (cid:52) / k ψ + ( ω − tr χ ) e ψ + ( ω − tr χ ) e ψ − ( η + η ) e θ ψ. (2.4.2) Equivalently, we have (cid:3) k ψ = − e e ψ + (cid:52) / k ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ, (cid:3) k ψ = − e e ψ + (cid:52) / k ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ. Remark 2.4.3.
Not that the terms ηe θ ψ, ηe θ ψ have to be interpreted as in Remark 2.1.23,i.e. ηe θ ψ = 12 η ( d/ k ψ − d (cid:63) / k +1 ψ ) . The term η d/ k ψ is the reduced form of a tensor product of (1+3) η with D / k (1+3) ψ while η d (cid:63) / k +1 ψ is the reduced form of a contraction between the form (1+3) η and k + 1 tensor D (cid:63) / k +1 (1+3) ψ . Remark 2.4.4.
Recall that (recall Definition 2.1.20), (cid:52) / k f := e θ ( e θ f ) + e θ (Φ) e θ f − k (cid:0) e θ (Φ) (cid:1) f. Thus, for a ψ ∈ s k , we have, (cid:52) / k ψ = (cid:52) / ψ − k (cid:0) e θ (Φ) (cid:1) ψ. Spacetime interpretation of Definition 2.4.2
The linearized equation verified by our main quantity q , which will be derived in the nextsubsection, has the form, (cid:3) ψ + V ψ = 0 . (2.4.3)with V a scalar potential. In what follows we give simple spacetime interpretation of theequation.Given a mixed spacetime tensor in T k M ⊗ T lS M of the form U µ ...µ k ,A ...A L where e µ is anorthonormal frame on M with ( e A ) A =1 , tangent to S . We define,˙ D µ U ν ...ν k ,A ...A L = e µ U ν ...ν k ,A ...A l − U D µ ν ...ν k ,A ...A l − . . . − U ν ... D µ ν k ,A ...A l − U ν ...ν k , ˙ D µ A ...A l − U ν ...ν k ,A ... ˙ D µ A l CHAPTER 2. PRELIMINARIES with ˙ D µ A denoting the projection of D e µ e A on S . One can easily check the commutatorformulae, ( ˙ D µ ˙ D ν − ˙ D ν ˙ D µ )Ψ A = R A B µν Ψ B , ( ˙ D µ ˙ D ν − ˙ D ν ˙ D µ )Ψ λA = R λ σ µν Ψ σA + R A B µν Ψ λB . Define, ˙ (cid:3) g Ψ := g µν ˙ D µ ˙ D ν Ψ . Consider the following Lagrangian for Ψ = Ψ AB ∈ S . L [Ψ] = g/ A B g/ A B (cid:16) g µν ˙ D µ Ψ A A ˙ D µ Ψ B B + V Ψ A A Ψ B B (cid:17) . Proposition 2.4.5.
The Euler- Lagrange equations for the Lagrangian L [Ψ] above aregiven by: ˙ (cid:3) Ψ = V Ψ (2.4.4) and its reduced for ψ = Ψ θθ is precisely (2.4.3) .Proof. Straightforward verification. α , α , and q We start with the wave equations for α and α , which are derived in a general null frame. Proposition 2.4.6.
The following identities hold true.1. The invariant quantity α ∈ s verifies the Teukolsky wave equation, (cid:3) α = − ωe ( α ) + (4 ω + 2 κ ) e ( α ) + V α + Err [ (cid:3) g α ] ,V = − ρ − e ( ω ) − ωω + 2 ω κ − κ ω + 12 κ κ, (2.4.5) where Err [ (cid:3) g α ] is given schematically byErr ( (cid:3) g α ) = Γ g e ( α ) + r − d ≤ (cid:16) ( η, Γ g )( α, β ) (cid:17) + ξ ( e ( β ) , r − d ˇ ρ ) + l.o.t.where l.o.t. denote terms which are quadratic and enjoy better decay properties orare higher order and decay at least as good. .4. INVARIANT WAVE EQUATIONS
2. The invariant quantity α ∈ s verifies the Teukolsky wave equation, (cid:3) α = − ωe ( α ) + (4 ω + 2 κ ) e ( α ) + V α + Err [ (cid:3) g α ] ,V = − ρ − e ( ω ) − ωω + 2 ωκ − κ ω + 12 κ κ, (2.4.6) where Err ( (cid:3) g α ) = r − d (Γ b α ) + d (Γ b β ) + l.o.t.Proof. See appendix A.13We may now state the wave equation satisfied by q . Theorem 2.4.7.
The invariant scalar quantity q defined in (2.3.10) verifies the equation, (cid:3) q + κκ q = Err [ (cid:3) q ] (2.4.7) where Err [ (cid:3) q ] is O ( (cid:15) ) .If q is defined relative to a null frame satisfying, in addition to the assumptions of section2.3.2, that η ∈ Γ g and ξ = 0 for r ≥ m , the error term is then given schematically byErr [ (cid:3) q ] = r d ≤ (Γ g · ( α, β )) + e (cid:16) r d ≤ (Γ g · ( α, β )) (cid:17) + d ≤ (Γ g · q ) + l.o.t. (2.4.8) Proof.
See appendix A.14.
Remark 2.4.8.
Note that the main frame used in this paper is an outgoing geodesic nullframe in r ≥ m so that ξ = 0 , but unfortunately, as it turns out, η ∈ Γ b . This wouldnot allow us to control the error term appearing in (2.4.7) . To overcome this problem, weare forced to define q relative to a different frame where ξ = 0 still holds for r ≥ m andfor which we have in addition η ∈ Γ g , see Proposition 3.5.5 for the existence of such aframe. See also the discussion at the beginning of section 3.4.6. The remark above leads us to the following.
Remark 2.4.9.
The quantity q we will be working with for the rest of the paper is defined,according to equation (2.3.10) , relative to the global frame of Proposition 3.5.5 for which η ∈ Γ g . It is only in such a frame that q verifies the correct decay estimates. CHAPTER 2. PRELIMINARIES hapter 3MAIN THEOREM
Note that all definitions below are consistent with the framework of Z -invariant polarizedspacetimes. Recall that m > (cid:15) close. Let δ H > Definition 3.1.1 (Initial data layer) . We consider a spacetime region ( L , g ) , sketchedbelow in figure 3.1, where • The metric g is a reduced metric from a Lorentzian spacetime metric g close toSchwarzschild in a suitable topology . • L = ( ext ) L ∪ ( int ) L . • The intersection ( ext ) L ∩ ( int ) L is non trivial. This topology will be specified in our initial data layer assumptions, see (3.3.5) as well as section3.2.4.
CHAPTER 3. MAIN THEOREM
Furthermore, our initial data layer ( L , g ) satisfies1. Boundaries.
The future and past boundaries of L are given by ∂ + L = A ∪ C (2 , L ) ∪ C (2 , L ) ,∂ − L = C (0 , L ) ∪ C (0 , L ) , where(a) The past outgoing null boundary of the far region ( ext ) L is denoted by C (0 , L ) .(b) The past incoming null boundary of the near region ( int ) L is denoted by C (0 , L ) .(c) ( ext ) L is unbounded in the future outgoing null directions.(d) The future outgoing null boundary of the far region ( ext ) L is denoted by C (2 , L ) .(e) The future incoming null boundary of the near region ( int ) L is denoted by C (2 , L ) .(f ) The future spacelike boundary of the near region ( int ) L is denoted by A .2. Foliations of L and adapted null frames. The spacetime L = ( ext ) L ∪ ( int ) L is foliated as follows(a) The far region ( ext ) L is foliated by two functions ( u L , ( ext ) s L ) such that • u L is an outgoing optical function on ( ext ) L whose leaves are denoted by C ( u L , L ) . • ( ext ) s L is an affine parameter along the level hypersurfaces of u L , i.e. ( ext ) L ( ( ext ) s L ) = 1 where ( ext ) L := − g ab ∂ b ( u L ) ∂ a . • We denote by ( ( ext ) ( e ) , ( ext ) ( e ) , ( ext ) ( e ) θ ) the null frame adapted to theoutgoing geodesic foliation ( u L , ( ext ) s L ) on ( ext ) L . • Let ( ext ) r L denote the area radius of the 2-spheres S ( u L , ( ext ) s L ) of thisfoliation. • The outgoing future null boundary C (2 , L ) corresponds precisely to u L = 1 and the outgoing past null boundary C (0 , L ) corresponds to u L = − . • The foliation by u L of ( ext ) L terminates at the time like boundary (cid:26) ( ext ) r L = 2 m (cid:18) δ H (cid:19)(cid:27) . (b) The near region ( int ) L is foliated by two functions ( u L , ( int ) s L ) such that .1. GENERAL COVARIANT MODULATED ADMISSIBLE SPACETIMES • u L is an ingoing optical function on ( int ) L whose leaves are denoted by C ( u L , L ) . • ( int ) s L is an affine parameter along the level hypersurfaces of u L , i.e. ( int ) L ( ( int ) s L ) = − where ( ext ) L := − g ab ∂ b ( u L ) ∂ a . • We denote by ( ( int ) ( e ) , ( int ) ( e ) , ( int ) ( e ) θ ) the null frame adapted to theoutgoing geodesic foliation ( u L , ( int ) s L ) on ( int ) L . • Let ( int ) r L denote the area radius of the 2-spheres S ( u L , ( int ) s L ) of thisfoliation. • The ( u L , ( int ) s ) foliation is initialized on ( ext ) r L = 2 m (1 + δ H ) as it willbe made precise below. • The foliation by u L , of ( int ) L terminates at the space like boundary A = (cid:8) ( int ) r L = 2 m (1 − δ H ) (cid:9) . where m and δ H have been defined above. • The ingoing future null boundary C (2 , L ) corresponds precisely to u L = 2 and the ingoing past null boundary C (0 , L ) corresponds to u L = 0 . • The foliation by u L of ( int ) L terminates at the time like boundary (cid:110) ( int ) r L = 2 m (1 + 2 δ H ) (cid:111) . Initializations of the ( u L , ( int ) s ) foliation. The ( u L , ( int ) s L ) foliation is initialized on ( ext ) r L = 2 m (1 + δ H ) by setting, u L = u L , ( int ) s L = ( ext ) s L and, with λ = ( ext ) λ = 1 − m ext ) r L , ( int ) ( e ) = λ ext ) ( e ) , ( int ) ( e ) = λ −
10 ( ext ) ( e ) , ( int ) ( e ) θ = ( ext ) ( e ) θ . Coordinates system on ( ext ) L ( ( ext ) r L ≥ m ) . In ( ext ) L ( ( ext ) r L ≥ m ) , thereexists adapted coordinates ( u L , ( ext ) s L , θ L , ϕ ) with b = 0 , see Proposition 2.2.20,such that the spacetime metric g takes the form, g = − du L d (cid:0) ( ext ) s L (cid:1) + Ω L ( du L ) + γ L (cid:18) dθ L − b L du L (cid:19) + e dϕ . (3.1.1)10 CHAPTER 3. MAIN THEOREM
Figure 3.1: The initial data layer L Recall that m > (cid:15) close, and that δ H > Definition 3.1.2 (GCM-admissible spacetime) . We consider a spacetime ( M , g ) , sketchedbelow in figure 3.2, where • The metric g is a reduced metric from a Lorentzian spacetime metric g close toSchwarzschild in a suitable topology . • M = ( ext ) M ∪ ( int ) M• T = ( ext ) M ∩ ( int ) M is a time-like hyper-surface. ( M , g ) is called a general covariant modulated admissible (or shortly GCM-admissible)spacetime if it is defined as follows This topology will be specified in our bootstrap assumptions, see (3.3.6) as well as section 3.2. .1. GENERAL COVARIANT MODULATED ADMISSIBLE SPACETIMES Boundaries.
The future and past boundaries of M are given by ∂ + M = A ∪ C ∗ ∪ C ∗ ∪ Σ ∗ ,∂ − M = C ∪ C , where(a) The past boundary C ∪ C is included in the initial data layer L , defined insection 3.1.1, in which the metric on M is specified to be a small perturbationof the Schwarzschild data.(b) The future spacelike boundary of the far region ( ext ) M is denoted by Σ ∗ .(c) The future outgoing null boundary of the far region ( ext ) M is denoted by C ∗ .(d) The future incoming null boundary of the near region ( int ) M is denoted by C ∗ .(e) The future spacelike boundary of the near region ( int ) M is denoted by A .(f ) The time-like boundary T , separating ( ext ) M from ( int ) M , starts at C ∩ C and terminates at C ∗ ∩ C ∗ .2. Foliations of M and adapted null frames. The spacetime M = ( ext ) M∪ ( int ) M is foliated as follows(a) The far region ( ext ) M is foliated by two functions ( u, ( ext ) s ) such that • u is an outgoing optical function on ( ext ) M , initialized on Σ ∗ , whose leavesare denoted by C ( u ) . • ( ext ) s is an affine parameter along the level hypersurfaces of u , i.e. L ( ( ext ) s ) = 1 where L := − g ab ∂ b u∂ a . • The ( u, ( ext ) s ) foliation is initialized on Σ ∗ as it will be made precise below. • We denote by ( ( ext ) e , ( ext ) e , ( ext ) e θ ) the null frame adapted to the outgoinggeodesic foliation ( u, ( ext ) s ) on ( ext ) M where ( ext ) e = L . • Let ( ext ) r and ( ext ) m respectively the area radius and the Hawking mass ofthe 2-spheres S ( u, ( ext ) s ) of this foliation. • The outgoing future null boundary C ∗ corresponds precisely to u = u ∗ andthe outgoing past null boundary C corresponds to u = 1 . • The foliation by u of ( ext ) M terminates at the time like boundary T = (cid:8) ( ext ) r = r T (cid:9) where r T satisfies m (cid:18) δ H (cid:19) ≤ r T ≤ m (cid:18) δ H (cid:19) . A specific choice of r T will be made in section 3.8.9, see (3.8.7), in the context of a Lebesgue pointargument needed to recover the top order derivatives. CHAPTER 3. MAIN THEOREM (b) The near region ( int ) M is foliated by two functions ( u, ( int ) s ) such that • u is an ingoing optical function on ( int ) M , initialized on T , whose leavesare denoted by C ( u ) . • ( int ) s is an affine parameter along the level hypersurfaces of u , i.e. L ( ( int ) s ) = − where L := − g ab ∂ b u∂ a . • The ( u, ( int ) s ) foliation is initialized on T as it will be made precise below. • We denote by ( ( int ) e , ( int ) e , ( int ) e θ ) the null frame adapted to the outgoinggeodesic foliation ( u, ( int ) s ) on ( int ) M where ( int ) e = L . • Let ( int ) r and ( int ) m respectively the area radius and the Hawking mass ofthe 2-spheres S ( u, ( int ) s ) of this foliation. • The foliation by u of ( int ) M terminates at the space like boundary A = (cid:8) ( int ) r = 2 m (1 − δ H ) (cid:9) where m and δ H have been defined above. • The ingoing future null boundary C ∗ corresponds precisely to u = u ∗ andthe ingoing past null boundary C corresponds to u = 1 .3. GCM foliation of Σ ∗ . The ( u, ( ext ) s ) -foliation of ( ext ) M restricted to the spacelikehypersurface Σ ∗ has the following properties(a) There exists a constant c Σ ∗ such that Σ ∗ := { u + ( ext ) r = c Σ ∗ } . (b) We have r (cid:29) u ∗ on Σ ∗ . (3.1.2) (c) ( ext ) s satisfies ext ) s = ( ext ) r on Σ ∗ . (d) We say that Σ ∗ is a general covariant modulated hypersurface (or shortly GCMhypersurface) if relative to the above defined null frame of ( ext ) M , the following See (3.3.4) for the precise condition. Recall that ( ext ) s satisfies on ( ext ) M the transport equation L ( ( ext ) s ) = 1 and thus needs to beinitialized on a hypersurface transversal to L , chosen here to be Σ ∗ . More generally, a GCM hypersurface is one with the property that we can specify, using the fullcovariance of the Einstein equations, a number of vanishing conditions (equal to the number of degreesof freedom of the diffeomorphism group) for well-chosen components of ˇΓ. .1. GENERAL COVARIANT MODULATED ADMISSIBLE SPACETIMES conditions hold along Σ ∗ κ = 2 r , d (cid:63) / d (cid:63) / κ = 0 , d (cid:63) / d (cid:63) / µ = 0 , (cid:90) S ηe Φ = 0 , (cid:90) S ξe Φ = 0 , a (cid:12)(cid:12) SP = 1 , (3.1.3) where a is the unique scalar function such that ν = e + ae is tangent to Σ ∗ ,and SP denotes the south poles of the spheres on Σ ∗ . Moreover we also assume (cid:90) S ∗ βe Φ = 0 , (cid:90) S ∗ e θ ( κ ) e Φ = 0 , with S ∗ := Σ ∗ ∩ C ∗ . (3.1.4) Note that the role of the GCM foliation of Σ ∗ is to initialize the ( u, ( ext ) s ) -foliation of ( ext ) M .(e) In view of the definition of ν and ς , we have ν ( u ) = e ( u ) + ae ( u ) = 2 /ς . ν being tangent to Σ ∗ , u is thus transported along Σ ∗ , and hence defined up to aconstant. To calibrate u on Σ ∗ , we fix the value u = 1 as follows S = Σ ∗ ∩ { u = 1 } is such that S ∩ C (1 , L ) ∩ SP (cid:54) = ∅ , (3.1.5) i.e. S is the unique sphere of Σ ∗ such that its south pole intersects the southpole of one of the sphere of the outgoing null cone C (1 , L ) of the initial datalayer.4. Initialization the ( u, ( int ) s ) -foliation on T . The ( u, ( int ) s ) foliation is initializedon T such that, u = u, ( int ) s = ( ext ) s In particular, the 2-spheres S ( u, ( int ) s ) coincide on T with S ( u, ( ext ) s ) and ( int ) r = ( ext ) r . Moreover, the null frame ( ( int ) e , ( int ) e , ( int ) e θ ) is defined on T by the fol-lowing renormalization, ( int ) e = λ ( ext ) e , ( int ) e = λ − ext ) e , ( int ) e θ = ( ext ) e θ on T where λ = ( ext ) λ = 1 − ( ext ) m ( ext ) r . Remark 3.1.3.
In Schwarzschild, u = t − r ∗ , u = t + r ∗ , with dr ∗ dr = Υ − , and ( ext ) e = Υ − ∂ t + ∂ r , ( ext ) e = ∂ t − Υ ∂ r , ( int ) e = ∂ t + Υ ∂ r , ( int ) e = Υ − ∂ t − ∂ r . The existence of such hypersurfaces is an essential part of our construction. CHAPTER 3. MAIN THEOREM H + C ⇤ C ⇤ AC C ⌃ ⇤ I +( ext ) M ( int ) MT H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ AC C ⌃ ⇤ I +( ext ) M ( int ) MT H + C ⇤ C ⇤ AC C ⌃ ⇤ I +( ext ) M ( int ) MT H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T Figure 3.2: The GCM admissible space-time M For convenience, we introduce in this section a notation for renormalized curvature com-ponents and Ricci coefficients.
Definition 3.1.4 (Renormalized curvature components and Ricci coefficients in ( ext ) M ) . We introduce the following notations in ( ext ) M ( ext ) ˇ R = (cid:110) α, β, ˇ ρ, ˇ µ, β, α (cid:111) , ( ext ) ˇΓ = (cid:110) ˇ κ, ϑ, ζ, η, ˇ κ, ϑ, ˇ ω, ξ (cid:111) , where, recall, ˇ ρ = ρ − ρ, ˇ µ = µ − µ, ˇ κ = κ − κ, ˇ κ = κ − κ, ˇ ω = ω − ω, and ξ = ω = 0 , η = − ζ. Note that all the above quantities are defined with respect to the outgoing geodesic foli-ation of ( ext ) M (see section 2.2.4), and that the averages are taken with respect to thatcorresponding 2-spheres. .2. MAIN NORMS Definition 3.1.5 (Renormalized curvature components and Ricci coefficients in ( int ) M ) . We introduce the following notations in ( int ) M ( int ) ˇ R = (cid:110) α, β, ˇ ρ, ˇ µ, β, α (cid:111) , ( int ) ˇΓ = (cid:110) ξ, ˇ ω, ˇ κ, ϑ, ζ, η, ˇ κ, ϑ (cid:111) , where we have defined ˇ ρ = ρ − ρ, ˇ µ = µ − µ, ˇ κ = κ − κ, ˇ κ = κ − κ, ˇ ω = ω − ω, and we recall that ξ = ω = 0 , η = ζ, µ − mr = 0 . Note that all the above quantities are defined with respect to the ingoing geodesic folia-tion of ( int ) M (see section 2.2.6), and that the averages are taken with respect to thatcorresponding 2-spheres. Remark 3.1.6.
In Schwarzschild, we have ( ext ) ˇ R = 0 , ( int ) ˇ R = 0 , ( ext ) ˇΓ = 0 , ( int ) ˇΓ = 0 . ( ext ) M All quantities appearing in this section are defined relative to the ( ext ) M frame adaptedto the ( u, ( ext ) s ) foliation. In particular, recall that with respect to this frame, we have ξ = ω = 0 , η = − ζ. Recall the definition (2.1.37) of higher order angular derivatives d / s of reduced scalars in s k . We introduce the notations d = { e , re , d / } . Definition 3.2.1.
We introduce the vectorfield T defined on ( ext ) M as T := 12 (cid:18)(cid:18) − mr (cid:19) e + e (cid:19) . (3.2.1) We also introduce the vectorfield N is defined on ( ext ) M by N := 12 (cid:18)(cid:18) − mr (cid:19) e − e (cid:19) . (3.2.2)16 CHAPTER 3. MAIN THEOREM
Remark 3.2.2.
In Schwarzschild, we have T = ∂ t , N = (cid:18) − m r (cid:19) ∂ r in the standard ( t, r, θ, ϕ ) coordinates. We are ready to introduce our norms in ( ext ) M . L curvature norms in ( ext ) M Let δ B > (cid:16) ( ext ) R ≥ m [ ˇ R ] (cid:17) := sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ m ) (cid:16) r δ B α + r β (cid:17) + (cid:90) Σ ∗ (cid:16) r δ B ( α + β ) + r ( ˇ ρ ) + r β + α (cid:17) + (cid:90) ( ext ) M ( r ≥ m ) (cid:16) r δ B ( α + β ) + r − δ B ( ˇ ρ ) + r − δ B β + r − − δ B α (cid:17) , (cid:16) ( ext ) R ≤ m [ ˇ R ] (cid:17) := (cid:90) ( ext ) M ( r ≤ m ) (cid:18) − mr (cid:19) | ˇ R | , and ( ext ) R [ ˇ R ] := ( ext ) R ≥ m [ ˇ R ] + ( ext ) R ≤ m [ ˇ R ] . For any nonzero integer k , we introduce the following higher derivatives norms (cid:16) ( ext ) R k [ ˇ R ] (cid:17) := (cid:16) ( ext ) R [ d ≤ k ˇ R ] (cid:17) + (cid:90) ( ext ) M ( r ≤ m ) (cid:16) | d ≤ k − N ˇ R | + | d ≤ k − ˇ R | (cid:17) . Remark 3.2.3.
Note that the derivative in the N direction, unlike all other first deriva-tives of ˇ R , appear in the spacetime integral (cid:82) ( ext ) M ( r ≤ m ) with top number of derivatives.This reflects the fact the N - derivatives do not degenerate at r = 3 m in the Morawetzestimate. .2. MAIN NORMS L Ricci coefficients norms in ( ext ) M For any k ≥
2, we introduce the following norms (cid:16) ( ext ) G ≥ m k (cid:2) ˇΓ (cid:3) (cid:17) := (cid:90) Σ ∗ (cid:34) r (cid:16) ( d ≤ k ϑ ) + ( d ≤ k ˇ κ ) + ( d ≤ k ζ ) + ( d ≤ k ˇ κ ) (cid:17) + ( d ≤ k ϑ ) + ( d ≤ k η ) + ( d ≤ k ˇ ω ) + ( d ≤ k ξ ) (cid:35) + sup λ ≥ m (cid:32) (cid:90) { r = λ } (cid:34) λ (cid:16) ( d ≤ k ϑ ) + ( d ≤ k ˇ κ ) + ( d ≤ k ζ ) (cid:17) + λ − δ B ( d ≤ k ˇ κ ) + ( d ≤ k ϑ ) + ( d ≤ k η ) + ( d ≤ k ˇ ω ) + λ − δ B ( d ≤ k ξ ) (cid:35)(cid:33) , (cid:16) ( ext ) G ≤ m k (cid:2) ˇΓ (cid:3) (cid:17) := (cid:90) ( ext ) M ( ≤ m ) (cid:12)(cid:12) d ≤ k (cid:0) ˇΓ (cid:1)(cid:12)(cid:12) , and ( ext ) G k (cid:2) ˇΓ (cid:3) := ( ext ) G ≤ m k (cid:2) ˇΓ (cid:3) + ( ext ) G ≥ m k (cid:2) ˇΓ (cid:3) . Decay norms in ( ext ) M Let δ dec > ( ext ) D [ α ] := sup ( ext ) M (cid:16) r (2 r + u ) δ dec + r (2 r + u ) + δ dec (cid:17) | α | , ( ext ) D [ β ] := sup ( ext ) M (cid:16) r (2 r + u ) δ dec + r (2 r + u ) + δ dec (cid:17) | β | , ( ext ) D [ ˇ ρ ] := sup ( ext ) M (cid:16) r u δ dec + r u + δ dec (cid:17) | ˇ ρ | , ( ext ) D [ˇ µ ] := sup ( ext ) M r u δ dec | ˇ µ | , ( ext ) D [ β ] := sup ( ext ) M r u δ dec | β | , ( ext ) D [ α ] := sup ( ext ) M ru δ dec | α | , CHAPTER 3. MAIN THEOREM and ( ext ) D [ ˇ R ] := ( ext ) D [ α ] + ( ext ) D [ β ] + ( ext ) D [ ˇ ρ ] + ( ext ) D [ˇ µ ] + ( ext ) D [ β ] + ( ext ) D [ α ] . Also, we introduce the following higher derivatives norms ( ext ) D [ ˇ R ] := ( ext ) D [ ˇ R ] + ( ext ) D [ d ˇ R ]+ sup ( ext ) M (cid:16) r (2 r + u ) δ dec + r (2 r + u ) + δ dec (cid:17) | e ( α ) | + sup ( ext ) M (cid:16) r u δ dec + r u + δ dec (cid:17) | e ( β ) | + sup ( ext ) M r u δ dec | e ( ˇ ρ ) | , and for any integer k ≥ ( ext ) D k [ ˇ R ] := ( ext ) D [ d ≤ k − ˇ R ] . Also, we define ( ext ) D [ˇ κ ] := sup ( ext ) M r u δ dec | ˇ κ | , ( ext ) D [ ϑ ] := sup ( ext ) M (cid:16) ru δ dec + r u + δ dec (cid:17) | ϑ | , ( ext ) D [ ζ ] := sup ( ext ) M (cid:16) ru δ dec + r u + δ dec (cid:17) | ζ | , ( ext ) D [ˇ κ ] := sup ( ext ) M (cid:16) ru δ dec + r u + δ dec (cid:17) | ˇ κ | , ( ext ) D [ ϑ ] := sup ( ext ) M ru δ dec | ϑ | , ( ext ) D [ η ] := sup ( ext ) M ru δ dec | η | + (cid:18)(cid:90) Σ ∗ u δ dec η (cid:19) , ( ext ) D [ˇ ω ] := sup ( ext ) M ru δ dec | ˇ ω | , ( ext ) D [ ξ ] := sup ( ext ) M ru δ dec | ξ | , and ( ext ) D [ˇΓ] := ( ext ) D [ˇ κ ] + ( ext ) D [ ϑ ] + ( ext ) D [ ζ ] + ( ext ) D [ˇ κ ] + ( ext ) D [ ϑ ] + ( ext ) D [ η ]+ ( ext ) D [ˇ ω ] + ( ext ) D [ ξ ] . Also, we introduce the following higher derivatives norms ( ext ) D [ˇΓ] := ( ext ) D [ d ˇΓ] + sup ( ext ) M r u δ dec | e ( ϑ, ζ, ˇ κ ) | .2. MAIN NORMS k ≥ ( ext ) D k [ˇΓ] := ( ext ) D [ d ≤ k − ˇΓ] . Remark 3.2.4.
The integral bootstrap assumption on Σ ∗ for η will only be needed in theproof of Proposition 3.4.6 and recovered in Proposition 7.3.5. In fact, other componentssatisfy an analog integral estimate on Σ ∗ : this is the case of ϑ , ξ and rβ , see Proposition7.3.5. But η is the only component for which we need to make this type of bootstrapassumption. ( int ) M All quantities appearing in this section are defined relative to the ( int ) M frame adaptedto the ( u, ( int ) s ) foliation. L based norms in ( int ) M We introduce the curvature norms, (cid:16) ( int ) R [ ˇ R ] (cid:17) := (cid:90) ( int ) M | ˇ R | . For any nonzero integer k , we introduce the following higher derivatives norms ( int ) R k [ ˇ R ] := ( int ) R [ d ≤ k ˇ R ] . For any k ≥
0, we introduce the following norms (cid:16) ( int ) G k [ˇΓ] (cid:17) := (cid:90) ( int ) M | d ≤ k ˇΓ | . Decay norms in ( int ) M We define ( int ) D [ ˇ R ] := sup ( int ) M u δ dec | ˇ R | , ( int ) D [ˇΓ] := sup ( int ) M u δ dec | ˇΓ | . Also, we introduce the following higher derivatives norms for any integer k ≥ ( int ) D k [ ˇ R ] := ( int ) D [ d ≤ k ˇ R ] , ( int ) D k [ˇΓ] := ( int ) D [ d ≤ k ˇΓ] . CHAPTER 3. MAIN THEOREM
We define the following norms M by combining our above norms on ( ext ) M and ( int ) M N ( En ) k := ( ext ) R k [ ˇ R ] + ( ext ) G k [ˇΓ] + ( int ) R k [ ˇ R ] + ( int ) G k [ˇΓ] , N ( Dec ) k := ( ext ) D k [ ˇ R ] + ( int ) D k [ˇΓ] + ( ext ) D k [ ˇ R ] + ( int ) D k [ˇΓ] . Recall the notations of section 3.1.1 concerning the initial data layer L . Recall that theconstant m > , I k := ( ext ) I k + ( int ) I k + I (cid:48) k where ( ext ) I := sup ( ext ) L (cid:20) r + δ B ( | α | + | β | ) + r (cid:12)(cid:12)(cid:12)(cid:12) ρ + 2 m r (cid:12)(cid:12)(cid:12)(cid:12) + r | β | + r | α | (cid:21) + sup ( ext ) L r (cid:32) | ϑ | + (cid:12)(cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12)(cid:12) + | ζ | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ + 2 (cid:0) − m r (cid:1) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) + sup ( ext ) L r (cid:16) | ϑ | + (cid:12)(cid:12)(cid:12) ω − m r (cid:12)(cid:12)(cid:12) + | ξ | (cid:17) + sup ( ext ) L ( ( ext ) r ≥ m ) (cid:18) r (cid:12)(cid:12)(cid:12) γr − (cid:12)(cid:12)(cid:12) + r | b | + | Ω + Υ | + | ς − | + r (cid:12)(cid:12)(cid:12)(cid:12) e Φ r sin θ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , ( int ) I := sup ( int ) L (cid:18) | α | + | β | + (cid:12)(cid:12)(cid:12)(cid:12) ρ + 2 m r (cid:12)(cid:12)(cid:12)(cid:12) + | β | + | α | (cid:19) + sup ( int ) L (cid:32) | ϑ | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ − (cid:0) − m r (cid:1) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | ζ | + (cid:12)(cid:12)(cid:12)(cid:12) κ + 2 r (cid:12)(cid:12)(cid:12)(cid:12) + | ϑ | + (cid:12)(cid:12)(cid:12) ω + m r (cid:12)(cid:12)(cid:12) + | ξ | (cid:33) , I (cid:48) := sup ( int ) L ∩ ( ext ) L (cid:0) | f | + | f | + | log( λ − λ ) | (cid:1) , λ = ( ext ) λ = 1 − m ext ) r L , Recall that the initial data layer foliations satisfy η + ζ = 0, as well as ξ = ω = 0 on ( ext ) L and η = ζ as well as ξ = ω = 0 on ( int ) L . .3. MAIN THEOREM I k the corresponding higher derivative norms obtained by replacing each componentby d ≤ k of it. In the definition of I (cid:48) above, ( f, f , λ ) denote the transition functions ofLemma 2.3.1 from the frame of the outgoing part ( ext ) L of the initial data layer to theframe of the ingoing part ( int ) L of the initial data layer in the region ( int ) L ∩ ( ext ) L . Remark 3.2.5.
Note that In the definition of ( ext ) I k we allow a higher power of r infront α , β and their derivatives than what it is consistent with the results of [17] and [37].The additional r δ B power, for δ B small, is consistent instead with the result of [38]. Before stating our main theorem, we first introduce the following constants that will beinvolved in its statement. • The constant m > • The integer k large which corresponds to the maximum number of derivatives of thesolution. • The size of the initial data layer norm is measured by (cid:15) > • The size of the bootstrap assumption norms are measured by (cid:15) > • δ H > | r − m | ≤ m δ H where the redshiftestimate holds and which includes in particular the region ( int ) M . • δ dec is tied to decay estimates in u , u for ˇΓ and ˇ R . • δ B is involved in the r -power of the r p weighted estimates for curvature.In what follows m is a fixed constant, δ H , δ B , and δ dec are fixed, sufficiently small,universal constants, and k large is a fixed, sufficiently large, universal constant, chosen suchthat 0 < δ H , δ dec , δ B (cid:28) min { m , } , δ B > δ dec , k large (cid:29) δ dec . (3.3.1)22 CHAPTER 3. MAIN THEOREM
Then, (cid:15) and (cid:15) are chosen such that (cid:15) , (cid:15) (cid:28) min (cid:26) δ H , δ dec , δ B , k large , m , (cid:27) (3.3.2)and (cid:15) = (cid:15) . (3.3.3)Using the definition of (cid:15) , we may now precise the behavior (3.1.2) of r on Σ ∗ inf Σ ∗ r ≥ (cid:15) − u ∗ . (3.3.4)From now on, in the rest of the paper, (cid:46) means bounded by a constant depending onlyon geometric universal constants (such as Sobolev embeddings, elliptic estimates,...) aswell as the constants m , δ H , δ dec , δ B , k large but not on (cid:15) and (cid:15) . We are now ready to give the following precise version of our main theorem.
Main Theorem (Main theorem, version 2) . There exists a sufficiently large integer k large and a sufficiently small constant (cid:15) > such that given an initial layer defined as in section3.1.1 and satisfying the bound I k large +5 ≤ (cid:15) , (3.3.5) there exists a globally hyperbolic development with a complete future null infinity I + anda complete future horizon H + together with foliations and adapted null frames verifyingthe admissibility conditions of section 3.1.2 such that following bound is satisfied N ( En ) k large + N ( Dec ) k small ≤ C(cid:15) (3.3.6) where C is a large enough universal constant and where k small is given by k small = (cid:22) k large (cid:23) + 1 . (3.3.7) In particular, .3. MAIN THEOREM • On ( ext ) M , we have | α | , | β | (cid:46) min (cid:40) (cid:15) r ( u + 2 r ) + δ dec , (cid:15) r ( u + 2 r ) δ dec (cid:41) , | ˇ ρ | (cid:46) min (cid:26) (cid:15) r u + δ dec , (cid:15) r u δ dec (cid:27) , | β | (cid:46) (cid:15) r u δ dec , | α | (cid:46) (cid:15) ru δ dec , and | ˇ κ | (cid:46) (cid:15) r u δ dec , | ϑ | , | ζ | , | ˇ κ | (cid:46) min (cid:26) (cid:15) r u + δ dec (cid:15) ru δ dec (cid:27) , | η | , | ϑ | , | ˇ ω | , | ξ | (cid:46) (cid:15) ru δ dec . • On ( int ) M we have, with ˇΓ = { ˇ κ, ϑ, ζ, η, ˇ κ, ϑ, ˇ ω, ξ } , ˇ R = { α, β, ˇ ρ, β, α } , | ˇΓ , ˇ R | (cid:46) (cid:15) u δ dec . • The Bondi mass converges as u → + ∞ along I + to the final Bondi mass which wedenote by m ∞ . The final Bondi mass verifies the estimate (cid:12)(cid:12)(cid:12)(cid:12) m ∞ m − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . In particular m ∞ > . • The Hawking mass m satisfies | m − m ∞ | m (cid:46) (cid:15) u δ dec on ( ext ) M ,(cid:15) u δ dec on ( int ) M . • The location of the future Horizon H + satisfies r = 2 m ∞ + O (cid:32) √ (cid:15) u δdec (cid:33) on H + . CHAPTER 3. MAIN THEOREM • On ( ext ) M , we have (cid:12)(cid:12)(cid:12)(cid:12) ρ + 2 m ∞ r (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) min (cid:26) (cid:15) r u + δ dec , (cid:15) r u δ dec (cid:27) , (cid:12)(cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r u δ dec , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ + 2 (cid:0) − m ∞ r (cid:1) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) min (cid:26) (cid:15) r u + δ dec (cid:15) ru δ dec (cid:27) , (cid:12)(cid:12)(cid:12) ω − m ∞ r (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ru δ dec . • On ( int ) M , we have. (cid:12)(cid:12)(cid:12)(cid:12) ρ + 2 m ∞ r (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) κ + 2 r (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ − (cid:0) − m ∞ r (cid:1) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ω + m ∞ r (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u δ dec . • On ( ext ) M , the space-time metric g is given in the ( u, r, θ, ϕ ) coordinates system by g = g m ∞ , ( ext ) M + O (cid:16) (cid:15) u δ dec (cid:17) (cid:16) ( dr, du, rdθ ) , r (sin θ ) ( dϕ ) (cid:17) where g m ∞ , ( ext ) M denotes the Schwarzschild metric of mass m ∞ > in outgoingEddington-Finkelstein coordinates, i.e. g m ∞ , ( ext ) M := − dudr − (cid:18) − m ∞ r (cid:19) ( du ) + r (cid:16) ( dθ ) + (sin θ ) ( dϕ ) (cid:17) . • On ( int ) M , the space-time metric g is given in the ( u, r, θ, ϕ ) coordinates system by g = g m ∞ , ( int ) M + O (cid:18) (cid:15) u δ dec (cid:19) (cid:16) ( dr, du, rdθ ) , r (sin θ ) ( dϕ ) (cid:17) where g m ∞ , ( ext ) M denotes the Schwarzschild metric of mass m ∞ > in ingoingEddington-Finkelstein coordinates, i.e. g m ∞ , ( int ) M := 2 dudr − (cid:18) − m ∞ r (cid:19) ( du ) + r (cid:16) ( dθ ) + (sin θ ) ( dϕ ) (cid:17) . Note that analog statements of the above estimates also hold for d k derivatives with k ≤ k small . .4. BOOTSTRAP ASSUMPTIONS AND FIRST CONSEQUENCES Remark 3.3.1.
In this paper, we choose to specify the closeness to Schwarzschild of ourinitial data in the context of the Characteristic Cauchy problem. Note that the conclusionsof our main theorem can be immediately extended to the case where the data are specifiedto be close to Schwarzschild on a spacelike hypersurface Σ . Indeed, one can reduce thislatter case to our situation by invoking • The results in [37] [38] which allow us to control the causal region between Σ andthe outgoing part of the initial data layer . • A standard local existence result which controls the finite causal region between Σ and the ingoing part of the initial data layer. Remark 3.3.2.
In the context of the previous remark, we note that the constant m > appearing in the initial data layer norm of the assumption (3.3.5) of our main theoremdoes not necessarily coincide with the ADM mass of the corresponding initial data set onthe spacelike hypersurface Σ . With respect to this ADM mass, we would recover the wellknown inequality stating that the final Bondi mass is smaller than the ADM mass. We assume that the combined norms N ( En ) k and N ( Dec ) k defined in section 3.2 verifies thefollowing bounds BA-E (Bootstrap Assumptions on energies and weighted energies) N ( En ) k large ≤ (cid:15), (3.4.1) BA-D (Bootstrap Assumptions on decay) N ( Dec ) k small ≤ (cid:15). (3.4.2)In the remaining of section 3.4.1, we state several simple consequences of the bootstrapassumptions which will be proved in Chapter 4. Note that the results of [38] are consistent with our initial data layer assumptions. CHAPTER 3. MAIN THEOREM
While the smallness constant involved in the bootstrap assumptions is (cid:15) >
0, we need thesmallness constant involved in the control of the initial data to be (cid:15) >
0. This is achievedin the theorem below.
Theorem M0.
Assume that the initial data layer L , as defined in section 3.1.1, satisfies I k large +5 ≤ (cid:15) . Then under the bootstrap assumptions
BA-D on decay, the following holds true on theinitial data hypersurface C ∪ C , max ≤ k ≤ k large (cid:40) sup C (cid:104) r + δ B (cid:0) | d k ( ext ) α | + | d k ( ext ) β | (cid:1) + r + δ B | d k − e ( ( ext ) α ) | (cid:105) + sup C (cid:20) r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) ( ext ) ρ + 2 m r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r | d k ( ext ) β | + r | d k ( ext ) α | (cid:21) (cid:41) (cid:46) (cid:15) , max ≤ k ≤ k large sup C (cid:34) | d k ( int ) α | + | d k ( int ) β | + (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) ( int ) ρ + 2 m r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + | d k ( int ) β | + | d k ( int ) α | (cid:35) (cid:46) (cid:15) , and sup C ∪C (cid:12)(cid:12)(cid:12)(cid:12) mm − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . The following two lemma are simple consequence of the bootstrap assumptions and willbe proved in section 4.2.
Lemma 3.4.1 (Control of averages) . Assume given a GCM admissible spacetime M as defined in section 3.1.2 verifying the bootstrap assumption for some sufficiently small (cid:15) > . Then, we have sup ( ext ) M u δ dec (cid:18) r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) , sup ( ext ) M u δ dec (cid:18) r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) κ + 2Υ r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12)(cid:12) d ≤ k small (cid:16) ω − mr (cid:17)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) , .4. BOOTSTRAP ASSUMPTIONS AND FIRST CONSEQUENCES ( ext ) M u + δ dec (cid:18) r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) , sup ( ext ) M u + δ dec (cid:18) r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) κ + 2Υ r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12)(cid:12) d ≤ k large (cid:16) ω − mr (cid:17)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) , sup ( int ) M u δ dec (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) , sup ( int ) M u δ dec (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) κ + 2 r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) d ≤ k small (cid:16) ω + mr (cid:17)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) , sup ( int ) M u + δ dec (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) , sup ( int ) M u + δ dec (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) κ + 2 r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) d ≤ k large (cid:16) ω + mr (cid:17)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) . Also, we have sup ( ext ) M (cid:16) u δ dec r (cid:12)(cid:12) d ≤ k small (cid:0) Ω + Υ (cid:1)(cid:12)(cid:12) + u + δ dec r (cid:12)(cid:12) d ≤ k large (cid:0) Ω + Υ (cid:1)(cid:12)(cid:12) (cid:17) (cid:46) (cid:15) , sup ( int ) M (cid:16) u δ dec (cid:12)(cid:12) d ≤ k small (cid:0) Ω − Υ (cid:1)(cid:12)(cid:12) + u + δ dec (cid:12)(cid:12) d ≤ k large (cid:0) Ω − Υ (cid:1)(cid:12)(cid:12) (cid:17) (cid:46) (cid:15) . Finally, recall that µ and µ are given by the following formula µ = 2 mr on ( ext ) M , µ = 2 mr on ( int ) M . Lemma 3.4.2 (Control of the Hawking mass) . Assume given a GCM admissible spacetime M as defined in section 3.1.2 verifying the bootstrap assumption for some sufficiently small (cid:15) > . Then, we have max ≤ k ≤ k large sup ( ext ) M u δ dec (cid:16) | d k e ( m ) | + r | d k e ( m ) | (cid:17) (cid:46) (cid:15) , max ≤ k ≤ k large sup ( int ) M u δ dec (cid:16) | d k e ( m ) | + | d k e ( m ) | (cid:17) (cid:46) (cid:15) . The e derivatives behave better in powers of r , max ≤ k ≤ k small sup ( ext ) M r u δ dec | d k e ( m ) | (cid:46) (cid:15) , max ≤ k ≤ k large sup ( ext ) M r u + δ dec | d k e ( m ) | (cid:46) (cid:15) . Moreover, sup M (cid:12)(cid:12)(cid:12)(cid:12) mm − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . CHAPTER 3. MAIN THEOREM
The following two propositions on the existence of a suitable coordinates system bothin ( ext ) M and in ( int ) M are also consequences of the bootstrap assumptions and will beproved in section 4.3. Proposition 3.4.3 (Control of a coordinates system on ( ext ) M ) . Let θ ∈ [0 , π ] be the Z -invariant scalar on M defined by (2.2.52) , i.e. θ = cot − ( re θ (Φ)) . (3.4.3) Consider the ( u, r, θ, ϕ ) coordinates system introduced in Proposition 2.2.23. Then, rela-tive to these ( u, r, θ, ϕ ) coordinates,1. The spacetime metric takes the form, g = − ςrκ dudr + ς ( κ + A ) κ du + γ (cid:18) dθ − ςbdu − b (cid:19) (3.4.4) where, b = e ( θ ) , b = e ( θ ) , γ = 1( e θ ( θ )) (3.4.5) and, Θ = 4 rκ dr − ς (cid:18) κ + Aκ (cid:19) du.
2. The reduced coordinates derivatives take the form, ∂ r = 2 rκ e − √ γrκ be θ ,∂ θ = √ γe θ ,∂ u = ς (cid:20) e − κ + Aκ e − √ γ (cid:18) b − (cid:18) κ + Aκ (cid:19) b (cid:19) e θ (cid:21) . (3.4.6)
3. The following estimates hold true: max ≤ k ≤ k small sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) (cid:16)(cid:12)(cid:12)(cid:12) d k (cid:16) γr − (cid:17)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12) d k b (cid:12)(cid:12)(cid:17) (cid:46) (cid:15), max ≤ k ≤ k small sup ( ext ) M u δ dec (cid:0)(cid:12)(cid:12) d k ˇΩ (cid:12)(cid:12) + (cid:12)(cid:12) d k ( ς − (cid:12)(cid:12) + r (cid:12)(cid:12) d k b (cid:12)(cid:12)(cid:1) (cid:46) (cid:15). Also, e Φ satisfies max ≤ k ≤ k small sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) e Φ r sin θ − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15). .4. BOOTSTRAP ASSUMPTIONS AND FIRST CONSEQUENCES Proposition 3.4.4 (Control of a coordinates system on ( int ) M ) . Let θ ∈ [0 , π ] be the Z -invariant scalar on M defined by (3.4.3) . Consider the ( u, r, θ, ϕ ) coordinates systemintroduced in Proposition 2.2.26. Then, relative to these ( u, r, θ, ϕ ) coordinates,1. The spacetime metric takes the form, g = − ςrκ dudr + ς ( κ + A ) κ du + γ (cid:18) dθ − ςbdu − b (cid:19) (3.4.7) where, b = e ( θ ) , b = e ( θ ) , γ = 1( e θ ( θ )) (3.4.8) and, Θ := 4 rκ dr − ς (cid:18) κ + Aκ (cid:19) du.
2. The reduced coordinates derivatives take the form, ∂ r = 2 rκ e − √ γrκ be θ ,∂ θ = √ γe θ ,∂ u = ς (cid:20) e − κ + Aκ e − √ γ (cid:18) b − (cid:18) κ + Aκ (cid:19) b (cid:19) e θ (cid:21) . (3.4.9)
3. The following estimates hold true: max ≤ k ≤ k small sup ( int ) M u δ dec (cid:16)(cid:12)(cid:12) d k ˇΩ (cid:12)(cid:12) + (cid:12)(cid:12) d k ( ς − (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) d k (cid:16) γr − (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) d k b (cid:12)(cid:12) + (cid:12)(cid:12) d k b (cid:12)(cid:12)(cid:17) (cid:46) (cid:15). Also, e Φ satisfies max ≤ k ≤ k small sup ( int ) M u δ dec (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) e Φ r sin θ − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15). We will need later to interpolate between the estimates provided by the bootstrap as-sumptions on decay and the bootstrap assumptions on energy. To this end, we will needthe following consequence of the bootstrap assumptions on weighted energies.30
CHAPTER 3. MAIN THEOREM
Proposition 3.4.5.
The Ricci coefficients and curvature components satisfy the followingpointwise estimates on M max k ≤ k large − sup M (cid:110) r + δB (cid:0) | d k α | + | d k β | (cid:1) + r (cid:0) | d k µ | + | d k ˇ ρ | (cid:1) + r (cid:0) | d k ˇ κ | + | d k ζ | + | d k ϑ | + | d k ˇ κ | + | d k β | (cid:1) + r (cid:0) | d k ϑ | + | d k ϑ | + | d k ˇ ω | + | d ξ | + | d k α | (cid:1)(cid:111) (cid:46) (cid:15). ( ext ) M Recall that the quantity q satisfies the following wave equation, see (2.4.7), (cid:3) q + κκ q = Err[ (cid:3) q ]where the nonlinear term Err[ (cid:3) q ] has the schematic structure exhibited in (2.4.8). Also,recall that according to our bootstrap assumption on decay and Proposition 3.4.5, η satisfies on ( ext ) M | d ≤ k small η | ≤ (cid:15)ru δ dec , | d ≤ k large − η | (cid:46) (cid:15)r . As discuss in Remark 2.4.8, this decay in r − is too weak to derive suitable decay for q . We thus need to provide another frame for ( ext ) M . This is the aim of the followingproposition. Proposition 3.4.6.
Let an integer k loss and a small constant δ > satisfying ≤ k loss ≤ δ dec k large − k small ) , δ := k loss k large − k small . (3.4.10) Let ( e , e , e θ ) the outgoing geodesic null frame of ( ext ) M . There exists another frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) of ( ext ) M provided by e (cid:48) = e + f e θ + 14 f e ,e (cid:48) θ = e θ + 12 f e ,e (cid:48) = e , Recall from (3.3.1) and (3.3.7) that we have0 < δ dec (cid:28) , δ dec k large (cid:29) , k small = (cid:22) k large (cid:23) + 1 . In particular, we have δ dec ( k large − k small ) (cid:29) k loss satisfying the requiredconstraints. .4. BOOTSTRAP ASSUMPTIONS AND FIRST CONSEQUENCES such that the Ricci coefficients and curvature components with respect to that frame satisfy ξ (cid:48) = 0 , max ≤ k ≤ k small + k loss sup ( ext ) M (cid:40)(cid:16) r u + δ dec − δ + ru δ dec − δ (cid:17) | d k Γ (cid:48) g | + ru δ dec − δ | d k Γ (cid:48) b | + r u δ dec − δ (cid:12)(cid:12)(cid:12)(cid:12) d k − e (cid:48) (cid:18) κ (cid:48) − r , κ (cid:48) + 2Υ r , ϑ (cid:48) , ζ (cid:48) , η (cid:48) , η (cid:48) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) r + δB + r u + δ dec − δ + r u δ dec − δ (cid:17)(cid:16) | d k α (cid:48) | + | d k β (cid:48) | (cid:17) + (cid:16) r + δB + r u δ dec + r u + δ dec − δ (cid:17) | d k − e (cid:48) ( α (cid:48) ) | + (cid:16) r u δ dec + r u + δ dec − δ (cid:17) | d k − e (cid:48) ( β (cid:48) ) | + (cid:16) r u + δ dec − δ + r ru δ dec − δ (cid:17) | d k ˇ ρ (cid:48) | + u δ dec − δ (cid:16) r | d k β (cid:48) | + r | d k α (cid:48) | (cid:17)(cid:41) (cid:46) (cid:15), where we have used the notation Γ (cid:48) g = (cid:26) rω (cid:48) , κ (cid:48) − r , ϑ (cid:48) , ζ (cid:48) , η (cid:48) , η (cid:48) , κ (cid:48) + 2Υ r , r − ( e (cid:48) ( r ) − , r − e (cid:48) θ ( r ) , e (cid:48) ( m ) (cid:27) , Γ (cid:48) b = (cid:110) ϑ (cid:48) , ω (cid:48) − mr , ξ (cid:48) , r − ( e (cid:48) ( r ) + Υ) , r − e (cid:48) ( m ) (cid:111) . Furthermore, f satisfies | d k f | (cid:46) (cid:15)ru + δ dec − δ + u δ dec − δ , for k ≤ k small + k loss + 2 on ( ext ) M , | d k − e (cid:48) f | (cid:46) (cid:15)ru δ dec − δ for k ≤ k small + k loss + 2 on ( ext ) M . (3.4.11) Remark 3.4.7.
The crucial point of Proposition 3.4.6 is that in the new frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) of ( ext ) M , η (cid:48) belongs to Γ (cid:48) g and thus displays a better decay in r − than η correspondingto the outgoing geodesic frame ( e , e , e θ ) of ( ext ) M . Here, r and m denote respectively the area radius and the Hawking mass of the outgoing geodesicfoliation of ( ext ) M , i.e. r = ( ext ) r and m = ( ext ) m . In particular, while e θ ( r ) = e θ ( m ) = 0, we have ingeneral e (cid:48) θ ( r ) (cid:54) = 0 and e (cid:48) θ ( m ) (cid:54) = 0. CHAPTER 3. MAIN THEOREM
In this section, we construct 2 smooth global frames on M by matching the frame of ( int ) M on the one hand with a renormalization of the frame on ( ext ) M , and on the otherhand, with a renormalization of the second frame of ( ext ) M given by Proposition 3.4.6. To construct the first global frame, we need to extend the frame ( ( int ) e , ( int ) e , ( int ) e θ )of ( int ) M slightly into ( ext ) M , and the frame ( ( ext ) e , ( ext ) e , ( ext ) e θ ) of ( ext ) M slightlyinto ( int ) M . We keep the same labels for the extended frame, i.e. ( ( int ) e , ( int ) e , ( int ) e θ )represents the extended frame of ( int ) M in ( ext ) M and vice versa. This convention alsoapplies to the Ricci coefficients, curvature components, area radius and Hawking mass ofthe extended frames.Note that these extensions require, in addition to the initialization of the frames on T , toinitialize1. ( ( ext ) e , ( ext ) e , ( ext ) e θ ) on C ∗ by( ( ext ) e , ( ext ) e , ( ext ) e θ ) = (( ( int ) Υ) − int ) e , ( int ) Υ ( int ) e , ( int ) e θ ) .
2. ( ( int ) e , ( int ) e , ( int ) e θ ) on C ∗ by( ( int ) e , ( int ) e , ( int ) e θ ) = ( ( ext ) Υ ( ext ) e , ( ( ext ) Υ) − ext ) e , ( ext ) e θ ) . We start with the definition of the region where the frame of ( int ) M and a conformalrenormalization of the frame of ( ext ) M will be matched. Definition 3.5.1.
We define the matching region as the spacetime regionMatch := (cid:18) ( ext ) M ∩ (cid:26) ( int ) r ≤ m (cid:18) δ H (cid:19)(cid:27)(cid:19) ∪ (cid:18) ( int ) M ∩ (cid:26) ( int ) r ≥ m (cid:18) δ H (cid:19)(cid:27)(cid:19) , where, as explained in the previous section, ( int ) r denotes the area radius of the ingoinggeodesic foliation of ( int ) M and its extension to ( ext ) M . .5. GLOBAL NULL FRAMES Proposition 3.5.2.
There exists a global null frame defined on ( int ) M ∪ ( ext ) M anddenoted by ( ( glo ) e , ( glo ) e , ( glo ) e θ ) such that(a) In ( ext ) M \
Match, we have ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( ext ) Υ ( ext ) e , ( ext ) Υ − ext ) e , ( ext ) e θ (cid:1) . (b) In ( int ) M \
Match, we have ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( int ) e , ( int ) e , ( int ) e θ (cid:1) . (c) In the matching region, we have max ≤ k ≤ k small − sup Match ∩ ( int ) M u δ dec (cid:12)(cid:12) d k ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k small − sup Match ∩ ( ext ) M u δ dec (cid:12)(cid:12) d k ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large − (cid:18)(cid:90) Match (cid:12)(cid:12) d k ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15), where ( glo ) ˇ R and ( glo ) ˇΓ are given by ( glo ) ˇ R = (cid:26) α, β, ρ + 2 mr , β, α (cid:27) , ( glo ) ˇΓ = (cid:26) ξ, ω + mr , κ − r , ϑ, ζ, η, η, κ + 2 r , ϑ, ω, ξ (cid:27) . (d) Furthermore, we may also choose the global frame such that, in addition, one of thefollowing two possibilities hold,i. We have on all ( ext ) M ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( ext ) Υ ( ext ) e , ( ext ) Υ − ext ) e , ( ext ) e θ (cid:1) . ii. We have on all ( int ) M ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( int ) e , ( int ) e , ( int ) e θ (cid:1) . Remark 3.5.3.
The global frame on M of Proposition 3.5.2 will be used to constructthe second global frame in the next section, see Proposition 3.5.5. It will also be used torecover high order derivatives in Theorem M8 (stated in section 3.6.2), see section 8.3.2. CHAPTER 3. MAIN THEOREM
We start with the definition of the region where first global frame of M (i.e. the one ofProposition 3.5.2) and a conformal renormalization of the frame second frame of ( ext ) M (i.e. the one of Proposition 3.4.6) will be matched. Definition 3.5.4.
We define the matching region as the spacetime regionMatch (cid:48) := ( ext ) M ∩ (cid:26) m ≤ ( ext ) r ≤ m (cid:27) , where ( ext ) r denotes the area radius of the outgoing geodesic foliation of ( ext ) M . Here is our main proposition concerning our second global frame.
Proposition 3.5.5.
Let an integer k loss and a small constant δ > satisfying (3.4.10) .There exists a global null frame ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) defined on ( int ) M ∪ ( ext ) M suchthat(a) In ( ext ) M ∩ { ( ext ) r ≥ m } , we have ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) = (cid:0) ( ext ) Υ ( ext ) e (cid:48) , ( ext ) Υ − ext ) e (cid:48) , ( ext ) e (cid:48) θ (cid:1) , where ( ( ext ) e (cid:48) , ( ext ) e (cid:48) , ( ext ) e (cid:48) θ ) denotes the second frame of ( ext ) M , i.e. the fame ofProposition 3.4.6.(b) In ( int ) M ∪ ( ( ext ) M ∩ { ( ext ) r ≤ m } ) , we have ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) = ( ( glo ) e , ( glo ) e , ( glo ) e θ ) , where ( ( glo ) e , ( glo ) e , ( glo ) e θ ) denotes the first global frame of M , i.e. the frame ofProposition 3.5.2.(c) In the matching region, we have max ≤ k ≤ k small + k loss sup Match (cid:48) u δ dec − δ (cid:12)(cid:12)(cid:12) d k ( ( glo (cid:48) ) ˇΓ , ( glo (cid:48) ) ˇ R ) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15), where ( glo (cid:48) ) ˇ R and ( glo (cid:48) ) ˇΓ are given by ( glo (cid:48) ) ˇ R = (cid:26) α, β, ρ + 2 mr , β, α (cid:27) , ( glo (cid:48) ) ˇΓ = (cid:26) ξ, ω + mr , κ − r , ϑ, ζ, η, η, κ + 2 r , ϑ, ω, ξ (cid:27) . with the Ricci coefficients and curvature components being the one associated to theframe ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) . .6. PROOF OF THE MAIN THEOREM (d) Furthermore, we may also choose the global frame such that, in addition, one of thefollowing two possibilities hold,i. We have on ( ext ) M ∩ { ( ext ) r ≥ m } ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) = (cid:0) ( ext ) Υ ( ext ) e (cid:48) , ( ext ) Υ − ext ) e (cid:48) , ( ext ) e (cid:48) θ (cid:1) . ii. We have on ( int ) M ∪ ( ( ext ) M ∩ { ( ext ) r ≤ m } )( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) = ( ( glo ) e , ( glo ) e , ( glo ) e θ ) . Remark 3.5.6.
The global frame on M of Proposition 3.5.5 will be needed to derive decayestimates for the quantity q in Theorem M1 (stated in section 3.6.1). We are ready to state our main intermediary results.
Theorem M1.
Assume given a GCM admissible spacetime M as defined in section 3.1.2verifying the bootstrap assumptions BA-E and
BA-D for some sufficiently small (cid:15) > .Then, if (cid:15) > is sufficiently small, there exists δ extra > δ dec such that we have thefollowing estimates in M , max ≤ k ≤ k small +20 sup ( ext ) M (cid:110)(cid:16) ru + δ extra + u δ extra (cid:17) | d k q | + ru δ extra | d k e q | (cid:111) + max ≤ k ≤ k small +20 sup ( int ) M u δ extra | d k q | (cid:46) (cid:15) . Moreover, q also satisfies the following estimate max ≤ k ≤ k small +21 u δ extra (cid:90) ( int ) M ( ≥ u ) | d k q | + max ≤ k ≤ k small +20 u δ extra (cid:90) Σ ∗ ( ≥ u ) | d k e q | (cid:46) (cid:15) . Theorem M2.
Under the same assumptions as above we have the following decay esti-mates for ( ext ) α max ≤ k ≤ k small +20 sup ( ext ) M (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17)(cid:16) | d k ( ext ) α | + r | d k e ext ) α | (cid:17) (cid:46) (cid:15) . Recall in particular that the conclusions of Theorem M0 hold under the bootstrap assumptions
BA-E and
BA-D . CHAPTER 3. MAIN THEOREM
Theorem M3.
Under the same assumptions as above we have the following decay esti-mates for α ( int ) D k small +16 [ α ] (cid:46) (cid:15) , max ≤ k ≤ k small +18 (cid:90) Σ ∗ u δ extra | d k α | (cid:46) (cid:15) . Theorem M4.
Under the same assumptions as above we also have the following decayestimates in ( ext ) M ( ext ) D k small +8 [ ˇ R ] + ( ext ) D k small +8 [ˇΓ] (cid:46) (cid:15) . Theorem M5.
Under the same assumptions as above we also have the following decayestimates for ˇ R and ˇΓ in ( int ) M ( int ) D k small +5 [ ˇ R ] + ( int ) D k small +5 [ˇΓ] (cid:46) (cid:15) . Note that, as an immediate consequence of Theorem M2 to Theorem M5 we have ob-tained, under the same assumptions as above, the following improvement of our bootstrapassumptions on decay N ( Dec ) k small +5 (cid:46) (cid:15) . (3.6.1) Definition 3.6.1 (Definition of ℵ ( u ∗ )) . Let (cid:15) > and (cid:15) > be given small constantssatisfying the constraint (3.3.3) . Let ℵ ( u ∗ ) be the set of all GCM admissible spacetimes M defined in section 3.1.2 such that • u ∗ is the value of u on the last outgoing slice C ∗ , • u ∗ satisfies (3.3.4) , • the bootstrap assumptions (3.4.1) (3.4.2) hold true, i.e., relative to the combinednorms defined in section 3.2.3, we have N ( En ) k large ≤ (cid:15), N ( Dec ) k small ≤ (cid:15). Definition 3.6.2.
Let U be the set of all values of u ∗ ≥ such that the spacetime ℵ ( u ∗ ) exists. The following theorem shows that U is not empty. .6. PROOF OF THE MAIN THEOREM Theorem M6.
There exists δ > small enough such that for sufficiently small constants (cid:15) > and (cid:15) > satisfying the constraints (3.3.3) (3.3.4) , we have [1 , δ ] ⊂ U . In view of Theorem M6, we may define U ∗ as the supremum over all value of u ∗ thatbelongs to U . U ∗ := sup u ∗ ∈U u ∗ . Assume by contradiction that U ∗ < + ∞ . Then, by the continuity of the flow, U ∗ ∈ U . Furthermore, according to the consequence(3.6.1) of Theorem M2 to Theorem M5, the bootstrap assumptions on decay (3.4.2) onany spacetime of ℵ ( U ∗ ) are improved by N ( Dec ) k small +5 (cid:46) (cid:15) . To reach a contradiction, we still need an extension procedure for spacetimes in ℵ ( u ∗ ) tolarger values of u , as well as to improve our bootstrap assumptions on weighted energies(3.4.1). This is done in two steps. Theorem M7.
Any GCM admissible spacetime in ℵ ( u ∗ ) for some < u ∗ < + ∞ suchthat N ( Dec ) k small +5 (cid:46) (cid:15) , has a GCM admissible extension (satisfying (3.3.4) ), i.e. u (cid:48)∗ > u ∗ , initialized by TheoremM0, which verifies N ( Dec ) k small (cid:46) (cid:15) . Remark 3.6.3.
Recall that the definition of a GCM admissible spacetime in section 3.1.2is such that T = { r = r T } for some r T satisfying m (cid:18) δ H (cid:19) ≤ r T ≤ m (cid:18) δ H (cid:19) . (3.6.2) All results obtained so far, in particular Theorems M0–M7, hold for any choice of r T satisfying (3.6.2) , see Remark 8.3.1 for a more precise statement. It is at this stage,in Theorem M8 below, that we need to make a specific choice of r T in the context of aLebesgue point argument required for the control of top order derivatives. This choice willbe made in (8.3.2) . CHAPTER 3. MAIN THEOREM
Theorem M8.
There exists a choice of r T satisfying (3.6.2) such that the GCM admis-sible spacetime exhibited in Theorem M7 satisfies in addition N ( En ) k large (cid:46) (cid:15) and therefore belongs to ℵ ( u (cid:48)∗ ) . In particular u (cid:48)∗ belongs to U . In view of Theorem M8, we have reached a contradiction, and hence U ∗ = + ∞ so that the spacetime may be continued forever. This concludes the proof of the maintheorem. The Penrose diagram of M Complete future null infinity.
We first deduce from our estimate that our spacetime M has a complete future null infinity I + . The portion of null infinity of M corresponds tothe limit r → + ∞ along the leaves C u of the outgoing geodesic foliation of ( ext ) M . As C u exists for all u ≥ u is an affine parameterof I + . To this end, recall from our main theorem that the estimates N ( Dec ) k small (cid:46) (cid:15) holdwhich implies in particular sup ( ext ) M ru δ dec (cid:16) | ξ | + (cid:12)(cid:12)(cid:12) ω − mr (cid:12)(cid:12)(cid:12) + r − | ς − | (cid:17) (cid:46) (cid:15) . (3.6.3)As | m − m | (cid:46) (cid:15) m , see Lemma 3.4.2, m is bounded. We infer thatlim C u ,r → + ∞ ξ, ω = 0 for all 1 ≤ u < ∞ . In view of the identity D e = − ωe + 2 ξe θ , we infer that e is a null geodesic generator of I + . Since we have e ( u ) = ς with | ς − | (cid:46) (cid:15) in view of (3.6.3), u is an affine parameter of I + so that I + is indeed complete. Using also Proposition 3.4.3 for the control of ς . .6. PROOF OF THE MAIN THEOREM Existence of a future event horizon.
Next, note that the estimates N ( Dec ) k small (cid:46) (cid:15) alsoimply sup ( int ) M u δ dec (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) κ + 2 r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ − (cid:0) − mr (cid:1) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) (cid:46) (cid:15) . In particular, considering the spacetime region r ≤ m (1 − δ H /
2) of ( int ) M , and in viewof the estimate | m − m | (cid:46) (cid:15) m , we infer, for all r ≤ m (1 − δ H / κ ≤ r − mr + O ( (cid:15) ) (cid:46) r ( r − m + 2 m − m ) + O ( (cid:15) ) (cid:46) m r ( − δ H + (cid:15) ) + O ( (cid:15) ) . Thus, since 0 < (cid:15) (cid:28) δ H (cid:28)
1, we deduce,sup ( int ) M (cid:16) r ≤ m (cid:16) − δ H (cid:17)(cid:17) κ ≤ − δ H m (cid:0) − δ H (cid:1) + O ( (cid:15) ) ≤ − δ H m . Thus, all 2-spheres S ( u, s ) of the ingoing geodesic foliation of ( int ) M which are locatedin the spacetime region r ≤ m (1 − δ H /
2) of ( int ) M are trapped. This implies that thepast of I + in M does not contain this region, and hence M contains the event horizon H + of a black hole in its interior. Moreover, since the timelike hyper surface T is foliatedby the outgoing null cones C u of ( ext ) M , it is in the past of I + . Hence, since T is one ofthe boundaries of ( int ) M , H + is actually located in the interior of the region ( int ) M . Asymptotic stationarity of M . Recall that we have introduced a vectorfield T in ( ext ) M as well as one in ( int ) M by T = e + Υ e in ( ext ) M , T = e + Υ e in ( int ) M . We can easily express all components of ( T ) π in terms of ˇΓ, e ( m ) , e m . Thus, making usof the estimate N ( Dec ) k small (cid:46) (cid:15) of our main theorem, we deduce, | ( T ) π | (cid:46) (cid:15) ru δ dec in ( ext ) M and | ( T ) π | (cid:46) (cid:15) u δ dec in ( int ) M . In particular, T is an asymptotically Killing vectorfield and hence our spacetime M isasymptotically stationary.The above conclusions regarding I + and H + allow us to draw the Penrose diagram of M ,see Figure 3.3.40 CHAPTER 3. MAIN THEOREM H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ AC C ⌃ ⇤ I +( ext ) M ( int ) MT H + C ⇤ C ⇤ AC C ⌃ ⇤ I +( ext ) M ( int ) MT H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T H + C ⇤ C ⇤ A C C ⌃ ⇤ I + ( e x t ) M ( i n t ) M T Figure 3.3: The Penrose diagram of the space-time M Limits at null infinity and Bondi mass
Recall the following formula for the derivative of the Hawking mass in ( ext ) M , see Propo-sition 2.2.16 e ( m ) = r π (cid:90) S (cid:18) − κϑ −
12 ˇ κϑϑ + 2ˇ κ ˇ ρ + 2 e θ ( κ ) ζ + 2 κζ (cid:19) . As a simple corollary of the decay estimates of our main theorem, i.e., N ( Dec ) k small (cid:46) (cid:15) , wededuce, | e ( m ) | (cid:46) (cid:15) r u δ dec . (3.6.4)Since r − is integrable, we infer the existence of a limit to m as r → + ∞ along C u M B ( u ) = lim r → + ∞ m ( u, r ) for all 1 ≤ u < + ∞ where M B ( u ) is the so-called Bondi mass. .6. PROOF OF THE MAIN THEOREM ( ext ) M , see Proposition 2.2.8 e ( ϑ ) + 12 κϑ = 2 d (cid:63) / ζ − κϑ + 2 ζ . In view of N ( Dec ) k small (cid:46) (cid:15) , we deduce | e ( rϑ ) | (cid:46) (cid:15) r u + δ dec . Since r − is integrable, we infer the existence of a limit to rϑ as r → + ∞ along C u Θ( u, · ) = lim r → + ∞ rϑ ( r, u, · ) for all 1 ≤ u < + ∞ . On the other hand, in view of N ( Dec ) k small (cid:46) (cid:15) again, r | ϑ | (cid:46) (cid:15) u δ dec , on ( ext ) M . We infer that | Θ( u, · ) | (cid:46) (cid:15) u δ dec for all 1 ≤ u < + ∞ . The spheres at null infinity are round
The Gauss curvature is given by the formula, K = − ρ − κκ + 14 ϑϑ. Thus, in view of our estimates in ( ext ) M , (cid:12)(cid:12)(cid:12)(cid:12) K − r (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r u + δ dec so that lim r → + ∞ r K = 1 . In particular the spheres at null infinity are round.42
CHAPTER 3. MAIN THEOREM
A Bondi mass formula
Using the formula for e ( m ) in ( ext ) M , see Proposition 2.2.16, together with the estimates N ( Dec ) k small (cid:46) (cid:15) , we deduce (cid:12)(cid:12)(cid:12)(cid:12) e ( m ) + r π (cid:90) S κϑ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ru +2 δ dec and hence (cid:12)(cid:12)(cid:12)(cid:12) e ( m ) + 18 | S | (cid:90) S ( rϑ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ru +2 δ dec . Letting r → + ∞ along C u , and using that the spheres at null infinity are round, we inferin view of the definition of M B and Θ e ( M B )( u ) = − (cid:90) S Θ ( u, · ) for all 1 ≤ u < + ∞ . Since e ( u ) = ς and e is orthogonal to the spheres foliating I + , we infer e = ς ∂ u . Thus,we obtain the following Bondi mass type formula ∂ u M B ( u ) = − ς (cid:90) S Θ ( u, · ) for all 1 ≤ u < + ∞ , with ς satisfying (3.6.3). Final Bondi mass
In view of the estimate | Θ( u, · ) | (cid:46) (cid:15) u δ dec for all 1 ≤ u < + ∞ , and the control for ς in (3.6.3), we infer that | ∂ u M B ( u ) | (cid:46) (cid:15) u δ dec for all 1 ≤ u < + ∞ . In particular, since u − − δ dec is integrable, the limit along I + exists M B (+ ∞ ) = lim u → + ∞ M B ( u )and is the so-called final Bondi mass. We denote it as m ∞ , i.e. m ∞ = M B (+ ∞ ). .6. PROOF OF THE MAIN THEOREM Control of m − m ∞ . We have as a consequence of the above estimate for ∂ u M B and thedefinition of m ∞ | M B ( u ) − m ∞ | (cid:46) (cid:15) u δ dec for all 1 ≤ u < + ∞ . Also, recall from (3.6.4) that we have obtained in ( ext ) M| e ( m ) | (cid:46) (cid:15) r u δ dec which yields, together with the definition of M B ( u ), by integration in r at fixed u | m ( r, u ) − M B ( u ) | (cid:46) (cid:15) ru δ dec in ( ext ) M . We infer sup ( ext ) M u δ dec | m − m ∞ | (cid:46) (cid:15) . (3.6.5)Also, recall the following formula for the derivative of the Hawking mass in ( int ) M , seeProposition 2.2.16 in the context of an outgoing geodesic foliation, e ( m ) = r π (cid:90) S (cid:18) − κϑ −
12 ˇ κϑϑ + 2ˇ κ ˇ ρ − e θ ( κ ) ζ + 2 κζ (cid:19) . Together with the estimates N ( Dec ) k small (cid:46) (cid:15) , we deduce | e ( m ) | (cid:46) (cid:15) u δ dec on ( int ) M and hence by integration in r at fixed u , for r ∈ [2 m (1 − δ H ) , r T ], (cid:12)(cid:12)(cid:12) m ( r, u ) − m (cid:16) r T , u (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u δ dec m δ H on ( int ) M . According to (3.6.5), since { r = r T } = T = ( ext ) M ∩ ( int ) M ⊂ ( ext ) M , and since u = u in T by the initialization of u , u δ dec (cid:12)(cid:12)(cid:12) m (cid:16) r T , u (cid:17) − m ∞ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . We deduce sup ( int ) M u δ dec | m − m ∞ | (cid:46) (cid:15) . (3.6.6)Combining (3.6.5) and (3.6.6) with the estimatesup M | m − m | (cid:46) (cid:15) m , in the statement of our main theorem (see also Lemma 3.4.2), we infer that | m ∞ − m | (cid:46) (cid:15) m . In particular we deduce that m ∞ > (cid:15) can be made arbitrarily small.44 CHAPTER 3. MAIN THEOREM
Coordinates systems on ( ext ) M and ( int ) M In view of Proposition 3.4.3, and together with the control of the averages κ , κ providedby Lemma 3.4.1, the control of ˇ κ provided by the estimates N ( Dec ) k small (cid:46) (cid:15) , and the controlof m − m ∞ obtained in (3.6.5), we infer for the space-time metric g on ( ext ) M in the( u, r, θ, ϕ ) coordinates system g = g m ∞ , ( ext ) M + O (cid:16) (cid:15) u δ dec (cid:17) (cid:16) ( dr, du, rdθ ) , r (sin θ ) ( dϕ ) (cid:17) where g m ∞ , ( ext ) M denotes the Schwarzschild metric of mass m ∞ > g m ∞ , ( ext ) M = − dudr − (cid:18) − m ∞ r (cid:19) ( du ) + r (cid:16) ( dθ ) + (sin θ ) ( dϕ ) (cid:17) . Also, in view of Proposition 3.4.4, and together with the control of the averages κ , κ provided by Lemma 3.4.1, the control of ˇ κ provided by the estimates N ( Dec ) k small (cid:46) (cid:15) , andthe control of m − m ∞ obtained in (3.6.6), we infer for the space-time metric g on ( int ) M in the ( u, r, θ, ϕ ) coordinates system g = g m ∞ , ( int ) M + O (cid:18) (cid:15) u δ dec (cid:19) (cid:16) ( dr, du, rdθ ) , r (sin θ ) ( dϕ ) (cid:17) where g m ∞ , ( ext ) M denotes the Schwarzschild metric of mass m ∞ > g m ∞ , ( int ) M = 2 dudr − (cid:18) − m ∞ r (cid:19) ( du ) + r (cid:16) ( dθ ) + (sin θ ) ( dϕ ) (cid:17) . Asymptotic of the future event horizon.
We show below that H + is located in thefollowing region of ( int ) M m (cid:18) − √ (cid:15) u δ dec (cid:19) ≤ r ≤ m (cid:32) √ (cid:15) u δdec (cid:33) on H + for any 1 ≤ u < + ∞ . (3.6.7)Note first that the lower bound follows from the fact thatsup ( int ) M (cid:18) r ≤ m (cid:18) − √ (cid:15) u δdec (cid:19)(cid:19) κ ≤ − √ (cid:15) u δdec m (cid:16) − √ (cid:15) u δdec (cid:17) + O (cid:18) (cid:15) u δ dec (cid:19) ≤ − √ (cid:15) m u δ dec < . .6. PROOF OF THE MAIN THEOREM S ( u ) := S u , r = 2 m √ (cid:15) u δdec , ≤ u < + ∞ (3.6.8)is in the past of I + . Since ( ext ) M is in the past of I + , it suffices to show that the forwardoutgoing null cone emanating from any 2-sphere (3.6.8) reaches ( ext ) M in finite time.Assume, by contradiction, that there exists an outgoing null geodesic, denoted by γ ,perpendicular to S ( u ), that does not reach ( ext ) M in finite time. Let e (cid:48) be the geodesicgenerator of γ . In view of Lemma 2.3.1 on general null frame transformation, and denotingby ( e , e , e θ ) the null frame of ( int ) M , we look for e (cid:48) under the form e (cid:48) = λ (cid:18) e + f e θ + 14 f e (cid:19) , and the fact that e (cid:48) is geodesic implies the following transport equations along γ for f and λ in view of Lemma 2.3.6 (applied with f = 0) λ − e (cid:48) ( f ) + (cid:16) κ ω (cid:17) f = − ξ + E ( f, Γ) ,λ − e (cid:48) (log( λ )) = 2 ω + E ( f, Γ) , where E and E are given schematically by E ( f, Γ) = − ϑf + l.o.t. ,E ( f, Γ) = f ζ − f ω − ηf − f κ + l.o.t.Here, l.o.t. denote terms which are cubic or higher order in f and Γ denotes the Riccicoefficients w.r.t. the original null frame ( e , e , e θ ) of ( int ) M .We then proceed as follows1. First, we initialize f and λ as follows on the γ ∩ S ( u ) f = 0 , λ = 1 on γ ∩ S ( u ) . Recall that we assume by contradiction that γ does not reach ( ext ) M and hence stays in ( int ) M . i.e. we keep the direction of e fixed. CHAPTER 3. MAIN THEOREM
2. Then, we initiate a continuity argument by assuming for some u < u < u + (cid:18) u (cid:15) (cid:19) δdec that we have | f | ≤ √ (cid:15) u + δ dec , Υ ≥ √ (cid:15) u δdec , < λ < + ∞ on γ ( u , u ) ∩ ( int ) M (3.6.9)where γ ( u , u ) denotes the portion of γ in u ≤ u ≤ u .3. We have λ − e (cid:48) ( u ) = e ( u ) + 14 f e ( u ) = 2 ς . Relying on our control of the ingoing geodesic foliation of ( int ) M , the above as-sumption for f and the transport equation for f , we obtain on γ ( u , u ) ∩ ( int ) M sup γ ( u ,u ) ∩ ( int ) M | f | (cid:46) (cid:15) u δ dec ( u − u ) (cid:46) (cid:15) − δdec u δdec which improves our assumption in (3.6.9) on f .4. We have in view of the control of fλ − e (cid:48) ( r ) = e ( r ) + 14 f e ( r ) = Υ + O (cid:18) (cid:15) u δ dec (cid:19) . This yields λ − e (cid:48) (log(Υ)) = mr e ( r ) − r λ − e (cid:48) ( m )Υ= mr Υ + O (cid:18) (cid:15) u δdec (cid:19) Υ . Thanks to our assumption on the lower bound of Υ, we infer λ − e (cid:48) (log(Υ)) = 2 mr (1 + O ( √ (cid:15) )) .6. PROOF OF THE MAIN THEOREM ( int ) M λ − e (cid:48) (log(Υ)) ≥ m . Integrating from u = u , we deduceΥ ≥ √ (cid:15) (1 + √ (cid:15) ) u δdec exp (cid:18) u − u m (cid:19) which is an improvement of our assumption in (3.6.9) on Υ.5. In view of the control of f and of the ingoing geodesic foliation of ( int ) M , we rewritethe transport equation for λ as λ − e (cid:48) (log( λ )) = 2 ω + E ( f, Γ)= − mr + O (cid:18) (cid:15) u δ dec (cid:19) . Since we have obtained above the other hand λ − e (cid:48) (log(Υ)) = 2 mr (1 + O ( √ (cid:15) ))we immediately infer λ − e (cid:48) (log( λ )Υ ) > , λ − e (cid:48) (log( λ ) √ Υ) < . Integrating from u = u , this yields (cid:32) √ (cid:15) (1 + √ (cid:15) ) u δdec (cid:33) Υ − ≤ λ ≤ (cid:32) √ (cid:15) (1 + √ (cid:15) ) u δdec (cid:33) Υ − . Since Υ has an explicit lower bounded in view of our previous estimate, as well asan explicit upper bound since we are in ( int ) M , this yields an improvement of ourassumptions in (3.6.9) for λ .6. Since we have improved all our bootstrap assumptions (3.6.9), we infer by a conti-nuity argument the following boundΥ ≥ √ (cid:15) (1 + √ (cid:15) ) u δdec exp (cid:18) u − u m (cid:19) on γ u , u + (cid:18) u (cid:15) (cid:19) δdec ∩ ( int ) M . Now, in this u interval, we may choose u := u + 3 m (cid:18) δ dec (cid:19) log (cid:18) u (cid:15) (cid:19) for which we have Υ ≥
1. This is a contradiction since Υ = O ( δ H ) in ( int ) M . Thus,we deduce that γ reaches ( ext ) M before u = u , a contradiction to our assumptionon γ . This concludes the proof of (3.6.7).48 CHAPTER 3. MAIN THEOREM
The role of this section is to give a short description of the results concerning our GeneralCovariant Modulation (GCM) procedure, which is at the heart of our proof. We willapply it in ( ext ) M under our main bootstrap assumptions BA-E BA-D . The proof of theresults stated in this section will be proved in Chapter 9.
To state our results, which are local in nature it is convenient to consider axially symmetricpolarized spacetime regions R foliated by two functions ( u, s ) such that • On R , ( u, s ) defines an outgoing geodesic foliation as in section 2.2.4. • We denote by ( e , e , e θ ) the null frame adapted to the outgoing geodesic foliation( u, s ) on R . • We denote by ◦ S a fixed sphere of R ◦ S := S ( ◦ u, ◦ s ) (3.7.1)and by ◦ r the area radius of ◦ S , where S ( u, s ) denote the 2-spheres of the outgoinggeodesic foliation ( u, s ) on R . • In adapted coordinates ( u, s, θ, ϕ ) with b = 0, see Proposition 2.2.20, the spacetimemetric g in R takes the form, with Ω = e ( s ) , b = e ( θ ), g = − ςduds + ς Ω du + γ (cid:18) dθ − ςbdu (cid:19) + e dϕ , (3.7.2)where θ is chosen such that b = e ( θ ) = 0. • The spacetime metric induced on S ( u, s ) is given by, g/ = γdθ + e dϕ . (3.7.3) • The relation between the null frame and coordinate system is given by e = ∂ s , e = 2 ς ∂ u + Ω ∂ s + b∂ θ , e θ = γ − / ∂ θ . (3.7.4) .7. THE GENERAL COVARIANT MODULATION PROCEDURE • We denote the induced metric on ◦ S by ◦ g/ = ◦ γ dθ + e dϕ . Definition 3.7.1.
Let < ◦ δ ≤ ◦ (cid:15) two sufficiently small constants. Let ( ◦ u, ◦ s ) real numbersso that ≤ ◦ u < + ∞ , m ≤ ◦ s < + ∞ . (3.7.5) We define R = R ( ◦ δ, ◦ (cid:15) ) to be the region R := (cid:110) | u − ◦ u | ≤ δ R , | s − ◦ s | ≤ δ R (cid:111) , δ R := ◦ δ (cid:0) ◦ (cid:15) (cid:1) − , (3.7.6) such that assumption A1-A3 below with constant ◦ (cid:15) on the background foliation of R , areverified. The smaller constant ◦ δ controls the size of the GCMS quantities as it will bemade precise below. Consider the renormalized Ricci and curvature components associated to the ( u, s ) geodesicfoliation of R ˇΓ : = (cid:26) ˇ κ, ϑ, ζ, η, κ − r , κ + 2Υ r , ˇ κ, ϑ, ξ, ˇ ω, ω − mr , ˇΩ , (cid:0) Ω + Υ (cid:1) , (cid:0) ς + 1 (cid:1)(cid:27) , ˇ R : = (cid:26) α, β, ˇ ρ, ρ + 2 mr , β, α (cid:27) . Since our foliation is outgoing geodesic we also have, ξ = ω = 0 , η + ζ = 0 . (3.7.7)We decompose ˇΓ = Γ g ∪ Γ b where,Γ g = (cid:26) ˇ κ, ϑ, ζ, ˇ κ, κ − r , κ + 2Υ r (cid:27) , Γ b = (cid:110) η, ϑ, ξ, ˇ ω, ω − mr , r − ˇΩ , r − ˇ ς, r − (cid:0) Ω + Υ (cid:1) , r − (cid:0) ς − (cid:1)(cid:111) . (3.7.8)Given an integer s max , we assume the following In applications, s max = k small + 4 in Theorem M7, and s max = k large + 5 in Theorem M0 andTheorem M6. CHAPTER 3. MAIN THEOREM
A1.
For k ≤ s max , we have on R (cid:107) Γ g (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − , (cid:107) Γ b (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − , (3.7.9)and, (cid:107) α, β, ˇ ρ, ˇ µ (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − , (cid:107) e ( α, β ) (cid:107) k − , ∞ (cid:46) ◦ (cid:15)r − , (cid:107) β (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − , (cid:107) α (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − . (3.7.10) A2.
We have, with m denoting the mass of the unperturbed spacetime,sup R (cid:12)(cid:12)(cid:12)(cid:12) mm − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15). (3.7.11) A3.
The metric coefficients are assumed to satisfy the following assumptions in R , forall k ≤ s max r (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) γr − , b, e Φ r sin θ − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k + (cid:107) Ω + Υ (cid:107) ∞ ,k + (cid:107) ς − (cid:107) ∞ ,k (cid:46) ◦ (cid:15) (3.7.12)We will assume, in addition, that the following small GCM conditions hold true on R , (cid:12)(cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) d k ˇ κ (cid:12)(cid:12) + r (cid:12)(cid:12) d k − d (cid:63) / d (cid:63) / κ (cid:12)(cid:12) + r (cid:12)(cid:12) d k − d (cid:63) / d (cid:63) / µ (cid:12)(cid:12) (cid:46) ◦ δr − for all k ≤ s max , (3.7.13) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ηe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ, r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ξe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. (3.7.14)Also, (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ς + Ω (cid:19) (cid:12)(cid:12)(cid:12) SP − − mr (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. (3.7.15)Additionally we may assume on R r (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. (3.7.16) .7. THE GENERAL COVARIANT MODULATION PROCEDURE Definition 3.7.2.
We say that S is an O ( ◦ (cid:15) ) Z -polarized deformation of ◦ S if there existsa map Ψ : ◦ S −→ S of the form, Ψ( ◦ u, ◦ s, θ, ϕ ) = (cid:16) ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ, ϕ (cid:17) (3.7.17) where U, S are smooth functions defined on the interval [0 , π ] of amplitude at most ◦ (cid:15) . Wedenote by ψ the reduce map defined on the interval [0 , π ] , ψ ( θ ) = ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) . (3.7.18) We restrict ourselves to deformations which fix the South Pole, i.e. U (0) = S (0) = 0 . (3.7.19) We consider general null transformations introduced in Lemma 2.3.1, e (cid:48) = λ (cid:18) e + f e θ + 14 f e (cid:19) ,e (cid:48) θ = (cid:18) f f (cid:19) e θ + 12 f e + 12 f (cid:18) f f (cid:19) e ,e (cid:48) = λ − (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19) . (3.7.20) Definition 3.7.3.
Given a deformation
Ψ : ◦ S −→ S we say that a new frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) ,obtained from the standard frame ( e , e , e θ ) via the transformation (3.7.20) , is S -adaptedif we have, e (cid:48) θ = e S θ = 1( γ S ) / ψ ( ∂ θ ) (3.7.21) where ψ ( ∂ θ ) is the push-forward defined by the deformation map ψ . The condition translates into the following relations between the functions
U, S defining52
CHAPTER 3. MAIN THEOREM the deformation and the transition functions ( f, f ). ς ∂ θ U = (cid:0) ( γ S ) (cid:1) / f (cid:18) f f ) (cid:19) ,∂ θ S − ς ∂ θ U = 12 (cid:0) ( γ S ) (cid:1) / f , ( γ S ) = γ + ( ς ) (cid:18) Ω + 14 b γ (cid:19) ( ∂ θ U ) − ς ∂ θ U ∂ θ S − ( γςb ) ∂ θ U,U (0) = S (0) = 0 . (3.7.22) Theorem 3.7.4 (GCMS-I) . Consider the region R as above, verifying the assumptions A1 – A3 and the small GCM conditions (3.7.13) . Let ◦ S denote the sphere ◦ S = S ( ◦ u, ◦ s ) .For any fix Λ , Λ ∈ R verifying, | Λ | , | Λ | (cid:46) ◦ δ (cid:0) ◦ r (cid:1) , (3.7.23)
1. There exists a unique GCM sphere S = S (Λ , Λ) , which is a deformation of ◦ S , and anadapted null frame e S , e S θ , e S , such that the following GCMS conditions are verified components. d/ S ,(cid:63) d/ S ,(cid:63) κ S = d/ S ,(cid:63) d/ S ,(cid:63) µ S = 0 , κ S = 2 r S . (3.7.24) In addition (cid:90) S f e Φ = Λ , (cid:90) S f e Φ = Λ , (3.7.25) where ( f, f ) belong to the triplet ( f, f , λ = e a ) which denote the change of framecoefficients from the frame of ◦ S to the one of S .2. The transition functions ( f, f , log λ ) verify, (cid:13)(cid:13) ( f, f , log λ ) (cid:13)(cid:13) h k ( S ) (cid:46) ◦ δ, k ≤ s max + 1 . (3.7.26) Here, the other assumptions (3.7.14) (3.7.15) are not needed. In the sense of Definition 3.7.2. Γ S , R S denote the Ricci and curvature components with respect to the adapted frame on S . .7. THE GENERAL COVARIANT MODULATION PROCEDURE
3. The area radius r S and Hawking mass m S of S verify, (cid:12)(cid:12) r S − ◦ r (cid:12)(cid:12) (cid:46) ◦ δ, (cid:12)(cid:12) m S − ◦ m (cid:12)(cid:12) (cid:46) ◦ δ. (3.7.27)The precise version of Theorem 3.7.4 and its proof are given in section 9.4. Theorem 3.7.5 (GCMS-II) . In addition to the assumptions of Theorem 3.7.4 we alsoassume that (3.7.16) holds true. Then,1. There exists a unique GCM sphere S , which is a deformation of ◦ S , such that inaddition to (3.7.24) the following GCMS conditions also hold true on S . (cid:90) S β S e Φ = 0 , (cid:90) S e S θ ( κ S ) e Φ = 0 . (3.7.28)
2. The transition functions ( f, f , log λ ) verify the estimates (3.7.26) .3. The area radius r S and Hawking mass m S of S verify (3.7.27) . The precise version of Theorem 3.7.5 and its proof are given in 9.7.
Theorem 3.7.6 (GCMH) . Consider the region R as above, verifying the assumptions A1 – A3 and the small GCM conditions (3.7.13) – (3.7.15) . Let S = S [ ◦ u, ◦ s, Λ , Λ ] thedeformation of ◦ S constructed in Theorem GCMS-I above.There exists a smooth spacelike hypersurface Σ ⊂ R passing through S , a scalar function u S defined on Σ , whose level surfaces are topological spheres denoted by S , and a smoothcollection of constants Λ S , Λ S verifying, Λ S = Λ , Λ S = Λ , such that the following conditions are verified:1. The surfaces S of constant u S verifies all the properties stated in Theorem GCMS-I for the prescribed constants Λ S , Λ S . In particular they come endowed with nullframes ( e S , e S θ , e S ) such thati. For each S the GCM conditions (3.7.24) , (3.7.25) hold with Λ = Λ S , Λ = Λ S .iii. The transversality conditions hold true on each S . ξ S = 0 , ω S = 0 , η S + ζ S = 0 . (3.7.29)54 CHAPTER 3. MAIN THEOREM
2. We have, for some constant c Σ , u S + r S = c Σ , along Σ . (3.7.30)
3. Let ν S be the unique vectorfield tangent to the hypersurface Σ , normal to S , andnormalized by g ( ν S , e S ) = − . There exists a unique scalar function a S on Σ suchthat ν S is given by ν S = e S + a S e S . The following normalization condition holds true at the South Pole SP of everysphere S , i.e. at θ = 0 , a S (cid:12)(cid:12)(cid:12) SP = − − m S r S . (3.7.31)
4. Under the additional transversality condition on Σ e S ( u S ) = 0 , e ( r s ) = r S κ S = 1 . (3.7.32) the Ricci coefficients η S , ξ S are well defined and verify, (cid:90) S η S e Φ = (cid:90) S ξ S e Φ = 0 . (3.7.33)
5. The transition functions ( f, f , log λ ) verify the estimates (3.7.26) .6. The area radius r S and Hawking mass m S of S verify (3.7.27) . The precise version of Theorem 3.7.6 and its proof are given in section 9.8.In addition to the three theorems stated above we also prove three rigidity statementswhich can be regarded as corollaries.
Corollary 3.7.7 (Rigidity I) . Assume S is a sphere in R which verifies the the GCMconditions κ S = 2 r S , d (cid:63) / S d (cid:63) / S κ S = d (cid:63) / S d (cid:63) / S µ S = 0 . (3.7.34) Here the average of κ S is taken on S . In view of the GCM conditions (3.7.24) we deduce e S ( r S ) = 1. .7. THE GENERAL COVARIANT MODULATION PROCEDURE Assume also that the background foliation verify on R the conditions A1-A3 . Then thetransition functions ( f, f , log λ ) from the background frame of R to to that of S verifiesthe estimates (cid:107) ( f, f , log( λ )) (cid:107) h smax +1 ( S ) (cid:46) ◦ δ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) . Moreover, The area radius r S and Hawking mass m S of S verify (3.7.27) . Corollary 3.7.8 (Rigidity II) . Assume given a sphere S ⊂ R endowed with a compatibleframe e S , e S θ , e S which verifies the GCM conditions κ S = 2 r S , d (cid:63) / S d (cid:63) / S κ S = d (cid:63) / S d (cid:63) / S µ S = 0 . (3.7.35) In addition, we assume1. We have, r (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S β S e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e S θ ( κ S ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. (3.7.36)
2. The background foliation verifies on R the conditions A1-A3 .Then the transition functions ( f, f , log λ ) from the background frame of R to that of S verifies the estimates (cid:107) ( f, f , log( λ )) (cid:107) h smax +1 ( S ) (cid:46) ◦ δ. Moreover, the area radius r S and Hawking mass m S of S verify (3.7.27) . Corollary 3.7.9 ( Rigidity III) . Assume given a GCM hypersurface Σ ⊂ R foliated bysurfaces S such that κ S = 2 r S , d (cid:63) / S d (cid:63) / S κ S = d (cid:63) / S d (cid:63) / S µ S = 0 , (cid:90) S η S e Φ = 0 , (cid:90) S ξ S e Φ = 0 . Assume in addition that for a specific sphere S on Σ we have the following additionalinformation:1. The transition functions f, f from the background foliation to S verify (cid:90) S f e Φ = O ( ◦ δ ) , (cid:90) S f e Φ = O ( ◦ δ ) . (3.7.37)56 CHAPTER 3. MAIN THEOREM
2. In R , the Ricci and curvature coefficients of the background foliation verify theassumptions A1–A3 .Then, for all derivatives of the transition functions along S , (cid:107) d ≤ s max +1 ( f, f , log λ ) (cid:107) L ( S ) (cid:46) ◦ δ. Moreover, The area radius r S and Hawking mass m S of S verify (3.7.27) . Both theorems GCMS-II and GCMH are based on Theorem GSMS-I. They are heavilybased on the transformation formulas for the Ricci and curvature coefficients recorded inProposition 2.3.4.
Sketch of the proof of Theorems GSMS-I and GSMS-II
A given deformation Ψ : ◦ S −→ S is fixed by the parameters U, S and transition functions F = ( f, f , λ ) connected by the system (3.7.22). Making use of the transformation formulasone can show that the GCMS conditions (3.7.24)-(3.7.25) are if and only if the transitionfunctions F verify a coercive nonlinear elliptic Hodge system of the form D Ψ F = B (Ψ),where the operator D Ψ depends on the deformation Ψ and the right hand side B , dependson on both Ψ and the background foliation (see Proposition 9.4.1 for the precise form of thesystem). To find a desired GSMS deformation we have to solve a coupled system betweenthe transport type equations in (3.7.22) and the elliptic coercive system D Ψ F = 0 ofProposition 9.4.1.The actual proof is thus based on an iteration procedure for a sequence of deformationspheres S ( n ) of ◦ S given by the maps Ψ ( n ) = ( U ( n ) , S ( n ) ) : ◦ S −→ S ( n ) and the corre-sponding transition functions log λ ( n ) , f ( n ) , f ( n ) . The iteration procedure for the quintets Q ( n ) = ( U ( n ) , S ( n ) , log λ ( n ) , f ( n ) , f ( n ) ), starting with the trivial quintet Q (0) correspondingto the zero deformation, is described in section 9.4.3. The main steps in the proof are asfollows.1. Given the triplet log λ ( n ) , f ( n ) , f ( n ) ) the pair ( U ( n ) , S ( n ) ) defines the deformationsphere S ( n ) and the corresponding pull back map n : ◦ S −→ S ( n ) according to theequation (3.7.22). .7. THE GENERAL COVARIANT MODULATION PROCEDURE ( n ) = ( U ( n ) , S ( n ) ) and the deformation sphere S ( n ) we define thetriplet (log λ ( n +1) , f ( n +1) , f ( n +1) ) by solving the corresponding elliptic system D Ψ( n ) F ( n +1) = B (Ψ ( n ) )This step is based on the crucial apriori estimates of section 9.4.1.3. Given the new pair ( f ( n +1) , f ( n +1) ) we make use of the equations (3.7.22) to find aunique new map ( U ( n +1) , S ( n +1) ) and thus the new deformation sphere S ( n + 1).4. The convergence of the iterates Q ( n ) , described in subsection 9.4.5 in the bounded-ness Proposition 9.4.8 and the contraction Proposition 9.4.9. The latter requires usto carefully compare the iterates Q ( n ) , Q ( n +1) by pulling them back to ◦ S . One hasto be particularly careful with the behavior of the iterates on the axis of symmetry.Theorem GSMS-II, which is an easy consequence of Theorem GSMS-I is proved in section9.7 and the transformation formulas which relate (cid:82) S β S e Φ to Λ = (cid:82) S f e Φ and (cid:82) S e S θ ( κ S ) e Φ to Λ = (cid:82) S f e Φ . One can show that there exist choices of Λ , Λ such that (cid:82) S β S e Φ = (cid:82) S e S θ ( κ S ) e Φ = 0 . Sketch of the proof of Theorem GCMH
The proof of Theorem GCMH makes use of Theorem GCMS-I to construct Σ as a unionof GCMS spheres. Step 1.
Theorem GCMS-I allows to construct, for every value of the parameters ( u, s )in R (i.e. such that the background spheres S ( u, s ) ⊂ R ) and every real numbers (Λ , Λ),a unique GCM sphere S [ u, s, Λ , Λ], as a Z -polarized deformation of S ( u, s ). In particular(3.7.24) and (3.7.25) are verified and S = S [ ◦ u, ◦ s, Λ , Λ ]. Step 2.
We look for functions Ψ( s ) , Λ( s ) , Λ( s ) such that1. We have, Ψ( ◦ s ) = ◦ u, Λ( ◦ s ) = Λ , Λ( ◦ s ) = Λ .
2. The resulting hypersurface Σ = ∪ s S [Ψ( s ) , s, Λ( s ) , Λ( s )] verifies u S + r S = c Σ , along Σ . CHAPTER 3. MAIN THEOREM
3. The additional GCM conditions (3.7.31) and (3.7.33) of Theorem GCMH are veri-fied.These conditions lead to a first order differential system for Ψ( s ) , Λ( s ) , Λ( s ), with pre-scribed initial conditions at ◦ s which allows us to determine the desired surface. The proofis given in detail in section 9.8. Sketch of the proof of the Rigidity Corollaries
The Rigidity I corollary, see Corollary 9.7.3, is proved in exactly the same way as Propo-sition 9.4.4 which is one of the main steps in the proof of Theorem GCMS-I. The rigidityII corollary, see Corollary 9.7.3, is based on Rigidity I and a simple variation of Lemma9.7.2. The proof of the rigidity III corollary, see Corollary 9.8.2, is essentially part of theproof of Theorem GCMH, once the existence part of the theorem has been established.
In this section, we provide a brief overview of the proof of Theorem M0-M8. In additionto the null frame adapted to the outgoing foliation of ( ext ) M and to the null frameadapted to the ingoing foliation of ( int ) M , we have also introduced 2 global frames on M = ( int ) M∪ ( ext ) M as well as associated scalars r and m in section 3.5. Unless otherwisespecified, when we discuss a particular spacetime region, i.e. ( ext ) M , ( int ) M or M , itshould be understood that the frame as well as r and m are the ones corresponding tothat region. Step 1.
Recall our GCM conditions on S ∗ = Σ ∗ ∩ C ∗ (cid:90) S ∗ e θ ( κ ) e Φ = 0 , (cid:90) S ∗ βe Φ = 0 . Recall that ν = e + a ∗ e is the unique tangent vectorfield to Σ ∗ which is orthogonal to e θ and normalized by g ( ν, e ) = −
2. Using the null structure equation for e ( κ ) and e ( β ),as well as e ( κ ) and e ( β ), we obtain transport equations along Σ ∗ in the ν direction for (cid:90) S e θ ( κ ) e Φ and (cid:90) S βe Φ = 0 . .8. OVERVIEW OF THE PROOF OF THEOREM M0-M8 ν , we propagate the control on S ∗ to Σ ∗ . Inparticular, we obtain the following estimates on S = Σ ∗ ∩ C , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) + (cid:15)r (cid:46) (cid:15) , (3.8.1)where we used in the last inequality the dominance condition of r on Σ ∗ , see (3.3.4). Step 2.
We consider the transition functions ( f, f , λ ) from the frame of the initial datalayer to the frame of ( ext ) M . Since • S is a sphere of ( ext ) M in the initial data layer, • S is a sphere of the GCM hypersurface Σ ∗ , • the estimate (3.8.1) holds on S ,we can invoke the GCM Corollary Rigidity II and III of section 3.7.4 which yields, togetherwith a Sobolev embedding on S sup S (cid:16) r | d ≤ k large +1 ( f, f , log λ ) | + | m − m | (cid:17) (cid:46) (cid:15) . (3.8.2) Step 3.
Relying on the transport equations in e for ( f, f , λ ), see Corollary 2.3.7, andProposition 2.2.16 for m , we propagate (3.8.2) to C , and then, proceeding similarly, inthe e direction to C which yieldssup C ∪C (cid:16) r | d ≤ k large +1 ( f, f , log λ ) | + | m − m | (cid:17) (cid:46) (cid:15) . Together with the control of the initial data layer foliation and the transformation formulasof Proposition 2.3.4, we then obtain the desired estimates on C ∪ C for the curvaturecomponents. Here are the main steps in the proof of Theorem M1.
Step 1.
Consider the global frame on M constructed in Proposition 3.5.5 and thedefinition of q on M with respect to that frame, see section 2.3.3 for the definition of q with respect to any null frame. According to Theorem 2.4.7 we have, (cid:3) q + V q = N, V = κκ (3.8.3)60 CHAPTER 3. MAIN THEOREM where the nonlinear term N = Err[ (cid:3) g q ] is a long expression of terms quadratic, orhigher order, in ˇΓ , ˇ R involving various powers of r . Making use of the symbolic notationintroduced in definition 2.3.8 we have, see (2.4.8),Err[ (cid:3) q ] = r d ≤ (Γ g · ( α, β )) + e (cid:16) r d ≤ (Γ g · ( α, β )) (cid:17) + d ≤ (Γ g · q ) + l.o.t.where the terms denoted by l.o.t. are higher order in (ˇΓ , ˇ R ). Remark 3.8.1.
Recall from Remark 2.4.8 that the above good structure of the error termErr [ (cid:3) q ] only holds in a frame for which ξ = 0 for r ≥ m and η ∈ Γ g . This is why,in Theorem M1, q is defined relative to the global frame of Proposition 3.5.5, see alsoRemark 2.4.9. Step 2.
We follow the Dafermos-Rodnianski version of the vector-field method to derivedesired decay estimates. We recall that, in the context of a wave equation of the form (cid:3) ( Sch ) ψ = 0 on Schwarzschild spacetime, their strategy consists in the following: • Start by deriving Morawetz-energy type estimates for ψ with nondegenerate fluxenergies and the usual degeneracy of bulk integrals at r = 3 m . • Derive r p weighted estimates for 0 < p < • The decay estimates obtained by using the standard r p weighted approach are tooweak to be useful in our nonlinear approach. We improve them by making use of a re-cent variation of the Dafermos-Rodnianski approach due to Angelopoulos, Aretakisand Gajic [4] which is based on first commuting the wave equation is (cid:3) ( Sch ) ψ = 0with r ( e + r − ) and then repeating the process described for the resulting newequation. This procedure allows to derive the improved decay estimates consistentwith our decay norms. • Derive estimates for higher derivatives by commuting with T , r d / , the red-shiftvectorfield, and re . Step 3.
The estimates mentioned in step 2 have to be adapted to the case of our equation(3.8.3). There are three main differences to take into account • The application of the vectorfield method to our context produces various nontrivialcommutator terms which have to be absorbed. This is taken care by our bootstrapassumption for ˇΓ , ˇ R , as well as, in some cases, by integration by parts. .8. OVERVIEW OF THE PROOF OF THEOREM M0-M8 • The presence of the potential V is mostly advantageous but various modificationshave to be nevertheless made, especially near the trapping region . • The presence of the nonlinear term N is the most important complication. Theprecise null structure of N is essential and various integrations by parts are needed. • The quadratic terms involving η in N can only be treated provided the definition of q is done with respect to the global frame on M constructed in Proposition 3.5.5,for with η behaves better in powers of r − . Recall from section 2.3.3 that q is defined with respect to a general null frame as follows q = r (cid:18) e ( e ( α )) + (2 κ − ω ) e ( α ) + (cid:18) − e ( ω ) + 8 ω − ω κ + 12 κ (cid:19) α (cid:19) which yields the following transport equation for αe ( e ( α )) + (2 κ − ω ) e ( α ) + (cid:18) − e ( ω ) + 8 ω − ω κ + 12 κ (cid:19) α = q r . Recall also that q , controlled in Theorem M1, is defined w.r.t. the global frame of Propo-sition 3.5.5 whose normalization is such that, in particular, ω is a small quantity. Also,since we have e ( r ) = r κ + l.o.t.we infer e ( e ( r α )) = q r + l.o.t.Integrating twice this transport equation from C where we control the initial data - andin particular α - in view of Theorem M0, and using the decay for q provided by TheoremM1, we deduce sup ( ext ) M (cid:18) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:19) | d ≤ k small +20 α | (cid:46) (cid:15) , sup ( ext ) M (cid:18) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:19) | d ≤ k small +19 e ( α ) | (cid:46) (cid:15) . At the linear level, on a Schwarzschild spacetime, this step was also treated (minus the improveddecay) in the paper [22]. Recall that δ extra has been introduced in Theorem M1 and satisfies δ extra > δ dec . CHAPTER 3. MAIN THEOREM
Now that we control α in the global frame of Proposition 3.5.5, we need to go back to theframe of ( ext ) M . By invoking the relationships between our various frame of ( ext ) M , seeProposition 3.5.5 and Proposition 3.4.6, and the transformation formula for α , we infer ( ext ) D k small +20 (cid:2) ( ext ) α (cid:3) (cid:46) (cid:15) and hence the conclusion of Theorem M2. Here are the main steps in the proof of Theorem M3.
Step 1.
To derive decay estimates for α in M , we first recall the following Teukolsky-Starobinski identity, see (2.3.15), e ( r e ( r q )) + 2 ωr e ( r q ) = r (cid:26) d (cid:63) / d (cid:63) / d/ d/ α + 32 κρe α − κρe ( α ) (cid:27) + l.o.t.where l.o.t. denotes terms which are quadratic of higher, and where all quantities aredefined w.r.t. the global frame of Proposition 3.5.5. Then, introducing the vectorfield (cid:101) T = e − κ (cid:16) κ + κ ˇΩ − ˇ κ ˇΩ (cid:17) e , we rewrite the identity as6 m (cid:101) T α + r d (cid:63) / d (cid:63) / d/ d/ α = 1 r (cid:16) e ( r e ( r q )) + 2 ωr e ( r q ) (cid:17) + l.o.t. (3.8.4)As it turns out, see Remark 6.2.3, this is a forward parabolic equation on each hypersurface of contant r in ( int ) M . Step 2.
Thanks to • the control in ( int ) M of the RHS of (3.8.4) which follows from the decay estimatesof Theorem M1 for q , as well as the bootstrap assumptions for the quadratic andhigher order terms, • the control of α on C - i.e. of the initial data of (3.8.4) - provided by Theorem M0, • parabolic estimates for the forward parabolic equation (3.8.4), .8. OVERVIEW OF THE PROOF OF THEOREM M0-M8 α in ( int ) M . Step 3.
It remains to control α on Σ ∗ . Recall that ν denotes the unique tangent vectorfieldto Σ ∗ which can be written as ν = e + ae . The Teukolsky-Starobinski identity of Step1 can then be written as6 mνα + r d (cid:63) / d (cid:63) / d/ d/ α = 1 r (cid:16) e ( r e ( r q )) + 2 ωr e ( r q ) (cid:17) + l.o.t. (3.8.5)where l.o.t. denotes terms which are quadratic of higher, as well as terms which are linearbut display additional decay in r . This is a forward parabolic equation along Σ ∗ . Toobtain the desired decay for α along Σ ∗ , one then proceeds as in Step 2, using in addition,for the linear term with extra decay in r , the behavior (3.3.4) of r on Σ ∗ . Here are the main steps in the proof of Theorem M4.
Step 1.
We derive decay estimates for the spacelike GCM hypersurface Σ ∗ . More pre-cisely, thanks to • the GCM conditions on Σ ∗ κ = 2 r , d (cid:63) / d (cid:63) / κ = 0 , d (cid:63) / d (cid:63) / µ = 0 , (cid:90) S ηe Φ = 0 , (cid:90) S ξe Φ = 0 , • the control of q in ( ext ) M , established in Theorem M1, and hence in particular onΣ ∗ , • the control of α of the outgoing geodesic foliation in ( ext ) M , established in TheoremM2, and hence in particular on Σ ∗ , • the control of α on Σ ∗ , established in Theorem M3, • the fact that the following condition holds on Σ ∗ r (cid:12)(cid:12) Σ ∗ ≥ (cid:15) − u , • the identity (2.3.11) relating q to derivatives of ρ , i.e. q = r (cid:18) d (cid:63) / d (cid:63) / ρ + 34 ρκϑ + 34 ρκϑ + · · · (cid:19) , CHAPTER 3. MAIN THEOREM • elliptic estimates for Hodge operators on the 2-spheres foliating Σ ∗ ,we infer the control with improved decay of all Ricci and curvature components on thespacelike hypersurface Σ ∗ . Step 2.
We derive decay estimates for the outgoing geodesic foliation of ( ext ) M . Moreprecisely: • First, we propagate the estimates involving only u − − δ dec decay in u from Σ ∗ to ( ext ) M . • We then focus on the harder to recover estimates, i.e. the ones involving u − − δ dec decay in u . We proceed as follows. – We first propagate the main GCM quantities ˇ κ , ˇ µ , and a renormalized quantityinvolving ˇ κ (see the quantity Ξ in Lemma 7.5.2) from Σ ∗ to ( ext ) M . – We then recover the estimates involving u − − δ dec decay in u on T . To this end,we use that we control the main GCM quantities, α from Theorem M3 (since T belongs both to ( ext ) M and ( int ) M ), q and α from Theorem M1–M2, andthe estimates are then derived somewhat in the spirit of the ones on Σ ∗ , inparticular by relying on elliptic estimates for Hodge operators on the 2-spheresfoliating T . – To recover the remaining estimates in ( ext ) M involving u − − δ dec decay in u , weintegrate the transport equations in e forward from T , which concludes theproof of Theorem M4. Here are the main steps in the proof of Theorem M5.
Step 1.
We first derive decay estimates for the ingoing geodesic foliation of ( int ) M onthe timelike hyper surface T . More precisely, thanks to • the fact that the null frame of ( int ) M is defined on T as a simple conformal renor-malization of the null frame of ( ext ) M in view of its initialization, see section 3.1.2, • the control of the outgoing geodesic foliation of ( ext ) M on T obtained in TheoremM4, .8. OVERVIEW OF THE PROOF OF THEOREM M0-M8 ( ext ) ˇ R, ( ext ) ˇΓ) to ( ( int ) ˇ R, ( int ) ˇΓ) on T . Step 2.
We derive on ( int ) M decay estimates for the ingoing geodesic foliation of ( int ) M .More precisely, thanks to • the improve decay estimates for α in ( int ) M derived in Theorem M3, • the improved decay estimates for ˇΓ and ˇ R on T derived in the Step 1, • the null structure equations and Bianchi identities,we infer O ( (cid:15) u − − δ dec ) decay estimates for ˇΓ and ˇ R corresponding to the ingoing geodesicfoliation of ( int ) M which concludes the proof of Theorem M5. Step 1.
Using(a) The control of the initial data layer,(b) Theorem GCMS-II of section 3.7.4,(c) Theorem GCMH of section 3.7.4,we produce a smooth hypersurface Σ ∗ in the initial data layer starting from a GCM sphere S ∗ , and satisfying all the required properties for the future spacelike boundary of a GCMadmissible spacetime, according to item 3 of definition 3.1.2. Step 2.
We then consider the outgoing geodesic foliation initialized on Σ ∗ which foliatesthe region we denote ( ext ) M , to the past of Σ ∗ , and included in the outgoing part ( ext ) L of the initial data layer. In order to control it, we consider the transition functions ( f, f , λ )from the background frame of the initial data layer to the frame of ( ext ) M . These functionssatisfy transport equations in e with right-hand side depending on ( f, f , λ ) and the Riccicoefficients of the background foliation. Integrating the transport equations from Σ ∗ ,where ( f, f , λ ) are under control as a by product of the use of Theorem GCMH in Step1, we obtain the control of ( f, f , λ ) in ( ext ) M . Using the transformation formulas ofProposition 2.3.4, and using the control of the initial data layer, we then infer the desiredcontrol (i.e. with (cid:15) smallness constant and suitable r -weights) for the Ricci coefficientsand curvature components of the foliation of ( ext ) M .66 CHAPTER 3. MAIN THEOREM
Step 3. ( ext ) M terminates on a timelike hypersurface T of constant area radius . Wethen consider the ingoing geodesic foliation initialized on T according to item 4 of defini-tion 3.1.2, which foliates the region we denote ( int ) M , included in the ingoing part ( int ) L of the initial data layer. Proceeding as in Step 2, relying on transport equations in e instead of e , we then derive the desired control (i.e. with (cid:15) smallness constant) for theRicci coefficients and curvature components of the foliation of ( int ) M , thus concludingthe proof of Theorem M6. From the assumptions of Theorem M7 we are given a GCM admissible spacetime M = M ( u ∗ ) ∈ ℵ ( u ∗ ) verifying the following improved bounds, for a universal constant C > N ( Dec ) k small +5 ( M ) ≤ C(cid:15) provided by Theorems M1-M5. We then proceed as follows. Step 1.
We extend M by a local existence argument, to a strictly larger spacetime M ( extend ) , with a naturally extended foliation and the following slightly increased bounds N ( Dec ) k small +5 ( M ( extend ) ) ≤ C(cid:15) . but which may not verify our admissibility criteria. Step 2.
Using(a) The control of the extended spacetime M ( extend ) ,(b) Theorem GCMS-II of section 3.7.4,(c) Theorem GCMH of section 3.7.4,we produce a small piece of smooth GCM hypersurface (cid:101) Σ ∗ in M ( extend ) \ M starting froma GCM sphere (cid:101) S ∗ . Step 3.
By a continuity argument based on a priori estimates, we extend (cid:101) Σ ∗ all the wayto the initial data layer, while ensuring that it remains in M ( extend ) \ M and satisfying allthe required properties for the future spacelike boundary of a GCM admissible spacetime,according to item 3 of definition 3.1.2. With respect to the foliation of ( ext ) M . .8. OVERVIEW OF THE PROOF OF THEOREM M0-M8 Step 4.
We then consider the outgoing geodesic foliation initialized on (cid:101) Σ ∗ which foliatesthe region we denote ( ext ) (cid:102) M , included in the outgoing part of M ( extend ) . In order tocontrol it, we consider the transition functions ( f, f , λ ) from the background frame of theinitial data layer to the frame of ( ext ) (cid:102) M . These functions satisfy transport equations in e with right-hand side depending on ( f, f , λ ) and the Ricci coefficients of the backgroundfoliation. Integrating the transport equations from (cid:101) Σ ∗ , where ( f, f , λ ) are under controlas a by product of the use of Theorem GCMH in Step 2, we obtain the control of ( f, f , λ )in ( ext ) (cid:102) M . Using the transformation formulas of Proposition 2.3.4, and using the controlof the initial data layer, we then derive the desired control (i.e. with (cid:15) smallness constantand suitable u and r weights) for the Ricci coefficients and curvature components of thefoliation of ( ext ) (cid:102) M . Step 5. ( ext ) (cid:102) M terminates on a timelike hypersurface (cid:101) T of constant area radius . Wethen consider the ingoing geodesic foliation initialized on (cid:101) T according to item 4 of defi-nition 3.1.2, which foliates the region we denote ( int ) (cid:102) M , included in the ingoing part of M ( extend ) . Proceeding as in Step 4, relying on transport equations in e instead of e , wethen derive the desired control (i.e. with (cid:15) smallness constant and suitable u -weights) forthe Ricci coefficients and curvature components of the foliation of ( int ) (cid:102) M , thus concludingthe proof of Theorem M7. So far, we have only improved our bootstrap assumptions on decay estimates. We nowimprove our bootstrap assumptions on energies and weighted energies for ˇ R and ˇΓ relyingon an iterative procedure recovering derivatives one by one . Step 0.
Let I m ,δ H the interval of R defined by I m ,δ H := (cid:20) m (cid:18) δ H (cid:19) , m (cid:18) δ H (cid:19)(cid:21) . (3.8.6)Recall that T = { r = r T } , where r T ∈ I m ,δ H , and note, see also Remark 3.6.3, that theresults of Theorems M0–M7 hold for any r T ∈ I m ,δ H .It is at this stage that we need to make a specific choice of r T in the context of a Lebesgue With respect to the foliation of ( ext ) (cid:102) M . See also [29] for a related strategy to recover higher order derivatives from the control of lower orderones. CHAPTER 3. MAIN THEOREM point argument. More precisely, we choose r T such that we have (cid:90) { r = r T } | d ≤ k large ˇ R | = inf r ∈ I m ,δ H (cid:90) { r = r } | d ≤ k large ˇ R | . (3.8.7)In view of this definition, and since T = { r = r T } , we infer that (cid:90) T | d ≤ k large ˇ R | (cid:46) (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d ≤ k large ˇ R | . (3.8.8) Remark 3.8.2.
From now on, we may thus assume that the spacetime M satisfies theconclusions of Theorem M0 and Theorem M7, as well as (3.8.8) , and our goal is to proveTheorem M8, i.e. to prove that N ( En ) k large (cid:46) (cid:15) holds. Step 1.
The O ( (cid:15) ) decay estimates derived in Theorem M7 imply in particular thefollowing (non sharp) consequence N ( En ) k small (cid:46) (cid:15) , where we recall N ( En ) k = ( ext ) R k [ ˇ R ] + ( ext ) G k [ˇΓ] + ( int ) R k [ ˇ R ] + ( int ) G k [ˇΓ] . This allows us to initialize our iteration scheme in the next step.
Step 2.
Next, for J such that k small ≤ J ≤ k large −
1, consider the iteration assumption N ( En ) J (cid:46) (cid:15) B [ J ] , (3.8.9)where (cid:15) B [ J ] := J (cid:88) j = k small − ( (cid:15) ) (cid:96) ( j ) B − (cid:96) ( j ) + (cid:15) (cid:96) ( J )0 B , (cid:96) ( j ) := 2 k small − − j , (3.8.10) B := (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d ≤ k large ˇ R | . (3.8.11)In view of Step 1, (3.8.9) holds for J = k small . From now on, we assume that (3.8.9) holdsfor J such that k small ≤ J ≤ k large −
2, and our goal is to show that this also holds for J + 1 derivatives. See sections 3.2.1 and 3.2.2 for the definition of our norms measuring energies for curvature compo-nents and Ricci coefficients. .8. OVERVIEW OF THE PROOF OF THEOREM M0-M8
Step 3.
Using the Teukolsky wave equations for α and α , as well as a wave equation forˇ ρ , see Proposition 8.4.1, we derive Morawetz type estimates for J + 1 derivatives of thesequantities in terms of O ( (cid:15) B [ J ] + (cid:15) N ( En ) J +1 ). Step 4.
Relying on Bianchi identities, we also derive Morawetz type estimates for J + 1derivatives for β and β . As a consequence, we obtain Morawetz type estimates for J + 1derivatives of all curvature components in terms of O ( (cid:15) B [ J ] + (cid:15) N ( En ) J +1 ). Step 5.
As a consequence of Step 4, we immediately obtain, for any r ≥ m , ( int ) R J +1 [ ˇ R ] + ( ext ) R J +1 [ ˇ R ] ≤ ( ext ) R ≥ r J +1 [ ˇ R ] + O ( r ( (cid:15) B [ J ] + (cid:15) N ( En ) J +1 )) . Step 6.
Relying on the Bianchi identities, we derive r p -weighted estimates for J + 1derivatives of curvature on r ≥ r with r ≥ m . We obtain ( ext ) R ≥ r J +1 [ ˇ R ] (cid:46) r δ B ext ) G ≥ r k [ˇΓ] + r ( (cid:15) B [ J ] + (cid:15) N ( En ) J +1 ) . Step 7.
Next, we estimate the Ricci coefficients of ( ext ) M . To control them, we relyon the null structure equations in ( ext ) M . Using the null structure equations in ( ext ) M and the GCM conditions on Σ ∗ , we derive the following weighted estimates for J + 1derivatives of the Ricci coefficients ( ext ) G J +1 [ˇΓ] (cid:46) ( ext ) R J +1 [ ˇ R ] + (cid:15) B [ J ] + (cid:15) N ( En ) J +1 . Together with the estimates of Step 5 and Step 6, we infer for a large enough choice of r ext ) G J +1 [ˇΓ] + ( int ) R J +1 [ ˇ R ] + ( ext ) R J +1 [ ˇ R ] (cid:46) (cid:15) B [ J ] + (cid:15) N ( En ) J +1 . Step 8.
Next, we estimate the Ricci coefficients of ( int ) M . Using the information on T induced by Step 7 and the null structure equations in ( int ) M , we derive ( int ) G J +1 [ˇΓ] (cid:46) ( int ) R J +1 [ ˇ R ] + (cid:15) B [ J ] + (cid:15) N ( En ) J +1 + (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) . We need to deal with the last term. Relying on a trace theorem in the spacetime region ( ext ) M ( r ∈ I m ,δ H ), and the fact that J + 2 ≤ k large , we obtain (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) (cid:46) (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | ( ( ext ) R J +1 [ ˇ R ]) + ( ext ) R J +1 [ ˇ R ] . CHAPTER 3. MAIN THEOREM
Step 9.
The last estimate of Step 7 and the 2 estimates of Step 8 yield, for (cid:15) > N ( En ) J +1 (cid:46) (cid:15) B [ J ] + (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | (cid:16) (cid:15) B [ J ] + (cid:15) N ( En ) J +1 (cid:17) . In view of the definition (3.8.10) of (cid:15) B [ J ], we infer that N ( En ) J +1 (cid:46) (cid:15) B [ J + 1]which is the iteration assumption (3.8.9) for J + 1 derivatives. We deduce that (3.8.9)holds for all J ≤ k large −
1, and hence N ( En ) k large − (cid:46) (cid:15) B [ k large − . Step 10.
Relying on the conclusion of Step 9, and arguing as in Step 3 to Step 7, weobtain the conclusion of Step 7 for J = k large −
1, i.e. ( ext ) G k large [ˇΓ] + ( int ) R k large [ ˇ R ] + ( ext ) R k large [ ˇ R ] (cid:46) (cid:15) B [ k large −
1] + (cid:15) N ( En ) k large . We then infer that (cid:15) B [ k large − (cid:46) (cid:15) + (cid:15) N ( En ) k large which yields, together with the last estimate of Step 9, ( ext ) G k large [ˇΓ] + ( int ) R k large [ ˇ R ] + ( ext ) R k large [ ˇ R ] (cid:46) (cid:15) + (cid:15) N ( En ) k large . Step 11.
It remains to recover ( int ) G k large [ˇΓ]. Arguing as for the first estimate of Step 8with J = k large −
1, we have ( int ) G k large [ˇΓ] (cid:46) ( int ) R k large [ ˇ R ] + (cid:15) B [ k large −
1] + (cid:15) N ( En ) k large + (cid:18)(cid:90) T | d k large ( ( ext ) ˇ R ) | (cid:19) . Thanks to the outcome of Step 10, we deduce that ( int ) G k large [ˇΓ] (cid:46) (cid:15) + (cid:15) N ( En ) k large + (cid:18)(cid:90) T | d k large ( ( ext ) ˇ R ) | (cid:19) and hence, for (cid:15) > N ( En ) k large (cid:46) (cid:15) + (cid:18)(cid:90) T | d k large ( ( ext ) ˇ R ) | (cid:19) . .9. STRUCTURE OF THE REST OF THE PAPER r T , or rather its consequence (3.8.8), which implies (cid:18)(cid:90) T | d k large ( ( ext ) ˇ R ) | (cid:19) (cid:46) (cid:15) + (cid:15) N ( En ) k large so that we finally obtain, for (cid:15) > N ( En ) k large (cid:46) (cid:15) which concludes the proof of Theorem M8. The rest of this paper is devoted to the proof of Theorem M0-M8, as well as our GCMprocedure. More precisely,1. Theorem M0, together with other first consequences of the bootstrap assumptions,is proved in Chapter 4.2. Theorem M1 is proved in Chapter 5.3. Theorems M2 and M3 are proved in Chapter 6.4. Theorems M4 and M5 are proved in Chapter 7.5. Theorems M6, M7 and M8 are proved in Chapter 8.6. Our GCM procedure is described in details in Chapter 9.7. Chapter 10 contains estimates for Regge-Wheeler type wave equations used in The-orem M1.8. Many of the long calculations are to be found in the appendix.72
CHAPTER 3. MAIN THEOREM hapter 4CONSEQUENCES OF THEBOOTSTRAP ASSUMPTIONS
According to the statement of Theorem M0 we consider given the initial layer L = ( ext ) L ∪ ( int ) L as defined in Definition 3.1.1. We also assume that the initial layer normverifies sup k ≤ k large +5 I k (cid:46) (cid:15) (4.1.1)where I k = ( ext ) I k + ( int ) I k + I (cid:48) k and, ( ext ) I = sup ( ext ) L (cid:20) r + δ B ( | α | + | β | ) + r (cid:12)(cid:12)(cid:12)(cid:12) ρ + 2 m r (cid:12)(cid:12)(cid:12)(cid:12) + r | β | + r | α | (cid:21) + sup ( ext ) L r (cid:32) | ϑ | + (cid:12)(cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12)(cid:12) + | ζ | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ + 2 (cid:0) − m r (cid:1) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) + sup ( ext ) L r (cid:16) | ϑ | + (cid:12)(cid:12)(cid:12) ω − m r (cid:12)(cid:12)(cid:12) + | ξ | (cid:17) + sup ( ext ) L ( ( ext ) r ≥ m ) (cid:18) r (cid:12)(cid:12)(cid:12) γr − (cid:12)(cid:12)(cid:12) + r | b | + | Ω + Υ | + | ς − | + r (cid:12)(cid:12)(cid:12)(cid:12) e Φ r sin θ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS ( int ) I = sup ( int ) L (cid:18) | α | + | β | + (cid:12)(cid:12)(cid:12)(cid:12) ρ + 2 m r (cid:12)(cid:12)(cid:12)(cid:12) + | β | + | α | (cid:19) + sup ( int ) L (cid:32) | ϑ | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ − (cid:0) − m r (cid:1) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | ζ | + (cid:12)(cid:12)(cid:12)(cid:12) κ + 2 r (cid:12)(cid:12)(cid:12)(cid:12) + | ϑ | + (cid:12)(cid:12)(cid:12) ω + m r (cid:12)(cid:12)(cid:12) + | ξ | (cid:33) , I (cid:48) = sup ( int ) L ∩ ( ext ) L (cid:0) | f | + | f | + | log( λ − λ ) | (cid:1) , λ = ( ext ) λ = 1 − m ext ) r L , with I k the corresponding higher derivative norms obtained by replacing each componentby d ≤ k of it. In the definition of I (cid:48) above, ( f, f , λ ) denote the transition functions ofLemma 2.3.1 from the frame of the outgoing part ( ext ) L of the initial data layer to theframe of the ingoing part ( int ) L of the initial data layer in the region ( int ) L ∩ ( ext ) L .We divide the proof of Theorem M0 in the following steps. Step 1.
We have the following lemma.
Lemma 4.1.1.
We have on ( ext ) M e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) = (cid:90) S (cid:32) κe θ ( κ ) − κe θ ( κ ) + 4 Kζ + 2 e θ ( ρ ) − e θ ( ϑ ϑ ) − ϑe θ ( κ ) + 2 e θ ( ζ ) (cid:33) e Φ ,e (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S (cid:18) − κβ + ζα − ϑβ (cid:19) e Φ ,e (cid:18)(cid:90) S βe Φ (cid:19) = − (cid:90) S e θ ( κκ ) e Φ + 3 ρ (cid:90) S ηe Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ + (cid:90) S (cid:32) κβ + 2 ωβ + 3 η ˇ ρ − ϑβ + ξα − ϑβ (cid:33) e Φ + Err (cid:20) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:21) , and e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) = κe (cid:18)(cid:90) S ζe Φ (cid:19) − κ (cid:90) S βe Φ + (cid:90) S (cid:32) − ˇ κβ − κ ζ + 6 ρξ − ωe θ ( κ ) − ϑ ( e θ ( κ ) − κζ ) + Err [ d/ d (cid:63) / ξ ] (cid:33) e Φ + Err (cid:20) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:21) + (cid:90) S ˇ κ (cid:18) e ( ζ ) + (cid:18) κ − ϑ (cid:19) ζ (cid:19) e Φ − ˇ κ (cid:90) S (cid:18) e ( ζ ) + (cid:18) κ − ϑ (cid:19) ζ (cid:19) e Φ − κ Err (cid:20) e (cid:18)(cid:90) S ζe Φ (cid:19)(cid:21) . .1. PROOF OF THEOREM M0 Proof.
We have in ( ext ) M , see Proposition 2.2.8, e ( κ ) + 12 κ κ = − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ . Together with the following commutation relation[ e θ , e ] = 12 ( κ + ϑ ) e θ , we infer e ( e θ ( κ )) + κe θ ( κ ) + 12 ϑe θ ( κ ) + 12 κe θ ( κ ) = 2 d (cid:63) / d/ ζ + 2 e θ ( ρ ) − e θ ( ϑ ϑ ) + 2 e θ ( ζ ) . Also, we have in view of Proposition 2.2.19 the following identity e ( e θ ( κ )) − κe ( ζ ) = − d/ d (cid:63) / ξ − κβ + κ ζ − κe θ κ + 6 ρξ − ωe θ ( κ ) + Err[ d/ d (cid:63) / ξ ] . Next, in view of Corollary 2.2.10, we have in ( ext ) M e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) = (cid:90) S (cid:18) e ( e θ ( κ )) + (cid:18) κ − ϑ (cid:19) e θ ( κ ) (cid:19) e Φ e (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S (cid:18) e ( β ) + (cid:18) κ − ϑ (cid:19) β (cid:19) e Φ ,e (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S (cid:18) e ( β ) + (cid:18) κ − ϑ (cid:19) β (cid:19) e Φ + Err (cid:20) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:21) , and e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) − κe (cid:18)(cid:90) S ζe Φ (cid:19) = (cid:90) S (cid:18) e ( e θ ( κ )) + (cid:18) κ − ϑ (cid:19) e θ ( κ ) (cid:19) e Φ + Err (cid:20) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:21) − κ (cid:90) S (cid:18) e ( ζ ) + (cid:18) κ − ϑ (cid:19) ζ (cid:19) e Φ − κ Err (cid:20) e (cid:18)(cid:90) S ζe Φ (cid:19)(cid:21) = (cid:90) S (cid:18) e ( e θ ( κ )) − κe ( ζ ) + (cid:18) κ − ϑ (cid:19) ( e θ ( κ ) − κζ ) (cid:19) e Φ + Err (cid:20) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:21) + (cid:90) S ˇ κ (cid:18) e ( ζ ) + (cid:18) κ − ϑ (cid:19) ζ (cid:19) e Φ − ˇ κ (cid:90) S (cid:18) e ( ζ ) + (cid:18) κ − ϑ (cid:19) ζ (cid:19) e Φ − κ Err (cid:20) e (cid:18)(cid:90) S ζe Φ (cid:19)(cid:21) . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Together with the above identities for e ( e θ ( κ )) and e ( e θ ( κ )), as well as the Bianchiidentities of Proposition 2.2.8 for e ( β ) and e ( β ), we infer e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) = (cid:90) S (cid:32) κe θ ( κ ) − κe θ ( κ ) + 2 d (cid:63) / d/ ζ + 2 e θ ( ρ ) − e θ ( ϑ ϑ ) − ϑe θ ( κ ) + 2 e θ ( ζ ) (cid:33) e Φ ,e (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S (cid:18) − κβ + d/ α + ζα − ϑβ (cid:19) e Φ ,e (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S (cid:32) κβ + e θ ( ρ ) + 2 ωβ + 3 ηρ − ϑβ + ξα − ϑβ (cid:33) e Φ +Err (cid:20) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:21) , and e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) − κe (cid:18)(cid:90) S ζe Φ (cid:19) = (cid:90) S (cid:32) − d/ d (cid:63) / ξ − κβ − κ ζ + 6 ρξ − ωe θ ( κ ) − ϑ ( e θ ( κ ) − κζ ) + Err[ d/ d (cid:63) / ξ ] (cid:33) e Φ +Err (cid:20) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:21) + (cid:90) S ˇ κ (cid:18) e ( ζ ) + (cid:18) κ − ϑ (cid:19) ζ (cid:19) e Φ − ˇ κ (cid:90) S (cid:18) e ( ζ ) + (cid:18) κ − ϑ (cid:19) ζ (cid:19) e Φ − κ Err (cid:20) e (cid:18)(cid:90) S ζe Φ (cid:19)(cid:21) . Using in particular the fact that d (cid:63) / ( e Φ ) = 0, that d (cid:63) / is the adjoint of d/ , and the identity d (cid:63) / d/ = d/ d (cid:63) / + 2 K , we deduce e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) = (cid:90) S (cid:32) κe θ ( κ ) − κe θ ( κ ) + 4 Kζ + 2 e θ ( ρ ) − e θ ( ϑ ϑ ) − ϑe θ ( κ ) + 2 e θ ( ζ ) (cid:33) e Φ ,e (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S (cid:18) − κβ + ζα − ϑβ (cid:19) e Φ ,e (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S e θ ( ρ ) e Φ + 3 ρ (cid:90) S ηe Φ + (cid:90) S (cid:32) κβ + 2 ωβ + 3 η ˇ ρ − ϑβ + ξα − ϑβ (cid:33) e Φ +Err (cid:20) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:21) , .1. PROOF OF THEOREM M0 e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) = κe (cid:18)(cid:90) S ζe Φ (cid:19) − κ (cid:90) S βe Φ + (cid:90) S (cid:32) − ˇ κβ − κ ζ + 6 ρξ − ωe θ ( κ ) − ϑ ( e θ ( κ ) − κζ ) + Err[ d/ d (cid:63) / ξ ] (cid:33) e Φ +Err (cid:20) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:21) + (cid:90) S ˇ κ (cid:18) e ( ζ ) + (cid:18) κ − ϑ (cid:19) ζ (cid:19) e Φ − ˇ κ (cid:90) S (cid:18) e ( ζ ) + (cid:18) κ − ϑ (cid:19) ζ (cid:19) e Φ − κ Err (cid:20) e (cid:18)(cid:90) S ζe Φ (cid:19)(cid:21) . Finally, from the identity (2.1.21) for e θ ( K ) and the formula for K , we have (cid:90) S e θ ( ρ ) e Φ = − (cid:90) S e θ ( κκ ) e Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ . We deduce e (cid:18)(cid:90) S βe Φ (cid:19) = − (cid:90) S e θ ( κκ ) e Φ + 3 ρ (cid:90) S ηe Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ + (cid:90) S (cid:32) κβ + 2 ωβ + 3 η ˇ ρ − ϑβ + ξα − ϑβ (cid:33) e Φ + Err (cid:20) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:21) which concludes the proof of Lemma 4.1.1. Step 2.
Using the transport equations of Lemma 4.1.1 and the bootstrap assumptionson decay for k = 0 , ( ext ) M , we infer in that region, and in particular onΣ ∗ (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)ru + δ dec , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r + δ dec , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κκ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ηe Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15)r + δ dec + (cid:15) ru δ dec , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S ζe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15)ru + δ dec + (cid:15) u δ dec . Next, we focus on the two last estimates. Recall the following GCM conditions κ = 2 r , (cid:90) S ηe Φ = 0 on Σ ∗ . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
We deduce on Σ ∗ (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15)r + δ dec + (cid:15) ru δ dec . Also, projecting both Codazzi on e Φ , using d (cid:63) / ( e Φ ) = 0 and the fact that d (cid:63) / is the adjointof d/ , and using also the GCM condition for κ on Σ ∗ , we have on Σ ∗ (cid:90) S βe Φ = − (cid:90) S d (cid:63) / κe Φ − (cid:90) S ζκe Φ + 12 (cid:90) S ϑ ζe Φ , (cid:90) S ζe Φ = r (cid:90) S βe Φ + r (cid:90) S ϑ ζe Φ . Together with the bootstrap assumptions on decay for k = 0 , ( ext ) M , weinfer on Σ ∗ (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S ζe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)u + δ dec . We have thus on Σ ∗ (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)ru + δ dec , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r + δ dec , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15)r + δ dec + (cid:15) ru δ dec , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)ru + δ dec + (cid:15) u δ dec . In view of the behavior (3.3.4) of r on Σ ∗ , we infer on Σ ∗ (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u δ dec , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ru δ dec , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15) ru δ dec , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u δ dec . Step 3.
Let ν ∗ the unique tangent vector to Σ ∗ which can be written as ν ∗ = e + ae .1. PROOF OF THEOREM M0 a is a scalar function on Σ ∗ . Recall that there exists a constant c ∗ such thatΣ ∗ = { u + r = c ∗ } . We infer ν ∗ ( u + r ) = 0 and hence0 = e ( u + r ) + ae ( u + r ) = 2 ς + r κ + A ) + a r κ which yields a = − ς + r ( κ + A ) r κ . In view of the GCM condition on κ and the definition of the Hawking mass m , we haveon Σ ∗ κ = 2 r , κ = − r and hence, we have on Σ ∗ a = − ς + Υ − r A. The bootstrap assumptions on decay for k = 0 derivatives in ( ext ) M , the definition (2.2.22)of A , and the estimates for ς and Ω yield the rough estimate | a | (cid:46) . Together with the fact that ν ∗ = e + ae and the estimates of Step 2, we infer (cid:12)(cid:12)(cid:12)(cid:12) ν ∗ (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u δ dec , (cid:12)(cid:12)(cid:12)(cid:12) ν ∗ (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15) ru δ dec . Step 4.
Now, recall that we have the following GCM on the last sphere S ∗ = Σ ∗ ∩ C ∗ ofΣ ∗ (cid:90) S ∗ e θ ( κ ) e Φ = (cid:90) S ∗ βe Φ = 0 . Integrating backward from S ∗ the estimate for e θ ( κ ) of Step 3 yields on Σ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u δ dec . The estimates for Ω and ς are proved later in Proposition 3.4.3. Since the proof does not rely onTheorem M0, we may use it here. CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Plugging in the estimate for β of Step 3, we infer on Σ ∗ (cid:12)(cid:12)(cid:12)(cid:12) ν ∗ (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ru δ dec . Integrating backward from S ∗ yields on Σ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ru δ dec . Thus, at the first sphere S = Σ ∗ ∩ C of Σ ∗ , we have obtained (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . (4.1.2) Remark 4.1.2.
Note that the only bootstrap assumptions used in the proof of Theorem M0are the bootstrap assumption
BA-D on decay for k = 0 , derivatives. Indeed, to obtain (4.1.2) , we have only used, in Steps 1–4, the bootstrap assumption BA-D on decay for k = 0 , derivatives, while, from now on, we will only rely on (4.1.2) and the assumptions (4.1.1) on the initial data layer. This observation will allow us to use the conclusions ofTheorem M0, not only for the bootstrap spacetime M in Theorem M1–M5, but also forthe extended spacetime in the proof of Theorem M8, where the only assumptions are theone on decay (which are established for the extended spacetime in Theorem M7). Step 5.
On the sphere S = Σ ∗ ∩ C of Σ ∗ , we have in view of the GCM conditions of Σ ∗ and (4.1.2) κ = 2 r , d (cid:63) / d (cid:63) / κ = 0 , d (cid:63) / d (cid:63) / µ = 0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . (4.1.3)We consider the transition functions ( f, f , λ ) from the frame of the outgoing part ( ext ) L of the initial data layer to the frame of ( ext ) M . Since • S is a sphere of ( ext ) M in the initial data layer, • S is a sphere of the GCM hypersurface Σ ∗ , • the estimate (4.1.3) holds on S ,we can invoke the GCM corollary Rigidity II and III of section 3.7.4 with the choice ◦ (cid:15) = ◦ δ = (cid:15) , s max = k large + 5, and with the background foliation being the one of theoutgoing part ( ext ) L of the initial data layer. We obtain (cid:107) d ≤ k large +6 ( f, f , log λ ) (cid:107) L ( S ) + sup S (cid:12)(cid:12)(cid:12)(cid:12) mm − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) .1. PROOF OF THEOREM M0 S , we deducesup S (cid:18) r | d ≤ k large +4 ( f, f , log λ ) | + (cid:12)(cid:12)(cid:12)(cid:12) mm − (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) r − ( ext ) r L (cid:12)(cid:12)(cid:19) (cid:46) (cid:15) . (4.1.4) Step 6.
Recall from Corollary 2.3.7 that ( f , f, log( λ )) satisfy the following transportequations along C λ − e (cid:48) ( rf ) = E (cid:48) ( f, Γ) ,λ − e (cid:48) (log( λ )) = E (cid:48) ( f, Γ) ,λ − e (cid:48) (cid:16) rf − r e (cid:48) θ (log( λ )) + rf Ω (cid:17) = E (cid:48) ( f, f , λ, Γ) , where, in view of the form of E (cid:48) , E (cid:48) , E (cid:48) in Corollary 2.3.7 and the estimates (4.1.1) forthe Ricci coefficients of the outgoing part ( ext ) L of the initial data layer, we have | d k E (cid:48) ( f, Γ) | + | d k E (cid:48) ( f, Γ) | (cid:46) (cid:15) r + | d ≤ k f | for k ≤ k large + 5 on C and | d k E (cid:48) ( f, f , λ, Γ) | (cid:46) (cid:15) r + | d ≤ k +1 f | + | d ≤ k +1 log( λ ) | for k ≤ k large + 5 on C . Next, recall from Lemma 2.2.14 the following commutator identity[ T , e ] = (cid:18)(cid:16) ω − mr (cid:17) − m r (cid:18) κ − r (cid:19) + e ( m ) r (cid:19) e + ( η + ζ ) e θ while from Lemma 2.2.13, we have schematically[ d /, e ] = (cid:0) ˇ κ, ϑ (cid:1) d / + (cid:0) ζ, rβ (cid:1) Together with the fact that λ − e (cid:48) = e + f e θ + f e , the commutator above identities for [ T, e ] and [ d /, e ], as well as the estimates (4.1.1) forthe Ricci coefficients and curvature components of the outgoing part ( ext ) L of the initialdata layer, we infer, for k ≤ k large + 5, | d k [ T, λ − e (cid:48) ] h | + | d k [ d /, λ − e (cid:48) ] h | (cid:46) (cid:15) r | d ≤ k +1 h | + 1 r | d ≤ k ( f d h ) | + 1 r | d ≤ k ( h d f ) | + | d ≤ k ( f d h ) | + | d ≤ k ( hf d f ) | . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
By commuting first the transport equations in the direction λ − e (cid:48) with ( T, d / ) k , and byusing these transport equations to recover the e derivatives, we deduce λ − e (cid:48) ( r d k f ) = E (cid:48) ,k ( f, Γ) ,λ − e (cid:48) ( d k log( λ )) = E (cid:48) ,k ( f, Γ) ,λ − e (cid:48) (cid:16) r d k (cid:0) f − re (cid:48) θ (log( λ )) + f Ω (cid:1)(cid:17) = E (cid:48) ,k ( f, f , λ, Γ) , where we have | E (cid:48) ,k ( f, Γ) | + | E (cid:48) ,k ( f, Γ) | (cid:46) (cid:15) r + | d ≤ k f | for k ≤ k large + 5 on C and | E (cid:48) ,k ( f, f , λ, Γ) | (cid:46) (cid:15) r + | d ≤ k +1 f | + | d ≤ k +1 log( λ ) | + | d ≤ k f | for k ≤ k large + 5 on C . This allows us to propagate the estimates for ( f, f , λ ) in (4.1.4) on S to any sphere on C , and hence sup C (cid:16) r | d ≤ k large +4 ( f, log λ ) | (cid:17) + sup C (cid:16) r | d ≤ k large +3 f | (cid:17) (cid:46) (cid:15) . (4.1.5) Step 7.
In view of (4.1.5), the change of frame formulas of Proposition 2.3.4, and theestimates (4.1.1) for the Ricci coefficients and curvature components of the outgoing part ( ext ) L of the initial data layer, we obtainmax ≤ k ≤ k large (cid:40) sup C (cid:104) r + δ B (cid:0) | d k ( ext ) α | + | d k ( ext ) β | (cid:1) + r + δ B | d k − e ( ( ext ) α ) | (cid:105) (4.1.6)+ sup C (cid:20) r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) ( ext ) ρ + 2 m r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r | d k ( ext ) β | + r | d k ( ext ) α | (cid:21) (cid:41) (cid:46) (cid:15) , as well as max ≤ k ≤ k large (cid:40) sup C (cid:104) r (cid:0) | d k ( ext ) (ˇ κ, ϑ, ζ, ˇ κ ) | + r | d k ( ext ) ϑ | (cid:1) (cid:105)(cid:41) (cid:46) (cid:15) . Now, according to Proposition 2.2.16, we have in ( ext ) M ( ext ) e ( ( ext ) m ) = ( ext ) r π (cid:90) S (cid:32) ( ext ) ˇ κ ( ext ) ˇ ρ + 2 ( ext ) e θ ( ( ext ) κ ) ( ext ) ζ − ( ext ) κ ( ( ext ) ϑ ) − ( ext ) ˇ κ ( ext ) ϑ ( ext ) ϑ + 2 ( ext ) κ ( ( ext ) ζ ) (cid:33) , .1. PROOF OF THEOREM M0 C r (cid:12)(cid:12)(cid:12) ( ext ) e ( ( ext ) m ) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . This allows us to propagate the estimates for ( ext ) m in (4.1.4) on S to any sphere on C ,and hence sup C (cid:12)(cid:12)(cid:12)(cid:12) ( ext ) mm − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . (4.1.7)Also, we have λ − e (cid:48) (cid:18) log (cid:18) ( ext ) r ( ext ) r L (cid:19)(cid:19) = λ − ext ) e ( ( ext ) r ) ( ext ) r − ( ext ) ( e ) ( ( ext ) r L ) ( ext ) r L − f ( ext ) ( e ) ( ( ext ) r L ) ( ext ) r = 12 (cid:16) λ − κ (cid:48) − κ (cid:17) + λ − κ (cid:48) −
12 ˇ κ − rf κ + A )= 12 (cid:16) d/ (cid:48) ( f ) + Err( κ, κ (cid:48) ) (cid:17) + λ − κ (cid:48) −
12 ˇ κ − rf κ + A )where we have denoted with primes the quantities with respect to the frame of ( ext ) M and without primes the quantities with respect to the frame of ( ext ) L . Together with theabove estimates, this yieldssup C r (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) log (cid:18) ( ext ) r ( ext ) r L (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . This allows us to propagate the estimates for ( ext ) r − ( ext ) r L in (4.1.4) on S to any sphereon C , and hence sup C (cid:12)(cid:12) ( ext ) r − ( ext ) r L (cid:12)(cid:12) (cid:46) (cid:15) . Step 8.
Recall that • ( ( ext ) e , ( ext ) e , ( ext ) e θ ) denotes the null frame of ( ext ) M , • ( ( int ) e , ( int ) e , ( ext ) e θ ) denotes the null frame of ( int ) M , • ( ( ext ) ( e ) , ( ext ) ( e ) , ( ext ) ( e ) θ ) denotes the null frame of ( ext ) L , • ( ( int ) ( e ) , ( int ) ( e ) , ( int ) ( e ) θ ) denotes the null frame of ( int ) L .84 CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Also, recall that the timelike hyper surface T is given by T = { ( ext ) r = r T } where 2 m (cid:18) δ H (cid:19) ≤ r T ≤ m (cid:18) δ H (cid:19) to that T ⊂ ( int ) L ∩ ( ext ) L , and recall that the frame of ( int ) M is initialed on T asfollows ( int ) e = λ ( ext ) e , ( int ) e = λ − ext ) e , ( int ) e θ = ( ext ) e θ on T where λ = ( ext ) λ = 1 − ( ext ) m ( ext ) r . Denoting • by ( f, f , λ ) the transition functions from the frame of the outgoing part ( ext ) L ofthe initial data layer to the frame of ( ext ) M as in Steps 5 to 7, • by ( f (cid:48) , f (cid:48) , λ (cid:48) ) the transition functions from the frame of the ingoing part ( int ) L ofthe initial data layer to the frame of ( int ) M , • by ( ˜ f, ˜ f , ˜ λ ) the transition functions on ( int ) L ∩ ( ext ) L from the frame outgoing part ( ext ) L of the initial data layer to the frame of the ingoing part ( int ) L of the initialdata layer,we obtain, using also that C ∩ C ⊂ T ,sup C ∩C (cid:16) | d ≤ k large +3 ( f (cid:48) , f (cid:48) , log( λ (cid:48) )) | (cid:17) (cid:46) sup C ∩C (cid:16) | d ≤ k large +3 ( f, f , log( λ )) | (cid:17) + sup C ∩C (cid:16) | d ≤ k large +3 ( ˜ f, ˜ f , log(Υ − ˜ λ )) | (cid:17) + sup C ∩C (cid:16) | d ≤ k large +3 log(Υ − Υ) | (cid:17) where we have denotedΥ = 1 − m ext ) r L , Υ = 1 − ( ext ) m ( ext ) r . Together with the control of ( ˜ f, ˜ f , log(Υ − ˜ λ )) provided on ( int ) L ∩ ( ext ) L by the estimates(4.1.1), the estimates (4.1.5) for ( f, f , λ ), and the estimates ( ext ) m − m and ( ext ) r − ( ext ) r L obtained in Step 7, we infersup C ∩C (cid:16) | d ≤ k large +3 ( f (cid:48) , f (cid:48) , log( λ (cid:48) )) | (cid:17) (cid:46) (cid:15) . .2. CONTROL OF AVERAGES AND OF THE HAWKING MASS Step 9.
Similarly to Step 6, we propagate the estimate for ( f (cid:48) , f (cid:48) , log( λ (cid:48) ) on C ∩ C provided by Step 8 to C using the analog of Corollary 2.3.7 in the ingoing direction e .We obtain the following analog of (4.1.5)sup C (cid:16) | d ≤ k large +3 ( f , log λ ) | (cid:17) + sup C (cid:16) r | d ≤ k large +2 f | (cid:17) (cid:46) (cid:15) . Together with the change of frame formulas of Proposition 2.3.4, and the estimates (4.1.1)for the Ricci coefficients and curvature components of the ingoing part ( int ) L of the initialdata layer, we obtainmax ≤ k ≤ k large sup C (cid:34) | d k ( int ) α | + | d k ( int ) β | + (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) ( int ) ρ + 2 m r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + | d k ( int ) β | + | d k ( int ) α | (cid:35) (cid:46) (cid:15) , (4.1.8)as well as max ≤ k ≤ k large (cid:40) sup C | d k ( int ) (ˇ κ, ϑ, ζ, ˇ κ, ϑ ) | (cid:41) (cid:46) (cid:15) . Also, since we have as a consequence of the initialization on T of the ingoing geodesicfoliation of ( int ) M ( int ) m = ( ext ) m on C ∩ C we infer from the control of ( ext ) m provided by Step 7 | ( int ) m − m | (cid:46) (cid:15) on C ∩ C . We then propagate, similarly to Step 7, this bound to C and obtainsup C (cid:12)(cid:12) ( int ) m − m (cid:12)(cid:12) (cid:46) (cid:15) . Together with (4.1.6), (4.1.7) and (4.1.8), this concludes the proof of Theorem M0.
In this section, we prove Lemma 3.4.1 and Lemma 3.4.2.86
CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Step 1.
We start with the control of ρ on M . Recall the identity (2.2.12) ρ + 2 mr = 14 ϑϑ. Thus, in view of the bootstrap assumptions
BA-D , BA-E , we have, (cid:12)(cid:12)(cid:12) ρ + 2 mr (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) min { r − u − − δ dec , r − u − − δ dec } in ( ext ) M , (cid:12)(cid:12)(cid:12) ρ + 2 mr (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec in ( int ) M . Differentiating the equation with respect to e , e we derive, e (cid:18) ρ + 2 mr (cid:19) = 14 e ( ϑ ) ϑ + ϑe ( ϑ ) + l.o.t. ,e (cid:18) ρ + 2 mr (cid:19) = 14 e ( ϑ ) ϑ + ϑe ( ϑ ) + l.o.t. ,e θ (cid:18) ρ + 2 mr (cid:19) = 0 . Taking higher derivatives in e , e and making use of the bootstrap assumptions BA-D , BA-E , we derive in ( ext ) M , (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) min { r − u − − δ dec , r − u − − δ dec } , (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − u − / − δ dec , and in ( int ) M , (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . In particular,sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) , sup ( int ) M u + δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + sup ( int ) M u + δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) ρ + 2 mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) .2. CONTROL OF AVERAGES AND OF THE HAWKING MASS Step 2.
Next, we proceed with the control of κ in ( ext ) M . Recalling Lemma 2.2.17, westart with e (cid:18) κ − r (cid:19) + 12 κ (cid:18) κ − r (cid:19) = − ϑ + 12 ˇ κ . (4.2.1)In view of Corollary 2.2.12 we deduce, from the first equation, e (cid:18) r (cid:18) κ − r (cid:19)(cid:19) = − r (cid:16) ϑ + ˇ κ (cid:17) . (4.2.2)Making use of the GCM condition κ = 2 r on Σ ∗ , which yields κ = 2 r on Σ ∗ , we deduce, integrating (4.2.2) with respect to r along C u from Σ ∗ ,sup ( ext ) M u δ dec r (cid:12)(cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) (cid:46) (cid:15) . Also, making use of the bootstrap assumptions
BA-D , BA-E we easily deduce,sup ( ext ) M u δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small +1 (cid:37) (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) (cid:46) (cid:15) , sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large +1 (cid:37) (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) (cid:46) (cid:15) . We next commute (4.2.2) with e and derive, e e (cid:18) r (cid:18) κ − r (cid:19)(cid:19) = e (cid:18) r (cid:18) ϑ + 12 ˇ κ (cid:19)(cid:19) − [ e , e ] (cid:18) r (cid:18) κ − r (cid:19)(cid:19) = e (cid:18) r (cid:18) ϑ + 12 ˇ κ (cid:19)(cid:19) − ω (cid:18) r (cid:18) κ − r (cid:19)(cid:19) − ζ (cid:18) r (cid:18) κ − r (cid:19)(cid:19) . It is thus easy to see that we can prove estimates of the typesup ( ext ) M u δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small +1 (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) (cid:46) (cid:15) , sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large +1 (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) (cid:46) (cid:15) , CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS provided that we can check that,sup ( ext ) M u δ dec r (cid:12)(cid:12)(cid:12)(cid:12) e ≤ k small +13 (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) (cid:46) (cid:15) , sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12)(cid:12) e ≤ k large +13 (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) (cid:46) (cid:15) . The difficulty in this case is to make sure that we can control terms of the type, e k +13 (cid:18) r (cid:18) e k +13 ( ϑ ) + 12 e k +13 (ˇ κ ) (cid:19)(cid:19) using only at most k derivatives of ˇΓ , ˇ R . To see this we note that, e ( ϑ ) = e ϑ − ( ˇΩ κ − ˇΩ ˇ κ ) ϑ + ˇ κ ˇ ϑ ,e (ˇ κ ) = e ˇ κ − ( ˇΩ κ − ˇΩ ˇ κ )ˇ κ + ˇ κ ˇ κ , and, e ( ϑ ) + 12 κϑ − ωϑ = − d (cid:63) / ζ − κϑ + 2 ζ ,e ˇ κ + 12 κ ˇ κ = − µ − κ ˇ κ + 2(ˇ ωκ + ω ˇ κ ) + ˇΩ κ κ + Err[ e ˇ κ ] , Err[ e ˇ κ ] : = 2( ζ − ζ ) + 2(ˇ ω ˇ κ − ˇ ω ˇ κ ) −
12 ˇ κ ˇ κ −
12 ˇ κ ˇ κ − ˇΩˇ κ κ. (4.2.3)We thus derive,sup ( ext ) M u δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small +1 (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large +1 (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Step 3.
We next estimate κ in ( ext ) M making use of the identity (2.2.14) derived inconnection to the Hawking mass κ + 2Υ r = 2Υ rκ (cid:18) κ − r (cid:19) − κ ˇ κ ˇ κ. Thus, in view of the estimates for κ derived in step 2 we easily infer that,sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) κ + 2Υ r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) κ + 2Υ r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . as desired. .2. CONTROL OF AVERAGES AND OF THE HAWKING MASS Step 4.
We estimate ω in ( ext ) M based on the following identity in Lemma 2.2.17 e (cid:18) κ − r (cid:19) + 12 κ (cid:18) κ − r (cid:19) = 2 ω (cid:18) κ − r (cid:19) + 4 r (cid:16) ω − mr (cid:17) + 2 (cid:18) ρ + 2 mr (cid:19) − κ (cid:18) κ − r (cid:19) ˇΩ + 2ˇ ω ˇ κ − ϑϑ + 2 ζ + 12 ˇΩ (cid:16) − ϑ + ˇ κ (cid:17) − ˇΩ( e (ˇ κ ) + κ ˇ κ ) + 12 ˇ κ ˇ κ − r ˇΩˇ κ, which we rewrite as ω − mr = r (cid:40) e (cid:18) κ − r (cid:19) + 12 κ (cid:18) κ − r (cid:19) − ω (cid:18) κ − r (cid:19) − (cid:18) ρ + 2 mr (cid:19) + 12 κ (cid:18) κ − r (cid:19) ˇΩ − ω ˇ κ + 12 ϑϑ − ζ −
12 ˇΩ (cid:16) − ϑ + ˇ κ (cid:17) + ˇΩ( e (ˇ κ ) + κ ˇ κ ) −
12 ˇ κ ˇ κ + 1 r ˇΩˇ κ (cid:41) . Using the estimates of ρ in Step 1, the estimates for κ in Step 2, as well as our bootstrapassumptions on decay and energy, we easily derivesup ( ext ) M u δ dec r (cid:12)(cid:12)(cid:12) d ≤ k small (cid:16) ω − mr (cid:17)(cid:12)(cid:12)(cid:12) + sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12) d ≤ k large (cid:16) ω − mr (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Remark 4.2.1.
It is to estimate k large derivatives of ω − mr − that we had to control k large + 1 derivatives of κ − /r is Step 2. Step 5.
We estimate Ω in ( ext ) M . First we need the control of Ω on Σ ∗ . To this end, werecall that s is initialized on Σ ∗ by s = r so that ν ( s − r ) = 0 on Σ ∗ , ν = e + ae , where the scalar function a is such that the vectorfield ν is tangent to Σ ∗ . On the otherhand, we have e ( s ) = 1 and e ( r ) = r κ = 1 on Σ ∗ where we used the GCM condition κ = 2 /r on Σ ∗ . We infer e ( s ) = e ( r ) on Σ ∗ andhence Ω = e ( r ) on Σ ∗ . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
This yields Ω = e ( r ) = rκ A, and hence, in view of the estimate for κ of step 3, the fact that A contains only quadraticterms in view of the formula for A , and in view of the bootstrap assumptions on decayand energy, we infersup Σ ∗ u δ dec r (cid:12)(cid:12)(cid:12) d ≤ k small (cid:16) Ω − mr (cid:17)(cid:12)(cid:12)(cid:12) + sup Σ ∗ u + δ dec r (cid:12)(cid:12)(cid:12) d ≤ k large (cid:16) Ω − mr (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Then, we use e (Ω) = − ω and Corollary 2.2.11 to obtain e (Ω) = − ω + ˇ κ ˇΩand hence e (Ω + Υ) = − (cid:16) ω − mr (cid:17) + mr (cid:18) κ − r (cid:19) + ˇ κ ˇΩ − e ( m ) r . Commuting with d , integrating from Σ ∗ where we have controlled Ω above, and using theestimates of Step 2 for κ , Step 4 for ω , the bootstrap assumptions, and the estimates for e ( m ) of Lemma 3.4.2 (which do not depend on the control of Ω), we infersup ( ext ) M u δ dec r (cid:12)(cid:12)(cid:12) d ≤ k small (cid:16) Ω − mr (cid:17)(cid:12)(cid:12)(cid:12) + sup ( ext ) M u + δ dec r (cid:12)(cid:12)(cid:12) d ≤ k large (cid:16) Ω − mr (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Step 6.
Next, we control ( int ) κ on the cylinder T . From the initialization of the frameof ( int ) M on T , we have ( int ) r = ( ext ) r, ( int ) κ = Υ ( ext ) κ, ( int ) κ = Υ − ext ) κ on T . Also, making use of the identity (2.2.14) derived in connection to the Hawking mass, wehave ( ext ) κ + 2Υ r = 2Υ r ( ext ) κ (cid:18) ( ext ) κ − r (cid:19) − ( ext ) κ ( ext ) ˇ κ ( ext ) ˇ κ. We deduce ( int ) κ + 2 r = Υ − (cid:18) ( ext ) κ + 2Υ r (cid:19) = 2 r ( ext ) κ (cid:18) ( ext ) κ − r (cid:19) − Υ − ext ) κ ( ext ) ˇ κ ( ext ) ˇ κ on T . .2. CONTROL OF AVERAGES AND OF THE HAWKING MASS T we remark that the vectorfield T T = e − e ( r ) e ( r ) e = e − κ + Aκ e , together with e θ , spans the tangent space to T . The transversal derivatives, on the otherhand, can be determined with help of the equation, e (cid:18) κ + 2 r (cid:19) + 12 κ (cid:18) κ + 2 r (cid:19) = − ϑ + 12 ˇ κ . adapted to the ( int ) M foliation. Making use of the estimates for ( ext ) κ in ( ext ) M derivedin Step 2 and the bootstrap assumptions, we infer that,sup T u + δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small +1 (cid:18) ( int ) κ + 2 r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + sup T u + δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large +1 (cid:18) ( int ) κ + 2 r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) + sup T u + δ dec (cid:12)(cid:12)(cid:12) d k large +1 (cid:16) ( ext ) ˇ κ ( ext ) ˇ κ (cid:17)(cid:12)(cid:12)(cid:12) Now, in view of the transport equations for ( ext ) e ( ( ext ) ˇ κ ), ( ext ) e ( ( ext ) ˇ κ ), ( ext ) e ( ( ext ) ˇ κ )and ( ext ) e ( ( ext ) ˇ κ ), as well as the bootstrap assumptions, we havesup T u + δ dec (cid:12)(cid:12)(cid:12) d k large +1 (cid:16) ( ext ) ˇ κ ( ext ) ˇ κ (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) + sup T u + δ dec (cid:12)(cid:12)(cid:12) ( ext ) ˇ κ d k large d/ ( ( ext ) ζ ) (cid:12)(cid:12)(cid:12) + sup T u + δ dec (cid:12)(cid:12)(cid:12) ( ext ) ˇ κ d k large d/ ( ( ext ) ζ ) (cid:12)(cid:12)(cid:12) + sup T u + δ dec (cid:12)(cid:12)(cid:12) ( ext ) ˇ κ d k large d/ ( ( ext ) ξ ) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) + sup T u + δ dec (cid:12)(cid:12)(cid:12) d (cid:63) / ext ) ˇ κ d k large ( ext ) ζ (cid:12)(cid:12)(cid:12) + sup T u + δ dec (cid:12)(cid:12)(cid:12) d (cid:63) / ext ) ˇ κ d k large ( ext ) ζ (cid:12)(cid:12)(cid:12) + sup T u + δ dec (cid:12)(cid:12)(cid:12) d (cid:63) / ext ) ˇ κ d k large ( ext ) ξ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) where we have integrated d/ by parts and used that d (cid:63) / is its adjoint. We infersup T u + δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small +1 (cid:18) ( int ) κ + 2 r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + sup T u + δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large +1 (cid:18) ( int ) κ + 2 r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Step 7.
From now on, we only work with the frame of ( int ) M . Starting with the equation, e (cid:18) κ + 2 r (cid:19) + 12 κ (cid:18) κ + 2 r (cid:19) = − ϑ + 12 ˇ κ . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Using the estimates of step 5 we can then proceed precisely as in Step 2 ( using the ( int ) M counterpart of the equations (4.2.3)) to derive,sup ( int ) M u δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small +1 (cid:18) κ + 2 r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + sup ( int ) M u + δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large +1 (cid:18) κ + 2 r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Step 8.
Finally, we estimate the remaining averages in ( int ) M , i.e. κ and ω . To estimate κ we make use once more of the identity, κ − r = − rκ (cid:18) κ + 2 r (cid:19) − κ ˇ κ ˇ κ. Making use of the estimates of κ in Step 5 as well as the bootstrap assumptions for ˇ κ andˇ κ we easily derive,sup ( int ) M u δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k small (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + sup ( int ) M u + δ dec (cid:12)(cid:12)(cid:12)(cid:12) d ≤ k large (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Step 9.
To estimate ω we proceed as in Step 4 by making use of the identity ω + mr = r (cid:40) e (cid:18) κ + 2 r (cid:19) + 12 κ (cid:18) κ + 2 r (cid:19) − ω (cid:18) κ + 2 r (cid:19) − (cid:18) ρ + 2 mr (cid:19) + 12 κ (cid:18) κ + 2 r (cid:19) ˇΩ − ω ˇ κ + 12 ϑϑ − ζ −
12 ˇΩ (cid:16) − ϑ + ˇ κ (cid:17) + ˇΩ( e (ˇ κ ) + κ ˇ κ ) −
12 ˇ κ ˇ κ + 1 r ˇΩˇ κ (cid:41) . Thus, in view of the estimates of ρ in Step 1, the estimates for κ in Step 5, the estimatesof κ above , as well as the bootstrap assumptions BA-D and
BA-E , we deduce,sup ( int ) M u δ dec (cid:12)(cid:12)(cid:12) d ≤ k small (cid:16) ω + mr (cid:17)(cid:12)(cid:12)(cid:12) + sup ( int ) M u + δ dec (cid:12)(cid:12)(cid:12) d ≤ k large (cid:16) ω + mr (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Step 10.
It remains to estimate Ω in ( int ) M . First we need the control of Ω on T . Tothis end, we recall that s is initialized on T by s = r so that T T ( s − r ) = 0 on T , T T = e − κ + Aκ e , It is to estimate k large derivatives of ω + m/r that we made sure to control k large + 1 derivatives of κ + 2 /r . .2. CONTROL OF AVERAGES AND OF THE HAWKING MASS T . On the otherhand, we have e ( s ) = − e ( r ) = rκ/
2, and henceΩ = e ( r ) + κ + Aκ ( − − e ( r )) = r κ + A ) (cid:18) κ − r κ (cid:19) on T This yields Ω = r κ + A ) (cid:18) κ − r κ (cid:19) on T , and hence, in view of the estimate for κ of step 7, the estimate for κ of step 8, the factthat A contains only quadratic terms in view of the formula for A , and in view of thebootstrap assumptions on decay and energy, we infersup T u δ dec (cid:12)(cid:12) d ≤ k small (cid:0) Ω − Υ (cid:1)(cid:12)(cid:12) + sup T u + δ dec (cid:12)(cid:12) d ≤ k large (cid:0) Ω − Υ (cid:1)(cid:12)(cid:12) (cid:46) (cid:15) . Then, we use the analog of the transport equation used to estimate Ω in ( ext ) M , i.e. e (Ω − Υ) = 2 (cid:16) ω + mr (cid:17) − mr (cid:18) κ + 2 r (cid:19) + ˇ κ ˇΩ + 2 e ( m ) r . Commuting with d , integrating from T where we have controlled Ω above, and using theestimates of Step 2 for κ , Step 4 for ω , the bootstrap assumptions, and the estimates for e ( m ) of Lemma 3.4.2 (which do not depend on the control of Ω), we infersup ( int ) M u δ dec (cid:12)(cid:12) d ≤ k small (cid:0) Ω − Υ (cid:1)(cid:12)(cid:12) + sup ( int ) M u + δ dec (cid:12)(cid:12) d ≤ k large (cid:0) Ω − Υ (cid:1)(cid:12)(cid:12) (cid:46) (cid:15) . This concludes the proof of Lemma 3.4.1.
Step 1.
We start with the control of e ( m ) and e ( m ) in ( ext ) M . According to Proposition2.2.16 we have in ( ext ) M e ( m ) = r π (cid:90) S Err , (4.2.4)94 CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS and e ( m ) = (cid:0) − ς − ˇ ς (cid:1) r π (cid:90) S Err + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) r π (cid:90) S Err + ς − r π (cid:90) S ˇ ς (cid:0) ρ ˇ κ + 2 ˇ ρκ + 2 κ d/ η + 2 κ d/ ξ + Err (cid:1) − ς − r π (cid:90) S (Ωˇ ς + ˇΩ ς ) (2 ρ ˇ κ + 2 ˇ ρκ − κ d/ ζ + Err ) − mr ς − (cid:104) − ˇ ς ˇ κ + Ω ˇ ς ˇ κ + ˇΩ ςκ (cid:105) , (4.2.5)where Err := 2ˇ κ ˇ ρ + 2 e θ ( κ ) ζ − κϑ −
12 ˇ κϑϑ + 2 κζ , Err := 2 ˇ ρ ˇ κ − e θ ( κ ) η − e θ ( κ ) ξ −
12 ˇ κϑϑ + 2 κη + 2 κ (cid:0) η − ζ (cid:1) ξ − κϑ , Err := 2 ˇ ρ ˇ κ − κϑ − κϑϑ + 2 κζ , Err := 2 ˇ ρ ˇ κ + κ (cid:18) η − ϑϑ (cid:19) + 2 κ (cid:0) η − ζ (cid:1) ξ − κϑ . Thus, according to the bootstrap assumption
BA-D on decay, we deduce, | e ( m ) | (cid:46) (cid:15) r − u − − δ dec , | e ( m ) | (cid:46) (cid:15) u − − δ dec . Moreover, differentiating the equations with respect to e , e and making use of bothbootstrap assumptions BA-D BA-E on decay and energy, and integrating by part oncethe e θ derivative for the terms involving e θ ( κ ) and e θ ( κ ) when they contain top orderderivatives, we infer that, max ≤ k ≤ k small sup ( ext ) M r u δ dec | d k e ( m ) | (cid:46) (cid:15) , max ≤ k ≤ k large sup ( ext ) M (cid:16) r u + δ dec + ru δ dec (cid:17) | d k e ( m ) | (cid:46) (cid:15) , as well as max ≤ k ≤ k small sup ( ext ) M u δ dec | d k e ( m ) | (cid:46) (cid:15) , max ≤ k ≤ k large sup ( ext ) M u δ dec | d k e ( m ) | (cid:46) (cid:15) , consistent with the statement of the lemma. .2. CONTROL OF AVERAGES AND OF THE HAWKING MASS Step 2.
We derive the estimates on ( int ) M . According to the analogue of Proposition2.2.16 in the situation of the incoming geodesic foliations of ( int ) M , and proceeding as inStep 1, we easily derive,max ≤ k ≤ k large sup ( int ) M u δ dec (cid:16) | d k e ( m ) | + | d k e ( m ) | (cid:17) (cid:46) (cid:15) (cid:46) (cid:15) . (4.2.6) Step 3.
We estimate m − m in ( ext ) M .First, recall from Theorem M0 that we havesup C ∪C | m − m | (cid:46) (cid:15) m . (4.2.7)We start with the control in ( ext ) M . Note that ( ext ) M is covered by integral curves of e starting from C . Thus, integrating the e m equation and making use of the estimatesup C | m − m | (cid:46) (cid:15) m as well as the fact that e ( u ) = 2, we easily deduce that,sup ( ext ) M | m − m | (cid:46) (cid:15) m + (cid:15) (cid:46) (cid:15) m . Step 4.
We estimate | m − m | on T .In view of our initialization of the ingoing geodesic foliation of ( int ) M on T , ( int ) κ ( int ) κ = ( ext ) κ ( ext ) κ on T . Since the spheres of both foliations agree on T , we infer from the definition of the Hawkingmass, ( int ) m = ( ext ) m on T . Using the estimate for ( ext ) m we infer thatsup T | ( int ) m − m | (cid:46) (cid:15) m . Step 5.
We estimate | m − m | on ( int ) M .Note first that in ( int ) M , e ( r ) + 1 = r κ + 1 = r (cid:18) κ + 2 r (cid:19) . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Thus, in view of the estimate for κ + r derived in Lemma 3.4.1sup ( int ) M | e ( r ) + 1 | (cid:46) (cid:15) . Thus integrating the estimate (4.2.6) in r ∈ [2 m (1 − δ H ) , r T ], where we recall that r T ≤ m (1 + 2 δ H ), we derive, sup ( int ) M | m − m | (cid:46) (cid:15) m . Since M = ( ext ) M ∪ ( int ) M we infer that,sup M | m − m | (cid:46) (cid:15) m . This concludes the proof of Lemma 3.4.2.
The goal of this section is to prove Propositions 3.4.3 and 3.4.4. In both cases, the firsttwo claims, on the form of the spacetime metric in the corresponding coordinates systemas well as on the expression of the coordinates vectorfield with respect to the null frame( e , e , e θ ), is already proved in Propositions 2.2.23 and 2.2.26. So we only focus on thethird claim, i.e. on estimating ˇΩ, ˇΩ, ς , ς , γ , b , b and e Φ . The proof of Propositions 3.4.3and 3.4.4 thus reduces to the proof of the following lemma. Lemma 4.3.1.
Let θ ∈ [0 , π ] be the Z -invariant scalar on M defined by (2.2.52) , i.e. θ = cot − ( re θ (Φ)) . (4.3.1) Let b = e ( θ ) , b = e ( θ ) , γ = 1( e θ ( θ )) . (4.3.2) Then, we have max ≤ k ≤ k small sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) (cid:16)(cid:12)(cid:12)(cid:12) d k (cid:16) γr − (cid:17)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12) d k b (cid:12)(cid:12)(cid:17) (cid:46) (cid:15), max ≤ k ≤ k small sup ( ext ) M u δ dec (cid:0)(cid:12)(cid:12) d k ˇΩ (cid:12)(cid:12) + (cid:12)(cid:12) d k ( ς − (cid:12)(cid:12) + r (cid:12)(cid:12) d k b (cid:12)(cid:12)(cid:1) (cid:46) (cid:15), max ≤ k ≤ k small sup ( int ) M u δ dec (cid:16)(cid:12)(cid:12) d k ˇΩ (cid:12)(cid:12) + (cid:12)(cid:12) d k ( ς − (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) d k (cid:16) γr − (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) d k b (cid:12)(cid:12) + (cid:12)(cid:12) d k b (cid:12)(cid:12)(cid:17) (cid:46) (cid:15). .3. CONTROL OF COORDINATES SYSTEMS Also, e Φ satisfies max ≤ k ≤ k small sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) e Φ r sin θ − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k small sup ( int ) M u δ dec (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) e Φ r sin θ − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15). Proof.
We prove the estimates in ( ext ) M . The proof in ( int ) M is similar and left to thereader. Step 1.
We start with the estimate for ˇΩ. Recall that d (cid:63) / ˇΩ = ξ so that the bootstrap assumptions for ξ imply on any 2-sphere of the foliation of ( ext ) M and for any k ≤ k small r (cid:107) d k r d (cid:63) / ˇΩ (cid:107) L ( S ) + (cid:107) d k r d (cid:63) / ˇΩ (cid:107) L ( S ) (cid:46) r sup S | d k ξ | (cid:46) (cid:15)ru − − δ dec . In view of the commutation formulas of Lemma 2.2.13 and of Proposition 2.1.25, togetherwith the bootstrap assumptions, we infer any k ≤ k small , schematically,[ d k , r d (cid:63) / ] = O ( (cid:15) ) d ≤ k + O (1) d ≤ k − , and hence, r (cid:107) r d (cid:63) / d k ˇΩ (cid:107) L ( S ) + (cid:107) r d (cid:63) / d k ˇΩ (cid:107) L ( S ) (cid:46) (cid:15)ru − − δ dec + (cid:15) (cid:107) d ≤ k ˇΩ (cid:107) L ( S ) + (cid:15)r (cid:107) d ≤ k ˇΩ (cid:107) L ( S ) + (cid:107) d ≤ k − ˇΩ (cid:107) L ( S ) + r (cid:107) d ≤ k − ˇΩ (cid:107) L ( S ) (cid:46) (cid:15)ru − − δ dec + (cid:15) (cid:107) r d (cid:63) / d ≤ k ˇΩ (cid:107) L ( S ) + (cid:15) (cid:107) d ≤ k ˇΩ (cid:107) L ( S ) + (cid:107) d ≤ k − ˇΩ (cid:107) L ( S ) + (cid:107) d ≤ k ˇΩ (cid:107) L ( S ) (cid:107) d ≤ k − ˇΩ (cid:107) L ( S ) , where we used Gagliardo-Nirenberg on S . Together with the Poincar´e inequality of Corol-lary 2.1.34 for d (cid:63) / , we deduce r (cid:107) r d (cid:63) / d k ˇΩ (cid:107) L ( S ) + (cid:107) r d (cid:63) / d k ˇΩ (cid:107) L ( S ) + (cid:107) d k ˇΩ (cid:107) L ( S ) (cid:46) (cid:15)ru − − δ dec + (cid:107) d ≤ k − ˇΩ (cid:107) L ( S ) . By iteration, and using again Gagliardo-Nirenberg on S , we infer on any 2-sphere of thefoliation of ( ext ) M and for any k ≤ k small (cid:107) r d (cid:63) / d k ˇΩ (cid:107) L ( S ) + (cid:107) d k ˇΩ (cid:107) L ( S ) (cid:46) (cid:15)r u − − δ dec , CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS and thus, by Sobolev embeddingmax ≤ k ≤ k small sup ( ext ) M u δ dec | d k ˇΩ | (cid:46) (cid:15) which is the desired estimate for ˇΩ. Step 2.
Next, we estimate ς . First, recall that we have e θ (log ς ) = η − ζ. Since the bootstrap assumptions for η − ζ are at least as good as for ξ , we obtain, arguingas in Step 1 the following analog of the above estimate for ˇΩmax ≤ k ≤ k small sup ( ext ) M u δ dec | d k ˇ ς | (cid:46) (cid:15). Now that we control ˇ ς , we turn to the estimate for ς . First, recall from the GCM on Σ ∗ that we have u + r = c Σ ∗ and a (cid:12)(cid:12) SP = − − mr , where ν = e + ae and ν is tangent to Σ ∗ , with c Σ ∗ a constant, and SP denoting the south pole of the spheres of Σ ∗ . We deduce onthe south poles of Σ ∗ ν ( u + r ) = e ( u ) + e ( r ) + ae ( r ) = 2 ς + e ( r ) − (cid:18) mr (cid:19) e ( r )and hence2 ς − − r (cid:18)(cid:18) κ + 2Υ r (cid:19) + A − (cid:18) mr (cid:19) (cid:18) κ − r (cid:19)(cid:19) on SP ∩ Σ ∗ . Together with the fact that ς = ς − ˇ ς , the above control of ˇ ς , the control of κ and κ provided by Lemma 3.4.1, the formula for A , the control for ˇΩ in Step 1, the bootstrapassumptions on decay, and the fact that ς is constant on the sphere, we infermax ≤ k ≤ k small sup Σ ∗ u δ dec | d k ( ς − | (cid:46) (cid:15). Using ς = ς + ˇ ς and the above estimates for ς and ˇ ς , we obtainmax ≤ k ≤ k small sup Σ ∗ u δ dec | d k ( ς − | (cid:46) (cid:15). .3. CONTROL OF COORDINATES SYSTEMS e ( ς ) = 0 . Commuting with d , using the bootstrap assumptions on decay and the above control for ς − ∗ , we infer max ≤ k ≤ k small sup ( ext ) M u δ dec | d k ( ς − | (cid:46) (cid:15). Remark 4.3.2. In ( int ) M , we analogously transport ς from the timelike hyper surface T .To estimate ς on T , one uses the following identity (in the frame of ( int ) M ) ς − − κ + A Υ κ (cid:18) ς − (cid:19) − A Υ κ − (cid:0) κ − r (cid:1) + Υ (cid:0) κ + r (cid:1) Υ κ on T . This identity follows from the definition of ς and ς , the identity for e ( r ) and e ( r ) in ( int ) M , the fact that u = u on T , and that T = { r = r T } so that the vectorfield T T = e − e ( r ) e ( r ) e = e − κ + Aκ e is tangent to T . Step 3.
We make the auxiliary bootstrap assumption which will be recovered at the endof Step 5 (cid:12)(cid:12) e Φ (cid:12)(cid:12) ≤ r, (cid:12)(cid:12) e θ ( e Φ ) (cid:12)(cid:12) ≤ . (4.3.3)We start with the estimate for e Φ . Recall from (2.2.53) that the following identity holds e Φ r sin θ = √ a . (4.3.4)where a has been introduced in (2.2.54) by a = e r + ( e θ ( e Φ )) − . In order to estimate e Φ , it thus suffices to estimate a . Step 4.
Now, recall from Lemma 2.2.30 that a verifies the following identities on ( ext ) M , e ( a ) = (ˇ κ − ϑ ) e r + 2 e θ ( e Φ ) (cid:16) β − e (Φ) ζ (cid:17) e Φ ,e θ ( a ) = 2 e θ (Φ) e (cid:18)(cid:18) ρ + 2 mr (cid:19) + 14 (cid:18) κκ + 4Υ r (cid:19) − ϑϑ (cid:19) ,e ( a ) = (cid:16) ˇ κ − A − ϑ (cid:17) e r + 2 e θ ( e Φ ) (cid:16) β + e (Φ) ζ + ξe (Φ) (cid:17) e Φ . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Together with our bootstrap assumptions on decay for in ( ext ) M for ˇ κ , ϑ , ˇ κ , ϑ , β , β , ρ , ζ , ξ and ˇΩ and the bootstrap assumption (4.3.3), we infermax ≤ k ≤ k small sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) (cid:12)(cid:12) d k a (cid:12)(cid:12) (cid:46) (cid:15). In particular, we deduce, sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) | ˇ a | (cid:46) (cid:15). Step 5.
To estimate a we make use of equation (2.1.13) according to which (cid:16) e θ ( e Φ ) (cid:17) = 1 on the axis of symmetry . Since e also vanishes there we infer that a = 0 on the axis. Therefore, on the axis,ˇ a = − a , i.e., a = − ˇ a | axisand therefore, | a | (cid:46) | ˇ a | (cid:46) (cid:15)ru + δ dec + u δ dec . We conclude that, max ≤ k ≤ k small sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) (cid:12)(cid:12) d k a (cid:12)(cid:12) (cid:46) (cid:15). (4.3.5)In view of (4.3.4) and (4.3.5), we immediately infermax ≤ k ≤ k small sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) e Φ r sin θ − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15). Together with (4.3.5) and the definition of a , this implies (cid:12)(cid:12) e Φ (cid:12)(cid:12) = (1 + O ( (cid:15) )) r sin θ ≤ r , (cid:12)(cid:12) e θ ( e Φ ) (cid:12)(cid:12) = (cid:114) − e r + a ≤ | cos θ | + O ( (cid:15) ) ≤ , (4.3.6)which is an improvement of the bootstrap assumption (4.3.3) which hence holds every-where on ( ext ) M . .3. CONTROL OF COORDINATES SYSTEMS Step 6.
We now prove the estimates for b , b and γ . Recall from Lemma 2.2.29 that θ defined by (4.3.1) satisfies re θ ( θ ) = 1 + r ( K − r )1 + ( re θ (Φ)) ,e ( θ ) = − rβ + r ( − ˇ κ + A + ϑ ) e θ (Φ) + rξe (Φ) + rζe (Φ)1 + ( re θ (Φ)) ,e ( θ ) = − rβ + r ( − ˇ κ + ϑ ) e θ (Φ) − rζe (Φ)1 + ( re θ (Φ)) . In view of the definition of b , b and γ , we infer r √ γ = 1 + r ( K − r )1 + ( re θ (Φ)) ,b = − rβ + r ( − ˇ κ + A + ϑ ) e θ (Φ) + rξe (Φ) + rζe (Φ)1 + ( re θ (Φ)) ,b = − rβ + r ( − ˇ κ + ϑ ) e θ (Φ) − rζe (Φ)1 + ( re θ (Φ)) . Also, we have in view of the definition of a re θ (Φ)) = 1 + ( e θ ( e Φ )) e r = r e (1 + a )and hence r √ γ = 1 + e r (cid:18) r ( K − r )1 + a (cid:19) ,b = − e r (cid:18) rβ + r ( − ˇ κ + A + ϑ ) e θ (Φ) + rξe (Φ) + rζe (Φ)1 + a (cid:19) ,b = − e r (cid:18) rβ + r ( − ˇ κ + ϑ ) e θ (Φ) − rζe (Φ)1 + a (cid:19) . The bootstrap assumptions on decay in ( ext ) M for ˇ κ , ϑ , ˇ κ , ϑ , β , β , ζ , ξ and ˇΩ, theestimate (4.3.5) for a , the estimate (4.3.6), and the identity K − r = − κκ + 14 ϑϑ − ρ − r = − (cid:18) κκ + 4Υ r (cid:19) − (cid:18) ρ + 2 mr (cid:19) + 14 ϑϑ CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS imply max ≤ k ≤ k small sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) r √ γ − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12) d k b (cid:12)(cid:12)(cid:19) (cid:46) (cid:15), and max ≤ k ≤ k small sup ( ext ) M ru δ dec (cid:12)(cid:12) d k b (cid:12)(cid:12) (cid:46) (cid:15). In particular, we also havemax ≤ k ≤ k small sup ( ext ) M (cid:16) ru + δ dec + u δ dec (cid:17) (cid:12)(cid:12)(cid:12) d k (cid:16) γr − (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15). These are the desired estimate for b , b and γ in ( ext ) M . This concludes the proof of thelemma.In this section, we also prove two useful lemmas concerning estimates on 2-spheres of ( ext ) M and ( int ) M . Lemma 4.3.3.
Let θ ∈ [0 , π ] be the Z -invariant scalar on M defined by (2.2.52) . Then,we have on M re θ (Φ) = (cid:36) sin θ where (cid:36) is a reduced 1-scalar satisfying sup M | (cid:36) | ≤ . Also, we have θ ≤ | re θ (Φ) | + 2 on M . Proof.
The proof is similar on ( ext ) M and ( int ) M so we focus on ( ext ) M . Recall from(4.3.6) that (cid:12)(cid:12) e θ ( e Φ ) (cid:12)(cid:12) ≤ . Furthermore, in view of Proposition 3.4.3, we have in particularsup ( ext ) M (cid:12)(cid:12)(cid:12)(cid:12) e Φ r sin θ − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15). .3. CONTROL OF COORDINATES SYSTEMS (cid:36) = r sin θe θ (Φ) , we deduce | (cid:36) | = r sin θe Φ | e θ ( e Φ ) | ≤
32 (1 + O ( (cid:15) )) ≤ , which is the desired estimate for (cid:36) .We now consider the upper bound for (sin θ ) − . Recall the definition (2.2.54) of aa = e r + ( e θ ( e Φ )) − . We infer r e θ (Φ) = r ( e θ ( e Φ )) e = 1 + a e r −
1= 1 + a − (sin θ ) (cid:16) (cid:16) e r (sin θ ) − (cid:17)(cid:17) (sin θ ) (cid:16) (cid:16) e r (sin θ ) − (cid:17)(cid:17) = (cos θ ) + a − (sin θ ) (cid:16) e r (sin θ ) − (cid:17) (sin θ ) (cid:16) (cid:16) e r (sin θ ) − (cid:17)(cid:17) and hence sin θ | re θ (Φ) | = (cid:114) (cos θ ) + a − (sin θ ) (cid:16) e r (sin θ ) − (cid:17)(cid:114) (cid:16) e r (sin θ ) − (cid:17) . Now, in view of (4.3.5), a satisfies in particularsup ( ext ) M | a | (cid:46) (cid:15). Together with sup ( ext ) M (cid:12)(cid:12)(cid:12)(cid:12) e Φ r sin θ − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15), CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS we infer sin θ | re θ (Φ) | = (cid:112) (cos θ ) + O ( (cid:15) ) (cid:112) O ( (cid:15) ) . Thus, we deducesin θ | re θ (Φ) | ≥ √
22 (1 + O ( (cid:15) )) ≥
12 for 0 ≤ θ ≤ π π ≤ θ ≤ π. On the other hand, we have sin θ ≥ √
22 on π ≤ θ ≤ π θ ≤ | re θ (Φ) | + 2 on 0 ≤ θ ≤ π which is the desired estimate. This concludes the proof of the lemma. Lemma 4.3.4.
Let θ ∈ [0 , π ] be the Z -invariant scalar on M defined by (2.2.52) . Then,for any reduced 1-scalar h , we have on any 2-sphere S on ( ext ) M and of ( int ) M sup S | h | e Φ (cid:46) r − sup S ( | h | + | d /h | ) and (cid:13)(cid:13)(cid:13)(cid:13) he Φ (cid:13)(cid:13)(cid:13)(cid:13) L ( S ) (cid:46) r − (cid:107) h (cid:107) h ( S ) . Proof.
The proof is similar on ( ext ) M and ( int ) M so we focus on ( ext ) M . Recall that the2-surface S is parametrized by the coordinate θ ∈ [0 , π ], and that the axis corresponds tothe 2 poles θ = 0 and θ = π . In view ofsup ( ext ) M (cid:12)(cid:12)(cid:12)(cid:12) e Φ r sin θ − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15), we have sup S ∩{ π ≤ θ ≤ π } | h | e Φ (cid:46) r − sup S | h | and (cid:13)(cid:13)(cid:13)(cid:13) he Φ (cid:13)(cid:13)(cid:13)(cid:13) L ( S ∩{ π ≤ θ ≤ π } ) (cid:46) r − (cid:107) h (cid:107) L ( S ) which is the desired estimate for π/ ≤ θ ≤ π/ ≤ θ ≤ π/ π/ ≤ θ ≤ π of S . These regionscan be treated analogously, so we focus on 0 ≤ θ ≤ π/
4. Recall from Remark 2.1.21 that .3. CONTROL OF COORDINATES SYSTEMS s k , for k ≥
1, must vanish on the axis of symmetry of Z , i.e. at thetwo poles. In particular, h must vanish at θ = 0. We deduce he Φ = he Φ e = (cid:82) θ ∂ θ ( e Φ h ) e = (cid:82) θ (cid:112) γ S e θ ( e Φ h ) e = (cid:82) θ √ γe Φ d/ he . Since we have | γ | (cid:46) r , we infer | h | e Φ (cid:46) (cid:82) θ e Φ | d /h | e and since sup ( ext ) M (cid:12)(cid:12)(cid:12)(cid:12) e Φ r sin θ − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15), we deduce | h | e Φ (cid:46) r − (cid:82) θ sin( θ (cid:48) ) | d /h | dθ (cid:48) (sin θ ) . This yields sup S ∩{ ≤ θ ≤ π } | h | e Φ (cid:46) r − sup S | d /h | which is the desired sup norm estimate for 0 ≤ θ ≤ π/ L -norm on 0 ≤ θ ≤ π/
4. We have in view of the above (cid:13)(cid:13)(cid:13)(cid:13) he Φ (cid:13)(cid:13)(cid:13)(cid:13) L ( S ∩{ ≤ θ ≤ π } ) (cid:46) r − (cid:90) π (cid:16)(cid:82) θ sin( θ (cid:48) ) | d /h | dθ (cid:48) (cid:17) (sin θ ) e Φ dθ (cid:46) r − (cid:90) π (cid:18)(cid:90) θ (sin( θ (cid:48) )) | d /h | dθ (cid:48) (cid:19) dθ (sin θ ) (cid:46) r − (cid:90) π (sin θ ) | d /h | (cid:32)(cid:90) π θ dθ (cid:48) (sin( θ (cid:48) )) (cid:33) dθ (cid:46) r − (cid:90) π | d /h | sin θdθ (cid:46) r − (cid:90) π | d /h | e Φ dθ (cid:46) r − (cid:107) d /h (cid:107) L ( S ) CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS and hence (cid:13)(cid:13)(cid:13)(cid:13) he Φ (cid:13)(cid:13)(cid:13)(cid:13) L ( S ∩{ ≤ θ ≤ π } ) (cid:46) r − (cid:107) d /h (cid:107) L ( S ) which is the desired L ( S ) estimate for 0 ≤ θ ≤ π/
4. This concludes the proof of thelemma.
The goal of this section is to prove Proposition 3.4.5. We deal first with the region r ≤ m as follows1. The curvature components and Ricci coefficients satisfy in view of the bootstrapassumptions on energymax k ≤ k large (cid:90) ( int ) M (cid:16) | ˇ R | + | ˇΓ | (cid:17) + max k ≤ k large − (cid:90) ( ext ) M ( r ≤ m ) (cid:16) | ˇ R | + | ˇΓ | (cid:17) ≤ (cid:15) .
2. We first take the trace on the ingoing null cones foliating ( int ) M and the outgoingnull cones foliating ( ext ) M ( r ≤ m ) which looses one derivative. We thus obtainmax k ≤ k large − sup ≤ u ≤ u ∗ (cid:90) C u (cid:16) | ˇ R | + | ˇΓ | (cid:17) + max k ≤ k large − sup ≤ u ≤ u ∗ (cid:90) C u ( r ≤ m ) (cid:16) | ˇ R | + | ˇΓ | (cid:17) (cid:46) (cid:15) .
3. We then take the trace on the 2-spheres S foliation the null cones in ( int ) M and ( ext ) M ( r ≤ m ) to infermax k ≤ k large − sup ( int ) M (cid:16) (cid:107) ˇ R (cid:107) L ( S ) + (cid:107) ˇΓ (cid:107) L ( S ) (cid:17) + max k ≤ k large − sup ( ext ) M ( r ≤ m ) (cid:16) | ˇ R | + | ˇΓ | (cid:17) (cid:46) (cid:15).
4. Finally, using the Sobolev embedding on the 2-sphere S , which looses 2 derivatives,we deducemax k ≤ k large − sup ( int ) M (cid:16) | ˇ R | + | ˇΓ | (cid:17) + max k ≤ k large − sup ( ext ) M ( r ≤ m ) (cid:16) | ˇ R | + | ˇΓ | (cid:17) (cid:46) (cid:15), which is the desired estimate in the region ( int ) M ∪ ( ext ) M ( r ≤ m ). .4. POINTWISE BOUNDS FOR HIGH ORDER DERIVATIVES ( ext ) M ( r ≥ m ). We proceed as follows Step 1 . The Ricci coefficients satisfy in view of the bootstrap assumptions on energymax k ≤ k large (cid:90) Σ ∗ (cid:34) r (cid:16) ( d ≤ k ϑ ) + ( d ≤ k ˇ κ ) + ( d ≤ k ζ ) + ( d ≤ k ˇ κ ) (cid:17) + ( d ≤ k ϑ ) + ( d ≤ k η ) + ( d ≤ k ˇ ω ) + ( d ≤ k ξ ) (cid:35) + sup λ ≥ m (cid:32) (cid:90) { r = λ } (cid:34) λ (cid:16) ( d ≤ k ϑ ) + ( d ≤ k ˇ κ ) + ( d ≤ k ζ ) (cid:17) + λ − δ B ( d ≤ k ˇ κ ) + ( d ≤ k ϑ ) + ( d ≤ k η ) + ( d ≤ k ˇ ω ) + λ − δ B ( d ≤ k ξ ) (cid:35)(cid:33) ≤ (cid:15) . We take the trace on the 2-spheres S foliating the timelike cylinders { r = r } , for r ≥ m , which looses a derivative, and infer in particularmax k ≤ k large − sup ( ext ) M ( r ≥ m ) (cid:110) r (cid:0) (cid:107) d k ˇ κ (cid:107) L ( S ) + (cid:107) d k ζ (cid:107) L ( S ) + (cid:107) d k ϑ (cid:107) L ( S ) (cid:1) + r − δB (cid:107) d k ˇ κ (cid:107) L ( S ) + (cid:107) d k η (cid:107) L ( S ) + (cid:107) d k ϑ (cid:107) L ( S ) + (cid:107) d k ˇ ω (cid:107) L ( S ) + r − δB (cid:107) d k ξ (cid:107) L ( S ) (cid:111) (cid:46) (cid:15). Also, we take the trace on the 2-spheres S foliating the spacelike hyper surface Σ ∗ , whichlooses a derivative, and infer in particularmax k ≤ k large − sup Σ ∗ r (cid:107) d k ˇ κ (cid:107) L ( S ) (cid:46) (cid:15). Step 2 . On can easily prove the following trace theoremmax k ≤ k large − (cid:18) sup r ≥ m r δ B (cid:90) S ( d k α ) (cid:19) (cid:46) sup ≤ u ≤ u ∗ (cid:90) C u r δ B ( d ≤ k large α ) , which together with the bootstrap assumptions on energy for α in ( ext ) M ( r ≥ m )implies max k ≤ k large − (cid:18) sup r ≥ m r δ B (cid:90) S ( d k α ) (cid:19) (cid:46) (cid:15) . Step 3 . Using the trace theoremmax k ≤ k large − (cid:18) sup r ≥ m r (cid:90) S ( d k β ) (cid:19) (cid:46) sup ≤ u ≤ u ∗ (cid:90) C u r ( d ≤ k large β ) , CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS we infer, together with the bootstrap assumptions on energy for β in ( ext ) M ( r ≥ m ),max k ≤ k large − (cid:18) sup r ≥ m r (cid:90) S ( d k β ) (cid:19) (cid:46) (cid:15) . (4.4.1)The power of r of the above estimate is not strong enough. To upgrade the estimate,recall that we have the Bianchi identity e ( β ) + 2 κβ = d/ α + ζα. This yields e (cid:18) r δ B (cid:90) S β (cid:19) = (cid:90) S r δ B (cid:18) βe ( β ) + κβ + b e ( r ) r β (cid:19) = (cid:90) S r δ B (cid:18) − − δ B κβ + 2 β ( r − d /α + ζα ) − δ B κβ (cid:19) and hence e (cid:18) r δ B (cid:90) S β (cid:19) + 1 − δ B (cid:90) S r δ B κβ = (cid:90) S r δ B (cid:18) β ( d /α + rζα ) − δ B κβ (cid:19) (cid:46) (cid:18)(cid:90) S r δ B ( d ≤ α ) (cid:19) (cid:18)(cid:90) S r δ B β (cid:19) + (cid:15) (cid:90) S r δ B β where we used the pointwise estimates of Step 1 for ˇ κ and ζ . We infer e (cid:18) r δ B (cid:90) S β (cid:19) + (cid:90) S r δ B β (cid:46) (cid:90) S r δ B ( d ≤ α ) . Integrating, from r ≥ m , we deducesup r ≥ m r δ B (cid:90) S β + sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ m ) r δ B β (cid:46) sup ≤ u ≤ u ∗ (cid:90) C u r δ B ( d ≤ α ) + (cid:90) S r =6 m β (cid:46) (cid:15) , where we used the bootstrap assumptions on energy for α in ( ext ) M ( r ≥ m ) and thenon sharp estimate (4.4.1) for β . Using again (4.4.1), we obtainsup r ≥ m r δ B (cid:90) S β + sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ m ) r δ B β (cid:46) (cid:15) . To discuss higher order derivatives, recall from Lemma 2.2.13 the following commutator,written in schematic form, [ d /, e ] = (ˇ κ, ϑ ) d / + ( ζ, rβ ) . .4. POINTWISE BOUNDS FOR HIGH ORDER DERIVATIVES T , e ] = (cid:16)(cid:0) ω − mr (cid:1) − m r (cid:0) κ − r (cid:1) + e ( m ) r (cid:17) e + ( η + ζ ) e θ . In view of the estimates of Step 1 for k large − κ , ϑ , ζ , η , ˇ ω , the pointwiseestimates for β in (4.4.1), the control of κ in Lemma 3.4.1, and the control of e ( m ) inLemma 3.4.2, we infer, schematically, (cid:13)(cid:13)(cid:13) d k (cid:16) [ d /, e ] β, [ T , e ] β (cid:17)(cid:13)(cid:13)(cid:13) L ( S ) (cid:46) O ( (cid:15)r − ) (cid:107) d ≤ k +1 β (cid:107) L ( S ) for k ≤ k large − . Thus, commuting the Bianchi identity for e ( β ) with T and d / together with the abovecommutator estimate, using the Bianchi identity to recover the e derivatives, we obtainfor higher order derivativesmax k ≤ k large − (cid:18) sup r ≥ m r δ B (cid:90) S ( d k β ) + sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ m ) r δ B ( d k β ) (cid:19) (cid:46) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ m ) r δ B ( d ≤ k large α ) (cid:46) (cid:15) . Step 4 . Recall from Proposition 2.2.18 that we have e ˇ ρ + 32 κ ˇ ρ + 32 ρ ˇ κ = d/ β + Err[ e ˇ ρ ] , Err[ e ˇ ρ ] = −
32 ˇ κ ˇ ρ + 12 ˇ κ ˇ ρ − (cid:18) ϑα + ζβ (cid:19) + (cid:18) ϑα + ζβ (cid:19) . This yields e (cid:18) r (cid:90) S ( ˇ ρ ) (cid:19) = (cid:90) S r (cid:18) ρe ( ˇ ρ ) + κ ˇ ρ + 4 e ( r ) r ˇ ρ (cid:19) = (cid:90) S r (cid:16) − ρ ˇ κ ˇ ρ + 2 ˇ ρ ( r − d /β + Err[ e ˇ ρ ]) + ˇ κ ˇ ρ (cid:17) and hence e (cid:34)(cid:18) r (cid:90) S ( ˇ ρ ) (cid:19) (cid:35) (cid:46) (cid:20)(cid:90) S r (cid:16) ( ρ ˇ κ ) + ( r − d /β ) + (Err[ e ˇ ρ ]) + ˇ κ ˇ ρ (cid:17)(cid:21) Using the estimates of Step 1, 2 and 3 for ˇ κ , ζ , ϑ , α and β , and the control of ρ in Lemma3.4.1, we infer e (cid:34)(cid:18) r (cid:90) S ( ˇ ρ ) (cid:19) (cid:35) (cid:46) (cid:15)r + δB + (cid:15)r (cid:18) r (cid:90) S ( ˇ ρ ) (cid:19) . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Integrating from r = 4 m , we control (cid:107) ˇ ρ (cid:107) L ( S ) from the control in r ≤ m , we infersup r ≥ m r (cid:90) S ˇ ρ (cid:46) (cid:15) . Next, commuting the equation for e ( ˇ ρ ) with T and d / together with the commutatorestimate of Step 3, using the equation for e ( ˇ ρ ) to recover the e derivatives, we obtainsimilarly for higher order derivativesmax k ≤ k large − sup r ≥ m r (cid:90) S ( d k ˇ ρ ) (cid:46) (cid:15) . Step 5 . Recall from Proposition 2.2.18 that we have the following transport equations inthe e direction, e ˇ κ + 12 κ ˇ κ + 12 ˇ κκ = − d/ ζ + 2 ˇ ρ + Err[ e ˇ κ ] , Err[ e ˇ κ ] = −
12 ˇ κ ˇ κ −
12 ˇ κ ˇ κ + (cid:18) − ϑϑ + 2 ζ (cid:19) − (cid:18) − ϑϑ + 2 ζ (cid:19) . This yields e (cid:18) r (cid:90) S (ˇ κ ) (cid:19) = (cid:90) S r (cid:18) κe (ˇ κ ) + κ ˇ κ + e ( r ) r ˇ κ (cid:19) = (cid:90) S r (cid:18) κ (cid:18) −
12 ˇ κκ − d/ ζ + 2 ˇ ρ + Err[ e ˇ κ ] (cid:19) + ˇ κ ˇ κ (cid:19) and hence, using the the estimates of Step 1 and 4 for ˇ κ , ζ , ϑ and ˇ ρ , and the control of κ and κ in Lemma 3.4.1, we infer e (cid:18) r (cid:90) S (ˇ κ ) (cid:19) (cid:46) (cid:15)r (cid:90) S r ˇ κ + (cid:15)r − (cid:18)(cid:90) S r ˇ κ (cid:19) and hence e (cid:32)(cid:18) r (cid:90) S (ˇ κ ) (cid:19) (cid:33) (cid:46) (cid:15)r (cid:18) r (cid:90) S (ˇ κ ) (cid:19) + (cid:15)r − Integrating backward from Σ ∗ , where ˇ κ under control in view of Step 1, we infersup r ≥ m r (cid:90) S ˇ κ (cid:46) (cid:15) . .4. POINTWISE BOUNDS FOR HIGH ORDER DERIVATIVES e (ˇ κ ) with T and d / together with the commutatorestimate of Step 3, using the equation for e (ˇ κ ) to recover the e derivatives, we obtainsimilarly for higher order derivativesmax k ≤ k large − sup r ≥ m r (cid:90) S ( d k ˇ κ ) (cid:46) (cid:15) . Step 6 . In view of Codazzi for ϑ , and the estimates of Step 1 on ζ , and ϑ and of Step 3on ˇ κ in ( ext ) M ( r ≥ m ), we infermax k ≤ k large − sup ( ext ) M ( r ≥ m ) r (cid:107) d k β (cid:107) L ( S ) (cid:46) (cid:15). Step 7 . In view of the null structure equation for e ( κ ), and the estimates of Step 1 onˇ ω , ζ , η and ϑ , and of Step 3 on ˇ κ in ( ext ) M ( r ≥ m ), we infermax k ≤ k large − sup ( ext ) M ( r ≥ m ) (cid:107) d k ξ (cid:107) L ( S ) (cid:46) (cid:15). Step 8 . In view of the Bianchi identity for e ( β ), and the estimates of Step 1 on ˇ ω , ζ ,and η , the estimates of Step 2 on ˇ ρ , of Step 3 on ˇ κ and of Step 5 on ξ in ( ext ) M ( r ≥ m ),we infer max k ≤ k large − sup ( ext ) M ( r ≥ m ) (cid:107) d k α (cid:107) L ( S ) (cid:46) (cid:15). Step 9 . Gathering the estimates for Step 1 to Step 8, we have obtainedmax k ≤ k large − sup ( ext ) M ( r ≥ m ) (cid:110) r + δB (cid:0) (cid:107) d k α (cid:107) L ( S ) + (cid:107) d k β (cid:107) L ( S ) (cid:1) + r (cid:0) (cid:107) d k ˇ κ (cid:107) L ( S ) + (cid:107) d k ζ (cid:107) L ( S ) + (cid:107) d k ϑ (cid:107) L ( S ) (cid:1) + (cid:107) d k ϑ (cid:107) L ( S ) + (cid:107) d k ϑ (cid:107) L ( S ) + (cid:107) d k ˇ ω (cid:107) L ( S ) (cid:111) + max k ≤ k large − sup ( ext ) M ( r ≥ m ) (cid:110) r (cid:0) (cid:107) d k µ (cid:107) L ( S ) + (cid:107) d k ˇ ρ (cid:107) L ( S ) (cid:1) + r (cid:0) (cid:107) d k ˇ κ (cid:107) L ( S ) + (cid:107) d k β (cid:107) L ( S ) (cid:1)(cid:111) + max k ≤ k large − sup ( ext ) M ( r ≥ m ) (cid:110) (cid:107) d ξ (cid:107) L ( S ) + (cid:107) d k α (cid:107) L ( S ) (cid:111) (cid:46) (cid:15). Using the Sobolev embedding on the 2-sphere S which looses 2 derivatives, and in viewof the previous estimate on ( ext ) M ( r ≤ m ), we infermax k ≤ k large − sup M (cid:110) r + δB (cid:0) | d k α | + | d k β | (cid:1) + r (cid:0) | d k ˇ µ | + | d k ˇ ρ | (cid:1) + r (cid:0) | d k ˇ κ | + | d k ζ | + | d k ϑ | + | d k ˇ κ | + | d k β | (cid:1) + r (cid:0) | d k ϑ | + | d k ϑ | + | d k ˇ ω | + | d ξ | + | d k α | (cid:1)(cid:111) (cid:46) (cid:15) CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS which is the desired estimate on ( ext ) M ( r ≥ m ). This concludes the proof of Proposition3.4.5. Let ( e , e , e θ ) the outgoing geodesic null frame of ( ext ) M . We will exhibit another frame( e (cid:48) , e (cid:48) , e (cid:48) θ ) of ( ext ) M provided by e (cid:48) = e + f e θ + 14 f e ,e (cid:48) θ = e θ + 12 f e ,e (cid:48) = e , (4.5.1)where f is such that f = 0 on Σ ∗ ∩ C ∗ , η (cid:48) = 0 on Σ ∗ , ξ (cid:48) = 0 on ( ext ) M . (4.5.2)The desired estimates for the Ricci coefficients and curvature components with respect tothe new frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) of ( ext ) M will be obtained using • the change of frame formulas of Proposition 2.3.4, applied to the change of framefrom ( e , e , e θ ) to ( e (cid:48) , e (cid:48) , e (cid:48) θ ), • the estimates for f on ( ext ) M , • the estimates for the Ricci coefficients and curvature components with respect to theoutgoing geodesic frame ( e , e , e θ ) of ( ext ) M provided by the bootstrap assumptionson decay and Proposition 3.4.5. Step 1 . We start by deriving an equation for f on ( ext ) M . In view of the condition ξ (cid:48) = 0 on ( ext ) M , see (4.5.2), in view of ξ = ω = 0 and η = − ζ satisfied by the outgoinggeodesic foliation of ( ext ) M , and in view of Lemma 2.3.5, we have e (cid:48) ( f ) + 12 κf = − f ϑ − f η − f ζ + 18 f κ + 12 f ω + 18 f ϑ + 18 f ξ on ( ext ) M . (4.5.3) .5. PROOF OF PROPOSITION 3.4.6 f on Σ ∗ . In view of the condition η (cid:48) = 0 on Σ ∗ , see (4.5.2),and in view of Lemma 2.3.5, we have e (cid:48) ( f ) = − η + 2 f ω + 12 f ξ on Σ ∗ . (4.5.4)Now, since u + r is constant on Σ ∗ , the following vectorfield ν (cid:48) Σ ∗ := e (cid:48) + a (cid:48) e (cid:48) , a (cid:48) := − e (cid:48) ( u + r ) e (cid:48) ( u + r ) , is tangent to Σ ∗ . We compute in view of the above ν (cid:48) Σ ∗ ( f ) = e (cid:48) ( f ) + a (cid:48) e (cid:48) ( f )= − η + 2 f ω + 12 f ξ + a (cid:48) (cid:40) − κf − f ϑ − f η − f ζ + 18 f κ + 12 f ω + 18 f ϑ + 18 f ξ (cid:41) . Using (4.5.1), we have a (cid:48) = − e (cid:48) ( u + r ) e (cid:48) ( u + r )= − e ( u + r ) (cid:0) e + f e θ + f e (cid:1) ( u + r )= − ς + r ( κ + A ) r κ + f (cid:0) ς + r ( κ + A ) (cid:1) and hence ν (cid:48) Σ ∗ ( f ) = − η + 2 f ω + 12 f ξ − ς + r ( κ + A ) r κ + f (cid:0) ς + r ( κ + A ) (cid:1) (cid:40) − κf − f ϑ − f η − f ζ + 18 f κ + 12 f ω + 18 f ϑ + 18 f ξ (cid:41) on ( ext ) M . (4.5.5) Step 2 . Next, we estimate f on Σ ∗ . Introducing an integer k loss and a small constant δ > ≤ k loss ≤ δ dec k large − k small ) , δ = k loss k large − k small , CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS we assume the following local bootstrap assumption | d ≤ k small + k loss +2 f | ≤ √ (cid:15)ru + δ dec − δ on u ≤ u ≤ u ∗ (4.5.6)where 1 ≤ u < u ∗ . Since f = 0 on Σ ∗ ∩ C ∗ in view of (4.5.2), (4.5.6) holds for u close enough to u ∗ , and ourgoal is to prove that we may in fact choose u = 1 and replace √ (cid:15) with (cid:15) in (4.5.6).In view of the estimates for the Ricci coefficients and curvature components with respectto the outgoing geodesic frame ( e , e , e θ ) of ( ext ) M provided by Proposition 3.4.5, (4.5.5)yields ν (cid:48) Σ ∗ ( f ) = − η + h, | d k h | (cid:46) r − ( | d ≤ k f | + | d ≤ k f | ) for k ≤ k large − . Using commutator identities, using also (4.5.3) and (4.5.4), and in view of (4.5.6), weinfer | ν (cid:48) Σ ∗ ( d / k f ) | (cid:46) | d / ≤ k η | + √ (cid:15)r u + δ dec − δ for k ≤ k small + k loss + 2 , u ≤ u ≤ u ∗ . Since f = 0 on Σ ∗ ∩ C ∗ in view of (4.5.2), and since ν (cid:48) Σ ∗ is tangent to Σ ∗ , we deduce onΣ ∗ , integrating along the integral curve of ν (cid:48) Σ ∗ | d / k f | (cid:46) (cid:90) u ∗ u | d / ≤ k η | + √ (cid:15)u + δ dec − δ (cid:90) u ∗ u ν (cid:48) Σ ∗ ( u (cid:48) ) r for k ≤ k small + k loss + 2 , u ≤ u ≤ u ∗ . Since ν (cid:48) Σ ∗ ( u ) = e (cid:48) ( u ) + a (cid:48) e (cid:48) ( u )= e ( u ) − ς + r ( κ + A ) r κ + f (cid:0) ς + r ( κ + A ) (cid:1) (cid:18) e + f e θ + 14 f e (cid:19) u = 2 ς − f ς ς + r ( κ + A ) r κ + f (cid:0) ς + r ( κ + A ) (cid:1) we have ν (cid:48) Σ ∗ ( u ) = 2 + O ( (cid:15) ) .5. PROOF OF PROPOSITION 3.4.6 | d / k f | (cid:46) (cid:90) u ∗ u | d / ≤ k η | + √ (cid:15)u + δ dec − δ (cid:90) u ∗ u r for k ≤ k small + k loss + 2 , u ≤ u ≤ u ∗ . Together with the behavior (3.3.4) of r on Σ ∗ , we infer | d / k f | (cid:46) (cid:90) u ∗ u | d / ≤ k η | + (cid:15)ru + δ dec − δ for k ≤ k small + k loss + 2 , u ≤ u ≤ u ∗ . Next, we estimate η . We have by interpolation, since k loss ≤ k large − k small , (cid:107) d / ≤ k small + k loss +4 η (cid:107) L ( S ) (cid:46) (cid:107) d / ≤ k small η (cid:107) − kloss +4 klarge − ksmall L ( S ) (cid:107) d / ≤ k large η (cid:107) kloss +4 klarge − ksmall L ( S ) , and hence, using δ >
0, we have (cid:90) Σ ∗ ( ≥ u ) | d / ≤ k small + k loss +4 η | (cid:46) (cid:18)(cid:90) Σ ∗ ( ≥ u ) u (cid:48) δ | d / ≤ k small + k loss +4 η | (cid:19) (cid:46) u + δ dec − δ (cid:18)(cid:90) Σ ∗ u (cid:48) δ dec | d / ≤ k small η | (cid:19) − kloss +42( klarge − ksmall ) (cid:18)(cid:90) Σ ∗ | d / ≤ k large η | (cid:19) kloss +42( klarge − ksmall ) . where we have used the fact that k loss + 4 k large − k small (1 + δ dec ) + δ (cid:18)(cid:18) k loss (cid:19) (1 + δ dec ) + 12 (cid:19) δ ≤ δ and 12 + δ dec − δ = 12 + δ dec − k loss k large − k small ≥ δ dec > ≤ k loss ≤ ( k large − k small ) and δ dec > η along Σ ∗ that we have (cid:90) Σ ∗ u δ dec | d ≤ k small η | + (cid:90) Σ ∗ | d ≤ k large η | ≤ (cid:15) . We deduce (cid:90) Σ ∗ ( ≥ u ) | d / ≤ k small + k loss +4 η | (cid:46) (cid:15)u + δ dec − δ . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Together with the Sobolev embedding on the 2-spheres S foliating Σ ∗ , as well as thebehavior (3.3.4) of r on Σ ∗ , we infer (cid:90) u ∗ u | d / ≤ k small + k loss +2 η | (cid:46) (cid:15)u + δ dec − δ . Plugging in the above estimate for f , we infer | d / k f | (cid:46) (cid:15)ru + δ dec − δ for k ≤ k small + k loss + 2 , u ≤ u ≤ u ∗ . Together with (4.5.3) and (4.5.4), we recover e and e derivatives to deduce | d k f | (cid:46) (cid:15)ru + δ dec − δ for k ≤ k small + k loss + 2 , u ≤ u ≤ u ∗ . This is an improvement of the bootstrap assumption (4.5.6). Thus, we may choose u = 1,and f satisfies the following estimate | d k f | (cid:46) (cid:15)ru + δ dec − δ for k ≤ k small + k loss + 2 on Σ ∗ . Together with (4.5.4), as well as the behavior (3.3.4) of r on Σ ∗ , we infer | d k − e (cid:48) f | (cid:46) | d k − η | + (cid:15)r (cid:46) (cid:15)ru δ dec − δ for k ≤ k small + k loss + 2 on Σ ∗ . Collecting the two above estimates, we obtain | d k f | (cid:46) (cid:15)ru + δ dec − δ , | d k − e (cid:48) f | (cid:46) (cid:15)ru δ dec − δ for k ≤ k small + k loss + 2 on Σ ∗ . (4.5.7) Step 3 . Next, we estimate f on ( ext ) M . We assume the following local bootstrapassumption | d ≤ k small + k loss +2 f | ≤ √ (cid:15)ru + δ dec − δ + u δ dec − δ on r ≥ r . (4.5.8)where r ≥ m . In view of the control of f on Σ ∗ provided by (4.5.7), (4.5.8) holds for r sufficiently large, and our goal is to prove that we may in fact choose r = 4 m andreplace √ (cid:15) with (cid:15) in (4.5.8).Recall (4.5.3) e (cid:48) ( f ) + 12 κf = − f ϑ − f η − f ζ + 18 f κ + 12 f ω + 18 f ϑ + 18 f ξ on ( ext ) M . .5. PROOF OF PROPOSITION 3.4.6 e , e , e θ ) of ( ext ) M provided by Proposition 3.4.5, (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) − f ϑ − f η − f ζ + 18 f κ + 12 f ω + 18 f ϑ + 18 f ξ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − | d ≤ k f | + r − ( | d ≤ k f | + | d ≤ k f | ) for k ≤ k large − . Using commutator identities, using also (4.5.3), and in view of (4.5.8), we infer e (cid:48) (cid:16) ( d /, T ) k f (cid:17) + 12 κ ( d /, T ) k f ≤ (cid:15)r u δ dec − δ for k ≤ k small + k loss + 2 , r ≥ r . Integrating backwards from Σ ∗ where we have (4.5.7), we deduce | ( d /, T ) k f | ≤ (cid:15)ru + δ dec − δ + u δ dec − δ for k ≤ k small + k loss + 2 , r ≥ r . Together with (4.5.3), we recover the e derivatives and obtain | d k f | ≤ (cid:15)ru + δ dec − δ + u δ dec − δ for k ≤ k small + k loss + 2 , r ≥ r . This is an improvement of the bootstrap assumption (4.5.8). Thus, we may choose r =4 m , and we have | d k f | (cid:46) (cid:15)ru + δ dec − δ + u δ dec − δ for k ≤ k small + k loss + 2 on ( ext ) M . Also, commuting once (4.5.3) with e (cid:48) , using the commutator identity [ e (cid:48) , e (cid:48) ] = 2 ω (cid:48) e (cid:48) − ω (cid:48) e (cid:48) + ( η (cid:48) − η (cid:48) ) e (cid:48) θ , and proceeding as above to integrate backward from Σ ∗ where e (cid:48) f isunder control from (4.5.7), we also obtain | d k − e (cid:48) f | (cid:46) (cid:15)ru δ dec − δ for k ≤ k small + k loss + 2 on ( ext ) M . Note that δ dec − δ = δ dec − k loss k large − k small ≥ δ dec > δ and the upper bound on k loss . Note that (4.5.7) yields | d k f | (cid:46) (cid:15)u δ dec − δ for k ≤ k small + k loss + 2 on Σ ∗ . in view of the behavior (3.3.4) of r on Σ ∗ . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Collecting the two above estimates, we obtain | d k f | (cid:46) (cid:15)ru + δ dec − δ + u δ dec − δ , for k ≤ k small + k loss + 2 on ( ext ) M , | d k − e (cid:48) f | (cid:46) (cid:15)ru δ dec − δ for k ≤ k small + k loss + 2 on ( ext ) M , (4.5.9)which is the desired estimate for f . Step 4 . In view of Proposition 2.3.4 applied to our particular case, i.e. a triplet ( f, , f , λ )with f = 0 and λ = 1, and the fact that the frame ( e , e , e θ ) is outgoing geodesic, wehave ξ (cid:48) = ξ,ζ (cid:48) = ζ − f κ − f ω − f ϑ + l.o.t. ,η (cid:48) = η + 12 e (cid:48) ( f ) − f ω + l.o.t. ,η (cid:48) = − ζ + 14 κf + 14 f ϑ + l.o.t. ,κ (cid:48) = κ + d/ (cid:48) ( f ) + f ( ζ + η ) − f κ − f ω + l.o.t. ,κ (cid:48) = κ + f ξ + l.o.t. ,ϑ (cid:48) = ϑ − d (cid:63) / (cid:48) ( f ) + f ( ζ + η ) − f ω + l.o.t. ϑ (cid:48) = ϑ + f ξ + l.o.t. ,ω (cid:48) = f ζ − κf − ωf + l.o.t. ,ω (cid:48) = ω + 12 f ξ, and α (cid:48) = α + 2 f β + 32 f ρ + l.o.t. ,β (cid:48) = β + 32 ρf + l.o.t. ,ρ (cid:48) = ρ + f β + l.o.t. ,β (cid:48) = β + 12 f α,α (cid:48) = α. (4.5.10) .5. PROOF OF PROPOSITION 3.4.6 ξ, ξ, ϑ, κ, η, η, ζ, κ, ϑ and α, β, ρ, β, α , and quadratic or higher order in f , and do not contain derivatives ofthe latter. Together with the estimates (4.5.9) for f on ( ext ) M , and the estimates for theRicci coefficients and curvature components with respect to the outgoing geodesic frame( e , e , e θ ) of ( ext ) M provided by the bootstrap assumptions on decay and Proposition3.4.5, we immediately infermax ≤ k ≤ k small + k loss +1 sup ( ext ) M (cid:40)(cid:16) r u + δ dec − δ + ru δ dec − δ (cid:17) | d k (Γ (cid:48) g \ { η (cid:48) } ) | + ru δ dec − δ | d k Γ (cid:48) b | + r u δ dec − δ (cid:12)(cid:12)(cid:12)(cid:12) d k − e (cid:48) (cid:18) κ (cid:48) − r , κ (cid:48) + 2Υ r , ϑ (cid:48) , ζ (cid:48) , η (cid:48) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) r ( u + 2 r ) + δ dec − δ + r ( u + 2 r ) δ dec − δ (cid:17)(cid:16) | d k α (cid:48) | + | d k β (cid:48) | (cid:17) + (cid:16) r (2 r + u ) δ dec + r (2 r + u ) + δ dec − δ (cid:17) | d k − e (cid:48) ( α (cid:48) ) | + (cid:16) r u δ dec + r u + δ dec − δ (cid:17) | d k − e (cid:48) ( β (cid:48) ) | + (cid:16) r u + δ dec − δ + r ru δ dec − δ (cid:17) | d k ˇ ρ (cid:48) | + u δ dec − δ (cid:16) r | d k β (cid:48) | + r | d k α (cid:48) | (cid:17)(cid:41) (cid:46) (cid:15) (4.5.11)where we have introduced the notationΓ (cid:48) g \ { η (cid:48) } = (cid:26) rω (cid:48) , κ (cid:48) − r , ϑ (cid:48) , ζ (cid:48) , η (cid:48) , κ (cid:48) + 2Υ r , r − ( e (cid:48) ( r ) − , r − e (cid:48) θ ( r ) , e (cid:48) ( m ) (cid:27) . Note also, in view of the above transformation formula for ω (cid:48) , i.e. ω (cid:48) = f ζ − κf − ωf + l.o.t. , that we have in fact a gain of r − for ω (cid:48) compared to (4.5.11), i.e.max ≤ k ≤ k small + k loss +1 sup ( ext ) M (cid:16) r u + δ dec − δ + r u δ dec − δ (cid:17) | d k ω (cid:48) | (cid:46) (cid:15). (4.5.12)We now focus on estimating η (cid:48) . Proceeding as for the other Ricci coefficients would yieldfor η (cid:48) the same behavior than η and hence a loss of r − compared to the desired estimate.Instead, we rely on the following null structure equation which follow from Proposition2.2.1 and the fact that ξ (cid:48) = 0 e (cid:48) ( η (cid:48) − ζ (cid:48) ) + 12 κ (cid:48) ( η (cid:48) − ζ (cid:48) ) = 2 d (cid:63) / (cid:48) ω (cid:48) − ϑ (cid:48) ( η (cid:48) − ζ (cid:48) ) . Next,20
CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS • we commute with d / (cid:48) and T (cid:48) , and we rely on the corresponding commutator identities, • we use the above equation for e (cid:48) ( η (cid:48) ) to recover the e (cid:48) derivatives, • we rely on the estimates (4.5.11), as well as the estimate (4.5.12) for ω (cid:48) ,which allows us to derive (cid:12)(cid:12)(cid:12)(cid:12) e (cid:48) ( d k ( η (cid:48) − ζ (cid:48) )) + 12 κ (cid:48) d k ( η (cid:48) − ζ (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r u + δ dec − δ + r u δ dec − δ + (cid:15)r | d ≤ k ( η (cid:48) − ζ (cid:48) ) | , k ≤ k small + k loss . Integrating backwards from Σ ∗ where η (cid:48) = 0 in view of (4.5.2), and using the control ζ (cid:48) provided by (4.5.11), we infermax ≤ k ≤ k small + k loss sup ( ext ) M (cid:16) r u + δ dec − δ + ru δ dec − δ (cid:17) | d k η (cid:48) | (cid:46) (cid:15) + max ≤ k ≤ k small + k loss sup ( ext ) M (cid:16) r u + δ dec − δ + ru δ dec − δ (cid:17) | d k ζ (cid:48) | (cid:46) (cid:15). Also, commuting first the equation for e (cid:48) ( η (cid:48) − ζ (cid:48) ) with e (cid:48) , using the commutator identity[ e (cid:48) , e (cid:48) ] = 2 ω (cid:48) e (cid:48) − ω (cid:48) e (cid:48) + ( η (cid:48) − η (cid:48) ) e (cid:48) θ , and proceeding as above to integrate backward fromΣ ∗ , we also obtain max ≤ k ≤ k small + k loss sup ( ext ) M r u δ dec − δ | d k − e (cid:48) η (cid:48) | (cid:46) (cid:15) + max ≤ k ≤ k small + k loss sup ( ext ) M r u δ dec − δ | d k − e (cid:48) ζ (cid:48) | (cid:46) (cid:15). .6. EXISTENCE AND CONTROL OF THE GLOBAL FRAMES ≤ k ≤ k small + k loss sup ( ext ) M (cid:40)(cid:16) r u + δ dec − δ + ru δ dec − δ (cid:17) | d k Γ (cid:48) g | + ru δ dec − δ | d k Γ (cid:48) b | + r u δ dec − δ (cid:12)(cid:12)(cid:12)(cid:12) d k − e (cid:48) (cid:18) κ (cid:48) − r , κ (cid:48) + 2Υ r , ϑ (cid:48) , ζ (cid:48) , η (cid:48) , η (cid:48) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) r ( u + 2 r ) + δ dec − δ + r ( u + 2 r ) δ dec − δ (cid:17)(cid:16) | d k α (cid:48) | + | d k β (cid:48) | (cid:17) + (cid:16) r (2 r + u ) δ dec + r (2 r + u ) + δ dec − δ (cid:17) | d k − e (cid:48) ( α (cid:48) ) | + (cid:16) r u δ dec + r u + δ dec − δ (cid:17) | d k − e (cid:48) ( β (cid:48) ) | + (cid:16) r u + δ dec − δ + r ru δ dec − δ (cid:17) | d k ˇ ρ (cid:48) | + u δ dec − δ (cid:16) r | d k β (cid:48) | + r | d k α (cid:48) | (cid:17)(cid:41) (cid:46) (cid:15). Together with the fact that ξ (cid:48) = 0 in view of (4.5.2), this concludes the proof of Proposition3.4.6. To match the frame of ( int ) M and a conformal renormalization of the frame of ( ext ) M ,we will need to introduce a cut-off function. Definition 4.6.1.
Let ψ : R → R a smooth cut-off function such that ≤ ψ ≤ , ψ = 0 on ( −∞ , and ψ = 1 on [1 , + ∞ ) . We define ψ m ,δ H as follows ψ m ,δ H ( r ) = (cid:40) if r ≥ m (cid:0) δ H (cid:1) , if r ≤ m (cid:0) δ H (cid:1) , and ψ m ,δ H ( r ) = ψ (cid:32) r − m (cid:0) δ H (cid:1) m δ H (cid:33) on m (cid:18) δ H (cid:19) ≤ r ≤ m (cid:18) δ H (cid:19) . We are now ready to define the global frame of the statement of Proposition 3.5.2.22
CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Definition 4.6.2 (Definition of the global frame) . We introduce a global null frame de-fined on ( ext ) M ∪ ( int ) M and denoted by ( ( glo ) e , ( glo ) e , ( glo ) e θ ) . The global frame is definedas follows1. In ( ext ) M \
Match, we have ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( ext ) Υ ( ext ) e , ( ext ) Υ − ext ) e , ( ext ) e θ (cid:1) .
2. In ( int ) M \
Match, we have ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( int ) e , ( int ) e , ( int ) e θ (cid:1) .
3. It remains to define the global frame on the matching region. We denote by ( f, f , λ ) the reduced scalars such that we have in the matching region ( ext ) e = λ (cid:18) ( int ) e + f ( int ) e θ + 14 f int ) e (cid:19) , ( ext ) e θ = (cid:18) f f (cid:19) ( int ) e θ + f ( int ) e + f (cid:18) f f (cid:19) ( int ) e , ( ext ) e = λ − (cid:32)(cid:18) f f + 116 f f (cid:19) ( int ) e + f (cid:18) f f (cid:19) ( int ) e θ + f ( int ) e (cid:33) , where we recall that the frame of ( ext ) M has been extended to ( int ) M , see section3.5.1. Then, in the matching region, the global frame is given by ( glo ) e = λ (cid:48) (cid:18) ( int ) e + f (cid:48) ( int ) e θ + 14 f (cid:48) int ) e (cid:19) , ( glo ) e θ = (cid:18) f (cid:48) f (cid:48) (cid:19) ( int ) e θ + f (cid:48) ( int ) e + f (cid:48) (cid:32) f (cid:48) f (cid:48) (cid:33) ( int ) e , ( glo ) e = λ (cid:48)− (cid:32)(cid:18) f (cid:48) f (cid:48) + 116 f (cid:48) f (cid:48) (cid:19) ( int ) e + f (cid:48) (cid:32) f (cid:48) f (cid:48) (cid:33) ( int ) e θ + f (cid:48) ( int ) e (cid:33) , where f (cid:48) = ψ m ,δ H ( ( int ) r ) f, f (cid:48) = ψ m ,δ H ( ( int ) r ) f ,λ (cid:48) = 1 − ψ m ,δ H ( ( int ) r ) + ψ m ,δ H ( ( int ) r ) ( ext ) Υ λ. (4.6.1) Remark 4.6.3.
Recall that the smooth cut-off function ψ in Definition 3.5.1, allowingto define ψ m ,δ H , is such that we have in particular ψ = 0 on ( −∞ , and ψ = 1 on [1 , + ∞ ) . The following two special cases correspond to the properties (d) i. and (d) ii. ofProposition 3.5.2. .6. EXISTENCE AND CONTROL OF THE GLOBAL FRAMES • If the cut-off ψ in Definition 3.5.1 is such that ψ = 1 on [1 / , + ∞ ) , then ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( ext ) Υ ( ext ) e , ( ext ) Υ − ext ) e , ( ext ) e θ (cid:1) on ( ext ) M . • If the cut-off ψ in Definition 3.5.1 is such that ψ = 0 on ( −∞ , / , then ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( int ) e , ( int ) e , ( int ) e θ (cid:1) on ( int ) M . Definition 4.6.4 (Global area radius and Hawking mass) . We definition an area radiusand a Hawking mass on ( ext ) M ∪ ( int ) M as follows • On ( ext ) M \
Match, we have ( glo ) r = ( ext ) r, ( glo ) m = ( ext ) m • On ( int ) M \
Match, we have ( glo ) r = ( int ) r, ( glo ) m = ( int ) m • On the matching region, we have ( glo ) r = (1 − ψ m ,δ H ( ( int ) r )) ( int ) r + ψ m ,δ H ( ( int ) r ) ( ext ) r, ( glo ) m = (1 − ψ m ,δ H ( ( int ) r )) ( int ) m + ψ m ,δ H ( ( int ) r ) ( ext ) m. The following two lemmas provide the main properties of the global frame.
Lemma 4.6.5.
We have in ( ext ) M \
Match the following relations between the quantitiesin the respective frames ( glo ) α = Υ ext ) α, ( glo ) β = Υ ( ext ) β, ( glo ) ρ + 2 mr = ( ext ) ρ + 2 mr , ( glo ) β = Υ − ext ) β, ( glo ) α = Υ − ext ) α, ( glo ) ξ = 0 , ( glo ) ξ = Υ − ext ) ξ, ( glo ) ζ = − ( glo ) η = ( ext ) ζ, ( glo ) η = ( ext ) η, ( glo ) ω + mr = − m r (cid:18) ( ext ) κ − r (cid:19) + e ( m ) r , ( glo ) ω = Υ − (cid:18) ( ext ) ω − mr + m r (cid:18) ( ext ) κ − r (cid:19) + m r (cid:16) ( ext ) ˇΩ ( ext ) κ − ( ext ) ˇΩ ( ext ) ˇ κ (cid:17) − e ( m )Υ r (cid:19) , ( glo ) κ − r = Υ (cid:18) ( ext ) κ − r (cid:19) , ( glo ) κ + 2 r = Υ − (cid:18) ( ext ) κ + 2Υ r (cid:19) , ( glo ) ˇ κ = Υ ( ext ) ˇ κ, ( glo ) ˇ κ = Υ − ext ) ˇ κ, ( glo ) ϑ = Υ ( ext ) ϑ, ( glo ) ϑ = Υ − ext ) ϑ. CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Proof.
The proof follows immediately from the change of frame formula with the choice( f = 0 , f = 0 , λ = Υ), the fact that e θ (Υ) = 0, and the fact that the frame of ( ext ) M isoutgoing geodesic and thus satisfies in particular ξ = ω = 0 and η = − ζ . Lemma 4.6.6 (Control of the global frame in the matching region) . In the matchingregion, the following estimates holds for the global frame max ≤ k ≤ k small − (cid:18) sup Match ∩ ( int ) M u δ dec (cid:12)(cid:12) d k ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12) + sup Match ∩ ( ext ) M u δ dec (cid:12)(cid:12) d k ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12)(cid:19) + max ≤ k ≤ k large − (cid:18)(cid:90) Match (cid:12)(cid:12) d k ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). and (cid:18)(cid:90) Match (cid:12)(cid:12) d k large ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15) + (cid:18)(cid:90) T (cid:12)(cid:12) d k large ( ( ext ) ˇ R ) (cid:12)(cid:12) (cid:19) . Remark 4.6.7.
The quantities associated to the global frame can be estimated as follows • In ( int ) M \
Match, the global frame coincides with the frame of ( int ) M , and hence,the quantities associated to the global frame satisfy the same estimates than thebootstrap assumptions for the frame of ( int ) M . • In ( ext ) M \
Match, estimates for the quantities associated to the global frame followfrom the identities of Lemma 4.6.5 together with the bootstrap assumptions for theframe of ( ext ) M . • In Match, the estimates for the quantities associated to the global frame are providedby Lemma 4.6.6.
The proof of Proposition 3.5.2 easily follows from Definition 4.6.2, Remark 4.6.3, andLemma 4.6.6. Thus, from now on, we focus on the proof of Lemma 4.6.6 which is carriedout in the next section.
In this section, we prove Lemma 4.6.6. To ease the exposition, the quantities associatedto the the frame of ( int ) M are unprimed, the quantities associated to the frame of ( ext ) M are primed, and the quantities associated to the the global frame are double-primed. We only need the first estimate for the proof of Proposition 3.5.2, but the second estimate will beneeded in the proof of Theorem M8. .6. EXISTENCE AND CONTROL OF THE GLOBAL FRAMES
Step 1.
Let ( e , e θ , e ) denote the frame of ( int ) M (and its extension) and ( e (cid:48) , e (cid:48) θ , e (cid:48) ) theframe of ( ext ) M (and its extension). We denote by ( f, f , λ ) the reduced scalars such that e (cid:48) = λ (cid:18) e + f e θ + 14 f e (cid:19) ,e (cid:48) θ = (cid:18) f f (cid:19) e θ + f e + f (cid:18) f f (cid:19) e ,e (cid:48) = λ − (cid:32)(cid:18) f f + 116 f f (cid:19) e + (cid:18) f + 14 f f (cid:19) e θ + f e (cid:33) . Together with the initialization of the frame of ( ext ) M and ( int ) M on T in section 3.1.2(where the spheres coincide), we have in particular f = f = 0 , λ = Υ − on T . (4.6.2)Also, recall from section 3.5.1 that in order for ( e (cid:48) , e (cid:48) θ , e (cid:48) ) to be defined everywhere on ( int ) M ∩
Match, we need - in addition to the above initialization of ( f, f , λ ) on T , toinitialize it also on C ∗ ∩ Match by f = f = 0 , λ = Υ − on C ∗ ∩ Match . (4.6.3) Step 2.
Next, we control the change of frame ( f, f , λ ) from ( e , e θ , e ) to ( e (cid:48) , e (cid:48) θ , e (cid:48) ) inthe region ( int ) M ∩
Match. To this end, we rely on the transport equation of Lemma 2.3.6together with the fact that ω (cid:48) = ξ (cid:48) = ζ (cid:48) + η (cid:48) = 0. Then, ( f , f, log( λ )) satisfy the followingtransport equations λ − e (cid:48) ( f ) + (cid:16) κ ω (cid:17) f = − ξ + E ( f, Γ) ,λ − e (cid:48) (log( λ )) = 2 ω + E ( f, Γ) ,λ − e (cid:48) ( f ) + κ f = − ζ + η ) + 2 e (cid:48) θ (log( λ )) + 2 f ω + E ( f, f , Γ) , where E , E and E are given by E ( f, Γ) = − ϑf + l.o.t. ,E ( f, Γ) = f ζ − f ω − ηf − f κ + l.o.t. ,E ( f, f , Γ) = − f e (cid:48) θ ( f ) − f ϑ + l.o.t. , Here, l.o.t. denote terms which are cubic or higher order in f, f (or in f only in the caseof E and E ) and ˇΓ and do not contain derivatives of these quantities, where Γ and ˇΓ26 CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS denotes the Ricci coefficients and renormalized Ricci coefficients w.r.t. the original nullframe ( e , e , e θ ). We rewrite the transport equation for log( λ ) as λ − e (cid:48) (log (Υ λ ))= λ − e (cid:48) (log( λ )) + λ − e (cid:48) (log(Υ))= 2 ω + E ( f, Γ) + 1Υ (cid:18) e + f e θ + 14 f e (cid:19) Υ= 2 (cid:16) ω + mr (cid:17) + E ( f, Γ) + 2Υ m ( e ( r ) − Υ) r − e ( m ) r − (cid:18) f e θ + 14 f e (cid:19) Υ . In view of the above transport equations for f , f and λ , the initialization (4.6.2) (4.6.3) for( f, f , λ ) on T ∪ ( C ∗ ∩ Match), and the control of Γ induced by the bootstrap assumptionson ( int ) M , we easily deducemax ≤ k ≤ k small sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k ( f, log(Υ λ )) (cid:12)(cid:12) + max ≤ k ≤ k small − sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k f (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ( f, log(Υ λ )) (cid:12)(cid:12) (cid:19) + max ≤ k ≤ k large − (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k f (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). Step 3.
We need to improve the number of derivatives in the top order estimate for( f, f , log( λ )). To this end, note first in view of the transformation formulas of Proposition2.3.4 and the control of ( f, f , log( λ )) provided by Step 2, we have in particularmax ≤ k ≤ k large − (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ˇ R (cid:48) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). Relying on this estimate, the control of the Ricci coefficients associated to the outgoingnull frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) on T ∪ ( ( int ) M ∩
Match), and the null structure equations, we infermax ≤ k ≤ k large − (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ˇΓ (cid:48) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). We refer to section 8.9 for a completely analogous proof where the Ricci coefficients arerecovered in ( int ) M based on the control of the curvature components.In view of the transformation formulas of Proposition 2.3.4, which can be written schemat-ically as ∂ (cid:16) f, f , log( λ ) (cid:17) = F ( f, f , λ, ˇΓ) , .6. EXISTENCE AND CONTROL OF THE GLOBAL FRAMES f, f , log( λ )) provided by Step 2, and the above control of Γ (cid:48) , we infermax ≤ k ≤ k large (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). Step 4.
We still need to control one more derivative of ( f, f , log( λ )). Repeating theprocess of Step 3, we use again the transformation formulas of Proposition 2.3.4 and thenthe final estimate of Step 3 for ( f, f , log( λ )) yields the following control for the curvaturecomponents max ≤ k ≤ k large (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ˇ R (cid:48) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). Arguing as in Step 3, we infer max ≤ k ≤ k large (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ˇΓ (cid:48) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15) + (cid:18)(cid:90) T (cid:12)(cid:12) d k large ( ( ext ) ˇ R ) (cid:12)(cid:12) (cid:19) . Using again the transformation formulas of Proposition 2.3.4, this yields the followingcontrol for ( f, f , log( λ ))max ≤ k ≤ k large +1 (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). We have finally obtained for ( f, f , λ ) in ( int ) M ∩
Matchmax ≤ k ≤ k small − sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15), (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k large +1 ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15) + (cid:18)(cid:90) T (cid:12)(cid:12) d k large ( ( ext ) ˇ R ) (cid:12)(cid:12) (cid:19) . Step 5.
In addition to the estimate of ( f, f , λ ) in ( int ) M ∩
Match of Step 4, we need toestimate ( f, f , λ ) in ( ext ) M ∩
Match. To this end, we first control in ( ext ) M ∩
Match the In Step 3, there is no term corresponding to the one integrated on T . This is due to the fact that for k ≤ k large −
1, we have thanks to the bootstrap assumptions on energy and a trace estimatemax ≤ k ≤ k large − (cid:18)(cid:90) T (cid:12)(cid:12)(cid:12) d k ( ( ext ) ˇ R ) (cid:12)(cid:12)(cid:12) (cid:19) (cid:46) (cid:15). CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS reduced scalar ( f (cid:48) , f (cid:48) , λ (cid:48) ) satisfying e = λ (cid:48) (cid:18) e (cid:48) + f (cid:48) e (cid:48) θ + 14 f (cid:48) e (cid:48) (cid:19) ,e θ = (cid:18) f (cid:48) f (cid:48) (cid:19) e (cid:48) θ + 12 f (cid:48) e (cid:48) + 12 (cid:18) f (cid:48) + 14 f f (cid:48) (cid:19) e (cid:48) ,e = λ (cid:48)− (cid:18)(cid:18) f (cid:48) f (cid:48) + 116 f (cid:48) f (cid:48) (cid:19) e (cid:48) + (cid:18) f (cid:48) + 14 f (cid:48) f (cid:48) (cid:19) e (cid:48) θ + 14 f (cid:48) e (cid:48) (cid:19) . Together with the initialization of the frame of ( ext ) M and ( int ) M on T in section 3.1.2(where the spheres coincide), we have in particular f (cid:48) = f (cid:48) = 0 , λ (cid:48) = Υ − on T . Also, recall from section 3.5.1 that in order for ( e , e θ , e ) to be defined everywhere on ( ext ) M ∩
Match, we need - in addition to the above initialization of ( f, f , λ ) on T , toinitialize it also on C ∗ ∩ Match by f (cid:48) = f (cid:48) = 0 , λ (cid:48) = Υ − on C ∗ ∩ Match . (4.6.4)Arguing similarly to Steps 1-4, we estimate ( f (cid:48) , f (cid:48) , λ (cid:48) ) and (ˇΓ , ˇ R ) in ( ext ) M ∩
Match. Weobtainmax ≤ k ≤ k small − sup ( ext ) M∩ Match u δ dec (cid:12)(cid:12) d k (ˇΓ , ˇ R ) (cid:12)(cid:12) + max ≤ k ≤ k large − (cid:18)(cid:90) ( ext ) M∩ Match (cid:12)(cid:12) d k (ˇΓ , ˇ R ) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15), (cid:18)(cid:90) ( ext ) M∩ Match (cid:12)(cid:12) d k large ˇ R (cid:12)(cid:12) (cid:19) (cid:46) (cid:15), max ≤ k ≤ k small − sup ( ext ) M∩ Match u δ dec (cid:12)(cid:12) d k ( f (cid:48) , f (cid:48) , log(Υ (cid:48) λ (cid:48) )) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large (cid:18)(cid:90) ( ext ) M∩ Match (cid:12)(cid:12) d k ( f (cid:48) , f (cid:48) , log(Υ (cid:48) λ (cid:48) )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15), and (cid:18)(cid:90) ( ext ) M∩ Match (cid:12)(cid:12) d k large ˇΓ (cid:12)(cid:12) (cid:19) (cid:46) (cid:15) + (cid:18)(cid:90) T (cid:12)(cid:12) d k large ( ( ext ) ˇ R ) (cid:12)(cid:12) (cid:19) , (cid:18)(cid:90) ( ext ) M∩ Match (cid:12)(cid:12) d k large +1 ( f (cid:48) , f (cid:48) , log(Υ (cid:48) λ (cid:48) )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15) + (cid:18)(cid:90) T (cid:12)(cid:12) d k large ( ( ext ) ˇ R ) (cid:12)(cid:12) (cid:19) . Step 6.
As mentioned above, in addition to the estimate of ( f, f , λ ) in ( int ) M ∩
Matchof Step 4, we need to estimate ( f, f , λ ) in ( ext ) M ∩
Match. To this end, we derive simple .6. EXISTENCE AND CONTROL OF THE GLOBAL FRAMES f, f , λ ) and ( f (cid:48) , f (cid:48) , λ (cid:48) ) of Step 5. On the one hand, we havefrom the definition of ( f, f , λ ) g ( e (cid:48) , e ) = − λ, g ( e (cid:48) , e θ ) = λf, g ( e (cid:48) θ , e ) = − f (cid:18) f f (cid:19) , g ( e (cid:48) θ , e ) = − f ,g ( e (cid:48) , e ) = − λ − (cid:18) f f f f (cid:19) , g ( e (cid:48) , e θ ) = λ − f (cid:18) f f (cid:19) . On the other hand, we have from the definition of ( f (cid:48) , f (cid:48) , λ (cid:48) ) g ( e , e (cid:48) ) = − λ (cid:48) , g ( e , e (cid:48) θ ) = λ (cid:48) f (cid:48) , g ( e θ , e (cid:48) ) = − f (cid:48) , g ( e θ , e (cid:48) ) = − f (cid:48) (cid:32) f (cid:48) f (cid:48) (cid:33) ,g ( e , e (cid:48) ) = − λ (cid:48)− (cid:32) f (cid:48) f (cid:48) f (cid:48) f (cid:48) (cid:33) , g ( e , e (cid:48) θ ) = λ (cid:48)− f (cid:48) (cid:32) f (cid:48) f (cid:48) (cid:33) . We immediately infer λ (cid:48) = λ, f (cid:48) = − λf, f (cid:48) = − λ − f . In view of the estimates of Step 5, we infermax ≤ k ≤ k small − sup ( ext ) M∩ Match u δ dec (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large (cid:18)(cid:90) ( ext ) M∩ Match (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15), and (cid:18)(cid:90) ( ext ) M∩ Match (cid:12)(cid:12) d k large +1 ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). Together with Step 4, this yieldsmax ≤ k ≤ k small − sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k small − sup ( ext ) M∩ Match u δ dec (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large (cid:18)(cid:90) Match (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15) + (cid:18)(cid:90) T (cid:12)(cid:12) d k large ( ( ext ) ˇ R ) (cid:12)(cid:12) (cid:19) , and max ≤ k ≤ k large +1 (cid:18)(cid:90) Match (cid:12)(cid:12) d k ( f, f , log(Υ λ )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15) + (cid:18)(cid:90) T (cid:12)(cid:12) d k large ( ( ext ) ˇ R ) (cid:12)(cid:12) (cid:19) . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Step 7.
Next, we estimate r (cid:48) − r and m (cid:48) − m . Note first the in view of the initialization ofthe foliations of ( ext ) M and ( int ) M on T , as well as the initializations (4.6.3) on C ∗ ∩ Matchand (4.6.4) on C ∗ ∩ Match, we have r (cid:48) = r, m (cid:48) = m on T ∪
Match . (4.6.5)We start with the region ( int ) M ∩
Match. We have e (cid:48) ( r (cid:48) ) = r (cid:48) κ (cid:48) = 1 + r (cid:48) (cid:18) κ (cid:48) − r (cid:48) (cid:19) , e (cid:48) ( r (cid:48) ) = r (cid:48) κ (cid:48) + A (cid:48) ) = − Υ (cid:48) + r (cid:48) (cid:18) κ (cid:48) + 2Υ (cid:48) r (cid:48) (cid:19) + r (cid:48) A (cid:48) , which together with the identities for e (cid:48) ( m (cid:48) ) and e (cid:48) ( m (cid:48) ) in the outgoing foliation of ( ext ) M and the control of the foliation of ( ext ) M in ( int ) M ∩
Match established in Step 4 yields,using also e (cid:48) θ ( r (cid:48) ) = e (cid:48) θ ( m (cid:48) ) = 0,max ≤ k ≤ k small − sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k ( e (cid:48) ( r (cid:48) ) − , e (cid:48) ( r (cid:48) ) + Υ (cid:48) , e (cid:48) θ ( r (cid:48) ) , e (cid:48) ( m (cid:48) ) , e (cid:48) ( m (cid:48) ) , e (cid:48) θ ( m (cid:48) )) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large − (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ( e (cid:48) ( r (cid:48) ) − , e (cid:48) ( r (cid:48) ) + Υ (cid:48) , e (cid:48) θ ( r (cid:48) ) , e (cid:48) ( m (cid:48) ) , e (cid:48) ( m (cid:48) ) , e (cid:48) θ ( m (cid:48) )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). On the other hand, we have in view of the decomposition of e (cid:48) , e (cid:48) and e (cid:48) θ of Step 1 e (cid:48) ( r ) = λ (cid:18) e + f e θ + 14 f e (cid:19) r = λ (cid:18) r κ + A ) + 14 f e ( r ) (cid:19) = 1 + (cid:16) λ Υ − (cid:17) + λ (cid:18) r (cid:18) κ − r (cid:19) + r A + 14 f e ( r ) (cid:19) ,e (cid:48) ( m ) = λ (cid:18) e + f e θ + 14 f e (cid:19) m = λ (cid:18) e ( m ) + 14 f e ( m ) (cid:19) ,e (cid:48) ( r ) = λ − (cid:32)(cid:18) f f + 116 f f (cid:19) e + (cid:18) f + 14 f f (cid:19) e θ + f e (cid:33) r = λ − (cid:32) e ( r ) + (cid:18) f f + 116 f f (cid:19) e ( r ) + f e ( r ) (cid:33) = − Υ + λ − ( λ Υ −
1) + λ − (cid:32) r (cid:18) κ + 2 r (cid:19) + (cid:18) f f + 116 f f (cid:19) e ( r ) + f e ( r ) (cid:33) , .6. EXISTENCE AND CONTROL OF THE GLOBAL FRAMES e (cid:48) ( m ) = λ − (cid:32)(cid:18) f f + 116 f f (cid:19) e + (cid:18) f + 14 f f (cid:19) e θ + f e (cid:33) m = λ − (cid:32)(cid:18) f f + 116 f f (cid:19) e ( m ) + f e ( m ) (cid:33) ,e (cid:48) θ ( r ) = (cid:18)(cid:18) f f (cid:19) e θ + f e + f (cid:18) f f (cid:19) e (cid:19) r = f e ( r ) + f (cid:18) f f (cid:19) e ( r ) , and e (cid:48) θ ( r ) = (cid:18)(cid:18) f f (cid:19) e θ + f e + f (cid:18) f f (cid:19) e (cid:19) m = f e ( m ) + f (cid:18) f f (cid:19) e ( m ) . Together with the identities for e ( m ) and e ( m ) in the ingoing foliation of ( int ) M , thefinal estimates of Step 6 for f and λ , and the bootstrap assumptions for the foliation of ( int ) M , we infermax ≤ k ≤ k small − sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k ( e (cid:48) ( r ) − , e (cid:48) ( r ) + Υ , e (cid:48) θ ( r ) , e (cid:48) ( m ) , e (cid:48) ( m ) , e (cid:48) θ ( m )) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large − (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ( e (cid:48) ( r ) − , e (cid:48) ( r ) + Υ , e (cid:48) θ ( r ) , e (cid:48) ( m ) , e (cid:48) ( m ) , e (cid:48) θ ( m )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). We deducemax ≤ k ≤ k small − sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k ( e (cid:48) ( r (cid:48) − r ) , e (cid:48) θ ( r − r (cid:48) ) , d ( m (cid:48) − m )) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large − (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ( e (cid:48) ( r (cid:48) − r ) , e (cid:48) θ ( r − r (cid:48) ) , d ( m (cid:48) − m )) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). In particular, we have sup ( int ) M∩ Match u δ dec | ( e (cid:48) ( r (cid:48) − r ) , e (cid:48) ( m (cid:48) − m )) | (cid:46) (cid:15), and together with the initialization (4.6.5), we integrate the transport equation from T ∪ ( ( int ) M ∩
Match) and obtainsup ( int ) M∩ Match u δ dec | ( r (cid:48) − r, m (cid:48) − m ) | (cid:46) (cid:15). CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
Together with the above estimates, and recovering the e (cid:48) ( r (cid:48) − r ) using e (cid:48) ( r (cid:48) − r ) = (cid:16) e (cid:48) ( r (cid:48) ) + Υ (cid:48) (cid:17) − (cid:16) e (cid:48) ( r ) + Υ (cid:17) + 2 (cid:18) m (cid:48) r (cid:48) − mr (cid:19) , we infer max ≤ k ≤ k small − sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k ( r (cid:48) − r, m (cid:48) − m ) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large (cid:18)(cid:90) ( int ) M∩ Match (cid:12)(cid:12) d k ( r (cid:48) − r, m (cid:48) − m ) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). Finally, arguing similarly in the region ( ext ) M ∩
Match, we infermax ≤ k ≤ k small − sup ( ext ) M∩ Match u δ dec (cid:12)(cid:12) d k ( r (cid:48) − r, m (cid:48) − m ) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large (cid:18)(cid:90) ( ext ) M∩ Match (cid:12)(cid:12) d k ( r (cid:48) − r, m (cid:48) − m ) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15), and hence max ≤ k ≤ k small − sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k ( r (cid:48) − r, m (cid:48) − m ) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k small − sup ( ext ) M∩ Match u δ dec (cid:12)(cid:12) d k ( r (cid:48) − r, m (cid:48) − m ) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large (cid:18)(cid:90) Match (cid:12)(cid:12) d k ( r (cid:48) − r, m (cid:48) − m ) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15). Step 8.
Recall from Definition 4.6.2 that we have defined the global null frame ( e (cid:48)(cid:48) , e (cid:48)(cid:48) , e (cid:48)(cid:48) θ )as • In ( int ) M \
Match, ( e (cid:48)(cid:48) , e (cid:48)(cid:48) , e (cid:48)(cid:48) θ ) = ( e , e , e θ ). • In ( ext ) M \
Match, ( e (cid:48)(cid:48) , e (cid:48)(cid:48) , e (cid:48)(cid:48) θ ) = (Υ e (cid:48) , Υ − e (cid:48) , e (cid:48) θ ). • In Match, ( e (cid:48)(cid:48) , e (cid:48)(cid:48) , e (cid:48)(cid:48) θ ) is given by the change of frame formula starting from ( e , e , e θ )and with change of frame coefficients ( f (cid:48)(cid:48) , f (cid:48)(cid:48) , λ (cid:48)(cid:48) ) given by f (cid:48)(cid:48) = ψf, f (cid:48)(cid:48) = ψf , λ (cid:48)(cid:48) = 1 − ψ + ψ Υ (cid:48) λ, see (4.6.1). .6. EXISTENCE AND CONTROL OF THE GLOBAL FRAMES r (cid:48)(cid:48) and m (cid:48)(cid:48) as r (cid:48)(cid:48) = (1 − ψ ) r + ψr (cid:48) , m (cid:48)(cid:48) = (1 − ψ ) m + ψm (cid:48) . Step 9.
In view of the transformation formulas of Proposition 2.3.4, we have schematically(ˇΓ (cid:48)(cid:48) , ˇ R (cid:48)(cid:48) ) = (ˇΓ , ˇ R ) + d ( f (cid:48)(cid:48) , f (cid:48)(cid:48) , λ (cid:48)(cid:48) −
1) + f (cid:48)(cid:48) + f (cid:48)(cid:48) + ( λ (cid:48)(cid:48) −
1) + ( r (cid:48)(cid:48) − r ) + ( m (cid:48)(cid:48) − m ) . In view of the definition of ( f (cid:48)(cid:48) , f (cid:48)(cid:48) , λ (cid:48)(cid:48) ) and ( r (cid:48)(cid:48) , m (cid:48)(cid:48) ) in Step 8, we infer(ˇΓ (cid:48)(cid:48) , ˇ R (cid:48)(cid:48) ) = (ˇΓ , ˇ R ) + d ( f, f , Υ λ −
1) + f + f + (Υ λ −
1) + ( r (cid:48) − r ) + ( m (cid:48) − m ) . Together with the bootstrap assumptions in ( int ) M for (Γ , ˇ R ), the estimates for (Γ , ˇ R ) in ( ext ) M provided by Step 5, the estimates for ( f, f , λ ) provided by Step 6 in Match, andthe estimates for r (cid:48) − r and m (cid:48) − m provided by Step 7, we deducemax ≤ k ≤ k small − sup ( int ) M∩ Match u δ dec (cid:12)(cid:12) d k (ˇΓ (cid:48)(cid:48) , ˇ R (cid:48)(cid:48) ) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k small − sup ( ext ) M∩ Match u δ dec (cid:12)(cid:12) d k (ˇΓ (cid:48)(cid:48) , ˇ R (cid:48)(cid:48) ) (cid:12)(cid:12) (cid:46) (cid:15), max ≤ k ≤ k large − (cid:18)(cid:90) Match (cid:12)(cid:12) d k (ˇΓ (cid:48)(cid:48) , ˇ R (cid:48)(cid:48) ) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15), (cid:18)(cid:90) Match (cid:12)(cid:12) d k large (ˇΓ (cid:48)(cid:48) , ˇ R (cid:48)(cid:48) ) (cid:12)(cid:12) (cid:19) (cid:46) (cid:15) + (cid:18)(cid:90) T (cid:12)(cid:12) d k large ( ( ext ) ˇ R ) (cid:12)(cid:12) (cid:19) . Since the double-primed quantities correspond to the quantities associated to the theglobal frame, this concludes the proof of Lemma 4.6.6.
To match the first global frame of M of Proposition 3.5.5 with a conformal renormalizationof the second frame of ( ext ) M of Proposition 3.4.6, we will need to introduce a cut-offfunction. Definition 4.6.8.
Let ψ : R → R a smooth cut-off function such that ≤ ψ ≤ , ψ = 0 on ( −∞ , and ψ = 1 on [1 , + ∞ ) . We define ψ m as follows ψ m ( r ) = (cid:40) if r ≥ m , if r ≤ m , and ψ m ( r ) = ψ (cid:32) (cid:0) r − m (cid:1) m (cid:33) on m ≤ ( ext ) r ≤ m . CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS
We are now ready to define the second global frame, i.e. the global frame of the statementof Proposition 3.5.5.
Definition 4.6.9 (Definition of the second global frame) . We introduce a global nullframe defined on ( ext ) M ∪ ( int ) M and denoted by ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) . The secondglobal frame is defined as follows1. In ( ext ) M ∩ { ( ext ) r ≥ m } , we have ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) = (cid:0) ( ext ) Υ e (cid:48) , ( ext ) Υ − e (cid:48) , e (cid:48) θ (cid:1) , where ( e (cid:48) , e (cid:48) , e (cid:48) θ ) denotes the second frame of ( ext ) M , i.e. the one constructed in ofProposition 3.4.6.2. In ( int ) M ∪ ( ( ext ) M ∩ { ( ext ) r ≤ m } ) , we have ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) = (cid:0) ( glo ) e , ( glo ) e , ( glo ) e θ (cid:1) , where (cid:0) ( glo ) e , ( glo ) e , ( glo ) e θ (cid:1) denotes the first global frame of M of Proposition 3.5.5.3. It remains to define the global frame on the matching region Match (cid:48) . We denote by f the reduced scalar introduced in Proposition 3.4.6 such that we have in ( ext ) M e (cid:48) = ( ext ) e + f ( ext ) e θ + 14 f ext ) e ,e (cid:48) θ = ( ext ) e θ + f ( ext ) e ,e (cid:48) = ( ext ) e . Then, in the matching region Match (cid:48) , the second global frame of M is given by ( glo (cid:48) ) e = Υ (cid:48) (cid:18) Υ (cid:48)− glo ) e + f (cid:48) ( glo ) e θ + 14 f (cid:48) Υ (cid:48) ( glo ) e (cid:19) , ( glo (cid:48) ) e θ = ( glo ) e θ + f (cid:48) (cid:48) ( glo ) e , ( glo (cid:48) ) e = ( glo ) e , where f (cid:48) = ψ m ( ( ext ) r ) f, Υ (cid:48) = 1 − ψ m ( ( ext ) r ) + ψ m ( ( ext ) r ) ( ext ) Υ . (4.6.6) Remark 4.6.10.
Recall that the smooth cut-off function ψ in Definition 3.5.4, allowingto define ψ m ,δ H , is such that we have in particular ψ = 0 on ( −∞ , and ψ = 1 on [1 , + ∞ ) . The following two special cases correspond to the properties (d) i. and (d) ii. ofProposition 3.5.5. .6. EXISTENCE AND CONTROL OF THE GLOBAL FRAMES • If the cut-off ψ in Definition 3.5.4 is such that ψ = 1 on [1 / , + ∞ ) , then ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) = (cid:0) ( ext ) Υ e (cid:48) , ( ext ) Υ − e (cid:48) , e (cid:48) θ (cid:1) on ( ext ) M (cid:18) ( ext ) r ≥ m (cid:19) . • If the cut-off ψ in Definition 3.5.4 is such that ψ = 0 on ( −∞ , / , then ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) = (cid:0) ( glo ) e , ( glo ) e , ( glo ) e θ (cid:1) on ( int ) M ∪ ( ext ) M (cid:18) ( ext ) r ≤ m (cid:19) . Remark 4.6.11.
When dealing with the second global frame ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) , thearea radius and Hawking mass that we use are the ones corresponding to the first globalframe, i.e. ( glo ) r and ( glo ) m . The following two lemmas provide the main properties of the second global frame of M . Lemma 4.6.12.
We have in ( ext ) M ( r ≥ m ) the following relations between the quan-tities in the second global frame of M , i.e. ( ( glo (cid:48) ) e , ( glo (cid:48) ) e , ( glo (cid:48) ) e θ ) , and the second frameof ( ext ) M , i.e. ( e (cid:48) , e (cid:48) , e (cid:48) θ ) , ( glo (cid:48) ) α = Υ α (cid:48) , ( glo (cid:48) ) β = Υ β (cid:48) , ( glo (cid:48) ) ρ + 2 mr = ρ (cid:48) + 2 mr , ( glo (cid:48) ) β = Υ − β (cid:48) , ( glo (cid:48) ) α = Υ − α (cid:48) , ( glo (cid:48) ) ξ = 0 , ( glo (cid:48) ) ξ = Υ − ξ (cid:48) , ( glo (cid:48) ) ζ = − ( glo (cid:48) ) η = ζ (cid:48) , ( glo (cid:48) ) η = η (cid:48) , ( glo (cid:48) ) ω + mr = Υ ω (cid:48) + mr (1 − e (cid:48) ( r )) + e (cid:48) ( m ) r , ( glo (cid:48) ) ω = Υ − (cid:18) ω (cid:48) − mr + mr (cid:18) − e (cid:48) ( r )Υ (cid:19) − e (cid:48) ( m )Υ r (cid:19) , ( glo (cid:48) ) κ − r = Υ (cid:18) κ (cid:48) − r (cid:19) , ( glo (cid:48) ) κ + 2 r = Υ − (cid:18) κ (cid:48) + 2Υ r (cid:19) , ( glo (cid:48) ) ϑ = Υ ϑ (cid:48) , ( glo (cid:48) ) ϑ = Υ − ϑ (cid:48) . Proof.
The proof follows immediately from the change of frame formula with the choice( f = 0 , f = 0 , λ = Υ), the fact that e θ (Υ) = 0, and the fact that the frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) issuch that ξ (cid:48) = 0 and η (cid:48) = − ζ (cid:48) . Lemma 4.6.13 (Control of the second global frame in the matching region) . In thematching region, the following estimates holds for the second global frame max ≤ k ≤ k small + k loss sup Match (cid:48) u δ dec − δ (cid:12)(cid:12)(cid:12) d k ( ( glo (cid:48) ) ˇΓ , ( glo (cid:48) ) ˇ R ) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15). Remark 4.6.14.
The quantities associated to the second global frame can be estimatedas follows CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS • In ( int ) M ∪ ( ext ) M ( ( ext ) r ≤ m ) , the second global frame coincides with the firstglobal frame, and hence, the quantities associated to the second global frame satisfythe same estimates than the corresponding quantities for the first global frame. • In ( ext ) M ( ( ext ) r ≥ m ) , estimates for the quantities associated to the second globalframe follow from the identities of Lemma 4.6.12 together with the estimates ofProposition 3.4.6 for the second frame of ( ext ) M . • In Match (cid:48) , the estimates for the quantities associated to the global frame are providedby Lemma 4.6.13.
The proof of Proposition 3.5.5 easily follows from Definition 4.6.9, Remark 4.6.10, andLemma 4.6.13. Thus, from now on, we focus on the proof of Lemma 4.6.13 which iscarried out below.
Proof of Lemma 4.6.13.
Recall from definition 4.6.9 that we have in the matching regionMatch (cid:48) ( glo (cid:48) ) e = Υ (cid:48) (cid:18) Υ (cid:48)− glo ) e + f (cid:48) ( glo ) e θ + 14 f (cid:48) Υ (cid:48) ( glo ) e (cid:19) , ( glo (cid:48) ) e θ = ( glo ) e θ + f (cid:48) (cid:48) ( glo ) e , ( glo (cid:48) ) e = ( glo ) e , where f (cid:48) = ψ m ( ( ext ) r ) f, Υ (cid:48) = 1 − ψ m ( ( ext ) r ) + ψ m ( ( ext ) r ) ( ext ) Υ . Now, since ( ext ) r ≥ m on Match (cid:48) , we also have in that region( ( glo ) e , ( glo ) e , ( glo ) e θ ) = ( ( ext ) Υ ( ext ) e , ( ( ext ) Υ) − ext ) e , ( ext ) e θ ) . We deduce on Match (cid:48) ( glo (cid:48) ) e = ( ext ) Υ (cid:18) ( ext ) e + f (cid:48)(cid:48) ( ext ) e θ + 14 f (cid:48)(cid:48) ext ) e (cid:19) , ( glo (cid:48) ) e θ = ( ext ) e θ + f (cid:48)(cid:48) ( ext ) e , ( glo (cid:48) ) e = ( ( ext ) Υ) − ext ) e , where f (cid:48)(cid:48) = Υ (cid:48) ( ( ext ) Υ) − f (cid:48) = (cid:16) − ψ m ( ( ext ) r ) + ψ m ( ( ext ) r ) ( ext ) Υ (cid:17) ( ( ext ) Υ) − ψ m ( ( ext ) r ) f. .6. EXISTENCE AND CONTROL OF THE GLOBAL FRAMES (cid:16) ( glo (cid:48) ) ˇΓ , ( glo (cid:48) ) ˇ R (cid:17) = (cid:16) ( ext ) ˇΓ , ( ext ) ˇ R (cid:17) + d f + f. Together with the bootstrap assumptions on decay and Proposition 3.4.5 for ( ( ext ) ˇΓ , ( ext ) ˇ R ),and the estimate (3.4.11) for f , we infermax ≤ k ≤ k small + k loss sup Match (cid:48) u δ dec − δ (cid:12)(cid:12)(cid:12) d k ( ( glo (cid:48) ) ˇΓ , ( glo (cid:48) ) ˇ R ) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) which concludes the proof of Lemma 4.6.13.38 CHAPTER 4. CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS hapter 5DECAY ESTIMATES FOR q (Theorem M1) The goal of the chapter is to prove Theorem M1, i.e. to derive decay estimates for thequantity q for k ≤ k small + 20 derivatives. To this end, we will make use of the waveequation satisfied by q (see (2.4.7)) (cid:3) q + κκ q = N, (5.0.1)where N contains only quadratic or higher order terms. Now, in order to have a suitableright-hand side N , recall from the discussion in Remarks 2.4.8 and 2.4.9 that q is definedrelative to the global null frame of Proposition 3.5.5 for which ξ = 0 for r ≥ m and η ∈ Γ g . For such a global fame, N is given schematically by, see (2.4.8), N = r d ≤ (Γ g · ( α, β )) + e (cid:16) r d ≤ (Γ g · ( α, β )) (cid:17) + d ≤ (Γ g · q ) + l.o.t. (5.0.2) Smallness constants
Recall from the beginning of section 3.3.2 the constant m and the main small constants δ H , δ B , δ dec , (cid:15) and (cid:15) such that • The constant m > CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) • The integer k large which corresponds to the maximum number of derivatives of thesolution. • The size of the initial data layer norm is measured by (cid:15) > • The size of the bootstrap assumption norms are measured by (cid:15) > • δ H > | r − m | ≤ m δ H where the redshiftestimate holds and which includes in particular the region ( int ) M . • δ dec is tied to decay estimates in u , u for ˇΓ and ˇ R . • δ B is involved in the r -power of the r p weighted estimates for curvature.Recall also that these constants satisfy in view of (3.3.1) (3.3.2) (3.3.3)0 < δ H , δ dec , δ B (cid:28) min { m , } , δ B > δ dec , k large (cid:29) δ dec ,(cid:15) , (cid:15) (cid:28) min { δ H , δ dec , δ B , m , } , and (cid:15) = (cid:15) . We will need the following additional small constants in this chapter • δ extra >
0, tied to the decay of q , and is chosen such that δ extra > δ dec , • δ > • δ > k ≤ k small derivatives of (ˇΓ , ˇ R ) and k ≤ k large derivatives of (ˇΓ , ˇ R ), see Lemma 5.1.1, • q > q in Theorem M1has an extra gain u − ( δ extra − δ dec ) compared from the expected behavior inferred fromthe bootstrap assumptions.We will choose δ extra such that δ dec < δ extra < δ dec , δ B ≥ δ extra , .1. PRELIMINARIES δ and δ such that 0 < (cid:15), (cid:15) (cid:28) δ, δ (cid:28) δ dec , δ extra , δ H , m , , (5.1.1)and q such that δ dec < q < δ dec − δ − δ. (5.1.2) M by τ Recall that the spacetime M is decomposed as M = ( int ) M ∪ ( ext ) M and that u is anoutgoing optical function on ( ext ) M while u is an ingoing optical function. In this chapter,we rely on the global frame ( e , e , e θ , e ϕ ) defined in section 3.5, and r and m denote thecorresponding scalar functions associated to it. Also, we define the trapping region region M trap as, M trap := (cid:26) m ≤ r ≤ m (cid:27) . (5.1.3)Also, let ( trap (cid:14) ) M = M \ ( trap ) M the complement of ( trap ) M in M .We foliate our spacetime domain M by Z invariant hypersurfaces Σ( τ ) which are: • Incoming null in ( int ) M , with e as null incoming generator. We denote this portion ( int ) Σ( τ ). • Strictly spacelike in ( trap ) M . We denote this portion by ( trap ) Σ. • Outgoing null in M > m . We denote this portion by Σ > m ( τ ). • The parameter τ of Σ( τ ) can be chosen, smoothly, such that τ := u in M > m ,u + r in M trap ,u in ( int ) M . (5.1.4) This will allow us to choose in the proof of Theorem M1, see (5.2.10), δ extra = q − δ δ extra > δ dec for δ > CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) • In particular, the unit normal in the region M trap , i.e. the normal to ( trap ) Σ,satisfies − ≤ g ( N Σ , e ) ≤ − , − ≤ g ( N Σ , e ) ≤ − M trap . (5.1.5) Recall from Remark 2.4.9 that q is defined, according to equation (2.3.10) in Lemma2.3.10, relative to the global frame of Proposition 3.5.5 for which η ∈ Γ g with the notationΓ g = Γ (0) g = (cid:110) ξ, ϑ, ω + mr , κ − r , η, η, ζ, A (cid:111) , Γ b = Γ (0) b = (cid:110) ϑ, κ + 2 r , A, ω, ξ (cid:111) , where we recall thatΥ = 1 − mr , A = 2 r e ( r ) − κ, A = 2 r e ( r ) − κ. Note also that ξ vanishes in ( ext ) M away from the matching region of Proposition 3.5.5,and in particular for r ≥ m .For higher derivatives we write,Γ (1) g = (cid:110) d ξ, d ϑ, re θ ω, re θ ( κ ) , d η, d η, d ζ, d A (cid:111) Γ (1) b = (cid:110) d ϑ, re θ ( κ ) , d ξ, d A, re θ ω, d ξ (cid:111) and for s ≥
2, Γ ( s ) g = d s − Γ (1) g , Γ ( s ) b = d s − Γ (1) b Moreover we denoteΓ ≤ sg = (cid:110) Γ (0) g , Γ (1) g , . . . Γ ( s ) g (cid:111) , Γ ≤ sb = (cid:110) Γ (0) b , Γ (1) b , . . . Γ ( s ) b (cid:111) . N Σ is given in view of its definition by N Σ = 1 √ (cid:112) e ( r )( e ( u ) + e ( r )) (cid:16) e ( r ) e + ( e ( u ) + e ( r )) e (cid:17) = 1 √ (cid:112) − Υ + O ( (cid:15) ) (cid:16) (1 + O ( (cid:15) )) e + (2 − Υ + O ( (cid:15) )) e (cid:17) where we used the bootstrap assumptions. .1. PRELIMINARIES Lemma 5.1.1.
Consider the global frame of Proposition 3.5.5 and the above definition of Γ g and Γ b . Let an integer k loss and a small constant δ > satisfying ≤ k loss ≤ δ dec k large − k small ) , δ = k loss k large − k small . (5.1.6) Then, the Ricci coefficients and curvature components with respect to the global frame ofProposition 3.5.5 satisfy ξ = 0 on r ≥ m , max ≤ k ≤ k small + k loss sup M (cid:40)(cid:16) r τ + δ dec − δ + rτ δ dec − δ (cid:17) | d k Γ g | + rτ δ dec − δ | d k Γ b | + (cid:16) r + δB + r τ + δ dec − δ + r τ δ dec − δ (cid:17)(cid:16) | d k α | + | d k β | (cid:17) + (cid:16) r τ + δ dec − δ + r τ δ dec − δ (cid:17) | d k ˇ ρ | + τ δ dec − δ (cid:16) r | d k β | + r | d k α | (cid:17)(cid:41) (cid:46) (cid:15), max ≤ k ≤ k small + k loss sup M (cid:40) r τ δ dec − δ | d k − e (Γ g ) | + r ( τ + 2 r ) δ dec − δ (cid:16) | d k − e ( α ) | + | d k e ( β ) | (cid:17)(cid:41) (cid:46) (cid:15). Proof. In r ≥ m , the global frame of Proposition 3.5.5 coincides with a conformalrenormalization of the second frame of ( ext ) M , see Proposition 3.4.6. The estimatesthere follow immediately from the ones of Proposition 3.4.6. In the matching region7 / m ≤ r ≤ m , the estimates are stated in Proposition 3.5.5. Finally, for ( ext ) M ( r ≤ / m ) and ( int ) M , the estimates follow directly from interpolation between the bootstrapassumptions on decay for k ≤ k small and the pointwise estimates of Proposition 3.4.5 for k ≤ k large − Recall in particular that the global frame of Proposition 3.5.5 is such that η ∈ Γ g . Recall that we have0 < δ dec (cid:28) , δ dec k large (cid:29) , k small = (cid:22) k large (cid:23) + 1 . In particular, we have δ dec ( k large − k small ) (cid:29) k loss satisfying therequired constraints. We will in fact choose k loss = 33, see (5.2.3). CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) The following lemma will be important in what follows.
Lemma 5.1.2.
For the solution q to the wave equation (5.0.1) , the structure of the errorterm N can be written schematically as follows N = N g + e ( rN g ) + N m [ q ] (5.1.7) where, N g = r d ≤ (Γ g · ( α, β )) ,N m [ q ] = d ≤ (Γ g · q ) . (5.1.8) Moreover, for every k ≤ k large − we have schematically, d k N = d ≤ k N g + e ( d k ( rN g )) + d k N m [ q ] . (5.1.9) Remark 5.1.3.
In fact, (5.1.7) and (5.1.9) also contain lower order terms which arestrictly better in powers of r and contain at most the same number of derivatives. Forconvenience, we drop them in the rest of the proof of Theorem M1.Proof. For k = 0, this is an immediate consequence of (5.0.2). For the higher derivativeswe write, d k ( e ( rN g )) = e ( d k ( rN g )) + [ d k , e ]( rN g ) . In view of the formula for [ e , d / ] of Lemma 2.2.13, and the commutator formula for [ e , e ],we have, schematically,[ e , e ] = 0 , [ d /, e ] = Γ b d + Γ b , [ re , e ] = (cid:18) r + Γ b (cid:19) d . In view of our assumptions. (cid:12)(cid:12) d i (Γ b ) (cid:12)(cid:12) ≤ r − (cid:15), i ≤ k large − , Γ b is at least as good as r − , and hence, we deduce, schematically,[ d , e ] = 1 r d + 1 r . On the other hand, we have, schematically,[ d , r ] = r .1. PRELIMINARIES k ≤ k large − d k , e ]( rN g ) = (cid:88) i + j ≤ k − d i (cid:18) r d + 1 r (cid:19) d j ( rN g )= d ≤ k N g as desired. We restrict our attention to the region M ( τ , τ ) = M ∩ { τ ≤ τ ≤ τ } . For a given ψ ∈ s ( M ) we introduce the following quantities, for 0 ≤ τ < τ ≤ τ ∗ . Morawetz bulk quantities
Consider the vectorfields, T := 12 ( e + Υ e ) , R := 12 ( e − Υ e ) . (5.1.10)Let θ a smooth bump function equal 1 on | Υ | ≤ δ H vanishing for | Υ | ≥ δ H and definethe modified vectorfields,˘ R := θ
12 ( e − e ) + (1 − θ )Υ − R = 12 (cid:104) ˘ θe − e (cid:105) , ˘ T := θ
12 ( e + e ) + (1 − θ )Υ − T = 12 (cid:104) ˘ θe + e (cid:105) , (5.1.11)where ˘ θ = θ + Υ − (1 − θ ). Note that,˘ θ = (cid:40) | Υ | ≤ δ H , Υ − for | Υ | ≥ δ H . (5.1.12) Remark 5.1.4.
Note that ˘ R + ˘ T = e , − ˘ R + ˘ T = e in ( int ) M and ˘ R + ˘ T = Υ − e , − ˘ R + ˘ T = e in M > m . CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) We define the quantitiesMor[ ψ ]( τ , τ ) : = (cid:90) M ( τ ,τ ) r | ˘ Rψ | + 1 r | ψ | + (cid:18) − mr (cid:19) r (cid:18) |∇ / ψ | + 1 r | ˘ T ψ | (cid:19) , Morr[ ψ ]( τ , τ ) : = Mor[ ψ ]( τ , τ ) + (cid:90) M > m ( τ ,τ ) r − − δ | e ( ψ ) | , (5.1.13)with m = m ( τ, r ) = m ( u, r ) the Hawking mass in M . The constant δ > ψ ]( τ , τ ) is given below,Morr[ ψ ]( τ , τ ) = (cid:90) ( trap ) M ( τ ,τ ) | Rψ | + r − | ψ | + (cid:18) − mr (cid:19) (cid:18) |∇ / ψ | + 1 r | T ψ | (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r − (cid:0) | e ψ | + r − | ψ | (cid:1) + r − |∇ / ψ | + r − − δ | e ψ | (5.1.14)where ( trap (cid:14) ) M denotes the complement of ( trap ) M . Weighted bulk quantities
Define, for 0 < p < B p ; R [ ψ ]( τ , τ ) : = (cid:90) M ≥ R ( τ ,τ ) r p − (cid:0) p | ˇ e ( ψ ) | + (2 − p ) |∇ / ψ | + r − | ψ | (cid:1) ,B p [ ψ ]( τ , τ ) : = Morr[ ψ ]( τ , τ ) + ˙ B p ; 4 m [ ψ ]( τ , τ ) . (5.1.15)The bulk quantity B p [ ψ ]( τ , τ ) is equivalent to B p [ ψ ]( τ , τ ) (cid:39) (cid:90) τ τ M p − [ ψ ]( τ ) dτ where, M p − [ ψ ]( τ ) = (cid:90) Σ ≤ m ( τ ) | ˘ Rψ | + r − | ψ | + (cid:18) − mr (cid:19) (cid:18) |∇ / ψ | + m r | ˘ T ψ | (cid:19) + (cid:90) Σ ≥ m ( τ ) r p − (cid:0) p | e ( ψ ) | + (2 − p ) |∇ / ψ | + r − | ψ | (cid:1) + (cid:90) Σ ≥ m ( τ ) r − − δ | e ψ | . This equivalence follows from the coarea formula and the fact that the lapse of the τ -foliation iscontrolled uniformly from above and below. .1. PRELIMINARIES Remark 5.1.5.
Note that, for δ ≤ p ≤ − δ , B p [ ψ ]( τ , τ ) : = Morr [ ψ ]( τ , τ ) + ˙ B p ; 4 m [ ψ ]( τ , τ ) is equivalent to, B p [ ψ ]( τ , τ ) (cid:39) Morr [ ψ ]( τ , τ ) + (cid:90) M ≥ m ( τ ,τ ) r p − (cid:0) | ˇ e ( ψ ) | + |∇ / ψ | + r − | e ψ | + r − | ψ | (cid:1) . Indeed, (cid:90) M ≥ m ( τ ,τ ) r p − | e ψ | (cid:46) (cid:90) M ≥ m ( τ ,τ ) r − − δ | e ψ | . Therefore, since r ( | ˇ e ( ψ ) | + |∇ / ψ | ) (cid:46) | d ψ | , we have, B p [ ψ ]( τ , τ ) (cid:39) Morr [ ψ ]( τ , τ ) + (cid:90) M ≥ m ( τ ,τ ) r p − (cid:0) | d ψ | + | ψ | (cid:1) . (5.1.16) Basic energy-flux quantity
The basic energy-flux quantity on a hypersurface Σ( τ ) is defined by E [ ψ ]( τ ) = (cid:90) Σ( τ ) (cid:18)
12 ( N Σ , e ) | e ψ | + 12 ( N Σ , e ) | e ψ | + |∇ / ψ | + r − | ψ | (cid:19) . (5.1.17)Here N Σ denotes a choice for the normal to Σ so that in particular we have N Σ = (cid:40) N Σ = e on ( int ) Σ ,N Σ = e on ( ext ) Σ , (5.1.18)and, in view of (5.1.5),( N Σ , e ) ≤ − N Σ , e ) ≤ − ( trap ) Σ . (5.1.19) Weighted Energy-Flux type quantities
We have ˙ E p ; R [ ψ ]( τ ) := (cid:90) Σ ≥ R ( τ ) r p (cid:16) | ˇ e ψ | + r − | ψ | (cid:17) for p ≤ − δ, (cid:90) Σ ≥ R ( τ ) r p (cid:16) | ˇ e ψ | + r − p − − δ | ψ | (cid:17) for p > − δ, (5.1.20)48 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) and E p [ ψ ]( τ ) := E [ ψ ]( τ ) + ˙ E p ; 4 m [ ψ ]( τ ) . (5.1.21)Here ˇ e denotes the first order operatorˇ e ψ = r − Υ − e ( rψ ) . (5.1.22) Remark 5.1.6.
To control the weighted quantities (5.1.21) , it will be convenient to in-troduce in ( ext ) M ( r ≥ m ) the following renormalized frame e (cid:48) = Υ − e , e (cid:48) = Υ e , e (cid:48) θ = e θ . In particular, this yields ˇ e ψ = r − e (cid:48) ( rψ ) . Note also that we have the following alternate form ˇ e ψ = e (cid:48) ψ + r − ψ + e (cid:48) ( r ) − r ψ where e (cid:48) ( r ) − − e ( r ) − O ( (cid:15)r − ) in view of our assumption on Γ g . Flux quantities
The boundary of M ( τ , τ ) is given by ∂ M ( τ , τ ) = Σ( τ ) ∪ Σ( τ ) ∪ A ( τ , τ ) ∪ Σ ∗ ( τ , τ ) . Our basic flux quantity along the spacelike hypersurfaces A and Σ ∗ is given by F [ ψ ]( τ , τ ) := (cid:90) A ( τ ,τ ) (cid:16) δ − H | e Ψ | + δ H | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) + (cid:90) Σ ∗ ( τ ,τ ) (cid:16) | e Ψ | + | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) , (5.1.23)with A ( τ , τ ) = A ∩ M ( τ , τ ) and Σ ∗ ( τ , τ ) = Σ ∗ ∩ M ( τ , τ ). Weighted flux quantities ˙ F p [ ψ ]( τ , τ ) := (cid:90) Σ ∗ ( τ ,τ ) r p (cid:16) | e ψ | + |∇ / ψ | + r − | ψ | (cid:17) ,F p [ ψ ]( τ , τ ) := F [ ψ ]( τ , τ ) + ˙ F p [ ψ ]( τ , τ ) . (5.1.24) .1. PRELIMINARIES Weighted quantities for the inhomogeneous term N Recall the decomposition (5.1.7) for the inhomogeneous term NN = N g + e ( rN g ) + N m [ q ] . We define, for p ≥ δ , I p [ N g ]( τ , τ ) = (cid:18) (cid:90) τ τ dτ (cid:107) N g (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p | N g | + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p | N g || e ( N g ) | + sup τ ∈ [ τ ,τ ] (cid:90) Σ( τ ) r p +2 (cid:12)(cid:12) N g (cid:12)(cid:12) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | e ( N g ) | . (5.1.25) Remark 5.1.7.
While N m [ q ] is present in the decomposition of the inhomogeneous term N , (5.1.25) only contains a norm for N g . In fact, N m [ q ] will always be absorbed by theleft hand side wherever it appears. Higher derivative quantities
We define the higher order derivative quantities E s [ ψ ] , Mor s [ ψ ] , Morr s [ ψ ] , E sp [ ψ ], B sb [ ψ ], M sp [ ψ ], F s [ ψ ], F sp [ ψ ], I sp [ N g ] by the obvious procedure, Q s [ ψ ] = (cid:88) k ≤ s Q [ d k ψ ] . Remark 5.1.8.
Note that in view of Remark 5.1.5 we can also write, equivalently, for p < − δ , B sp [ ψ ]( τ , τ ) = M orr s [ ψ ]( τ , τ ) + (cid:90) M > m ( τ ,τ ) r p − | d ≤ s ψ | . (5.1.26)50 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) Decay Norms
We introduce, E sp,d [ ψ ] : = sup ≤ τ ≤ τ ∗ (1 + τ ) d E sp [ ψ ]( τ ) , B sp,d [ ψ ] : = sup ≤ τ ≤ τ ∗ (1 + τ ) d B sp [ ψ ]( τ, τ ∗ ) , (cid:39) sup ≤ τ ≤ τ ∗ (1 + τ ) d (cid:90) τ ∗ τ M sp − [ ψ ]( τ (cid:48) ) dτ (cid:48) , F sp,d [ ψ ] : = sup ≤ τ ≤ τ ∗ (1 + τ ) d F sp [ ψ ]( τ, τ ∗ ) , I sp,d [ N g ] : = sup ≤ τ ≤ τ ∗ (1 + τ ) d I sp [ N g ]( τ, τ ∗ ) . (5.1.27) Recall that we have to prove for k ≤ k small + 20 | d k q | (cid:46) (cid:15) r − (1 + τ ) − − δ extra , | d k q | (cid:46) (cid:15) r − (1 + τ ) − − δ extra , | d k e ( q ) | (cid:46) (cid:15) r − (1 + τ ) − − δ extra , and (cid:90) ( int ) M ( τ,τ ∗ ) | d k e q | + (cid:90) Σ ∗ ( τ,τ ∗ ) | d k e q | (cid:46) (cid:15) (1 + τ ) − − δ extra , for some constant δ extra such that δ dec < δ extra < δ dec . q The following result establishes decay of flux estimates for q . Theorem 5.2.1.
Let < q < be a fixed number and s ≤ k small +25 . Then, for all δ > we have, with a constant C depending only on s , δ and q such that for all δ ≤ p ≤ − δ ,we have E sp, q − p [ q ] + B sp, q − p [ q ] + F sp, q − p [ q ] (cid:46) E s +2 q [ˇ q ](0) + E s +42 − δ [ q ](0) + I s +5 q +2 , [ N g ] + I s +5 δ, q − δ [ N g ] , (5.2.1) .2. PROOF OF THEOREM M1 where we recall that the decay norms I sp,d [ N g ] are defined by, I sp,d [ N g ] = sup ≤ τ ≤ τ ∗ (1 + τ ) d I sp [ N g ]( τ, τ ∗ ) . Theorem 5.2.1 will be proved in section 5.4.3.To prove Theorem M1 we have to eliminate the norms I sp,d [ N g ] on the right hand side ofTheorem 5.2.1. Proposition 5.2.2.
Let s ≤ k small + 30 and assume q < δ dec − δ (5.2.2) where δ = 33 k large − k small = 33 k large − (cid:98) k large (cid:99) − is the small constant appearing in Lemma 5.1.1. Then, the following estimates hold true, I sq +2 , [ N g ] + I sδ, q − δ [ N g ] (cid:46) (cid:15) . The proof of Proposition 5.2.2 is postponed to section 5.2.3. Together with Theorem5.2.1, Proposition 5.2.2 immediately yields the proof of the following corollary.
Corollary 5.2.3.
In addition to the assumptions of Theorem 5.2.1 we assume δ dec < q < δ dec − δ (5.2.4) where δ > is given by (5.2.3) . Then for a sufficiently small bootstrap constant (cid:15) > ,for all s ≤ k small + 25 and for all δ ≤ p ≤ − δ , we have E sp, q − p [ q ] + B sp, q − p [ q ] + F sp, q − p [ q ] (cid:46) E s +2 q [ˇ q ](0) + E s +42 − δ [ q ](0) + (cid:15) . Since (cid:15) = (cid:15) / , and in view of the control on q at τ = 0 provided by Theorem M0,we immediately deduce from Corollary 5.2.3, For all 0 < q ≤ q , δ ≤ p ≤ − δ , and s ≤ k small + 25, E sp, q − p [ q ] + B sp, q − p [ q ] + F sp, q − p [ q ] (cid:46) (cid:15) . (5.2.5)We will also need the following two propositions concerning L estimates on spheres.52 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) Proposition 5.2.4.
On any S = S ( τ, r ) ⊂ Σ( τ ) , for s ≤ k small + 25 , (1 + τ ) q (cid:90) S r | q ( s ) | (cid:46) (cid:0) E s δ, q − δ [ q ] (cid:1) (cid:0) E s − δ, q + δ [ q ] (cid:1) (5.2.6) and, r − (1 + τ ) q − δ (cid:90) S r | q ( s ) | (cid:46) E sδ, q − δ [ q ] . (5.2.7) Proposition 5.2.5.
We have for s ≤ k small + 25(1 + τ ) q − δ (cid:90) Σ ∗ ( τ,τ ∗ ) | e d ≤ s q | (cid:46) F sδ, q − δ [ q ] . (5.2.8) Also, on any S = S ( τ, r ) ⊂ Σ( τ ) , for s ≤ k small + 23 , we have (1 + τ ) q − δ (cid:90) S r | e d ≤ s q | (cid:46) (cid:15) + F s +1 δ, q − δ [ q ] + E s +2 δ, q − δ [ q ] . (5.2.9)The proof of Proposition 5.2.4 is postponed to section 5.4.4, and the proof of Proposition5.2.5 is postponed to section 5.4.5.We now conclude the proof of Theorem M1. Indeed, in view of (5.2.5), Proposition 5.2.4and Proposition 5.2.5, we infer for s ≤ k small + 25(1 + τ ) q − δ (cid:90) ( int ) M ( τ,τ ∗ ) | d ≤ s +1 q | (cid:46) (cid:15) , (1 + τ ) q (cid:90) S r | q ( s ) | (cid:46) (cid:15) ,r − (1 + τ ) q − δ (cid:90) S r | q ( s ) | (cid:46) (cid:15) , (1 + τ ) q − δ (cid:90) Σ ∗ ( τ,τ ∗ ) | e d ≤ s q | (cid:46) (cid:15) , and for s ≤ k small + 23 (1 + τ ) q − δ (cid:90) S | d s e q | (cid:46) (cid:15) . In view of the standard Sobolev inequality on the 2-surfaces S i.e., (cid:107) ψ (cid:107) L ∞ ( S ) (cid:46) r − (cid:107) ( r ∇ / ) ≤ ψ (cid:107) L ( S ) , .2. PROOF OF THEOREM M1 s ≤ k small + 23 | q ( s ) | (cid:46) (cid:15) r − (1 + τ ) − − q , | q ( s ) | (cid:46) (cid:15) r − (1 + τ ) − − q − δ , and for s ≤ k small + 21 | d s e ( q ) | (cid:46) (cid:15) r − (1 + τ ) − − q − δ . Recall that q > δ dec and that δ > q − δ > δ dec . In particular, we may choose δ extra := q − δ , δ extra > δ dec , (5.2.10)which together with the above estimates for q implies for s ≤ k small + 25(1 + τ ) q − δ (cid:90) ( int ) M ( τ,τ ∗ ) | d ≤ s +1 q | (cid:46) (cid:15) , (1 + τ ) δ extra (cid:90) Σ ∗ ( τ,τ ∗ ) | e d ≤ s q | (cid:46) (cid:15) , for s ≤ k small + 23 | q ( s ) | (cid:46) (cid:15) r − (1 + τ ) − − δ extra , | q ( s ) | (cid:46) (cid:15) r − (1 + τ ) − − δ extra , and for s ≤ k small + 21 | d s e ( q ) | (cid:46) (cid:15) r − (1 + τ ) − − δ extra as desired. This concludes the proof of Theorem M1. Recall that, I p [ N g ]( τ , τ ) = (cid:18) (cid:90) τ τ dτ (cid:107) N g (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p | N g | + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p | N g || e ( N g ) | + sup τ ∈ [ τ ,τ ] (cid:90) Σ( τ ) r p +2 (cid:12)(cid:12) N g (cid:12)(cid:12) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | e ( N g ) | CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) and, I sp,d [ N g ] = sup ≤ τ ≤ τ ∗ (1 + τ ) d I sp [ N g ]( τ, τ ∗ ) . Since we have r δ (1 + τ ) q − δ (cid:46) r q + (1 + τ ) q , and (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r | d ≤ s e ( N g ) || d ≤ s N g | (cid:46) (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r | d ≤ s N g | + (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r | d ≤ s e ( N g ) | , we infer I sq +2 , [ N g ] + I sδ, q − δ [ N g ] (5.2.11) (cid:46) sup ≤ τ ≤ τ ∗ (cid:34) (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r q | d ≤ s +1 N g | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r q (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) +(1 + τ ) q (cid:32) (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r | d ≤ s N g | + (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r δ | d ≤ s e ( N g ) | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) (cid:33) + (1 + τ ) q (cid:18) (cid:90) τ ∗ τ dτ (cid:48) (cid:107) d ≤ s N g (cid:107) L ( ( trap ) Σ( τ (cid:48) )) (cid:19) (cid:35) . In order to prove Proposition 5.2.2, it suffices to estimate the right-hand side of (5.2.11).To this end, we will estimate separately the terms with highest power of r , i.e. the firsttwo terms, and the terms with highest power the τ , i.e. the four last terms. Terms with highest power of r in (5.2.11)We estimate the first two terms of (5.2.11). Recall from Lemma 5.1.2 that N g = r d ≤ (Γ g · ( α, β )) . Recall from Lemma 5.1.1 we havemax ≤ k ≤ k large − sup ( ext ) M ( r ≥ m ) r + δB (cid:16) | d k α | + | d k β | (cid:17) (cid:46) (cid:15). We infer for s ≤ k large − | d ≤ s +1 N g | (cid:46) (cid:15)r − − δB | r d ≤ s +3 Γ g | .2. PROOF OF THEOREM M1 s ≤ k large −
6, we deduce (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r q | d ≤ s +1 N g | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r q (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) (cid:46) (cid:15) (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r − − δ B + q ( r d ≤ s +3 Γ g ) + (cid:15) sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r − − δ B + q ( r d ≤ s +2 Γ g ) (cid:17) . Since we also have for s ≤ k large − r ≥ m (cid:90) { r = r } ( r d ≤ s +3 Γ g ) (cid:46) (cid:15) , (cid:90) M r ≤ m ( d ≤ s +3 Γ g ) (cid:46) (cid:15) , sup M | r d ≤ s +2 Γ g | (cid:46) (cid:15), we deduce (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r q | d ≤ s +1 N g | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r q (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) (cid:46) (cid:15) (cid:18) (cid:90) r ≥ m drr δ B − q (cid:19) . Since q < δ dec and δ B ≥ δ dec , we have q < δ B and hence, we obtain for s ≤ k large − (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r q | d ≤ s +1 N g | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r q (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) (cid:46) (cid:15) . This is the desired control of the terms with highest power of r in (5.2.11). Terms with highest power of τ in (5.2.11)We estimate the four last terms of (5.2.11). In view of Lemma 5.1.1 with k loss = 33, sothat δ = 33 k large − k small − k large − (cid:98) k large (cid:99) − , we have (cid:12)(cid:12)(cid:12) d ≤ k small +33 Γ g (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − τ − / − δ dec +2 δ , (cid:12)(cid:12)(cid:12) d ≤ k small +33 Γ g (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − τ − − δ dec +2 δ , (cid:12)(cid:12)(cid:12) d ≤ S +32 e Γ g (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − [ τ − − δ dec ] − δ (cid:46) (cid:15)r − τ − − δ dec +2 δ , (cid:12)(cid:12)(cid:12) d ≤ k small +33 ( α, β ) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − ( τ + r ) − / − δ dec +2 δ , (cid:12)(cid:12)(cid:12) d ≤ k small +33 ( α, β ) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − ( τ + r ) − − δ dec +2 δ , (cid:12)(cid:12)(cid:12) d ≤ S +32 e ( α, β ) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − − δ [ τ − − δ dec ] − δ (cid:46) (cid:15)r − τ − − δ dec +2 δ . CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) In particular, together with the bootstrap assumption for k ≤ k small , the pointwise bound | d ≤ k large − α | + | d ≤ k large − β | (cid:46) (cid:15)r − − δB and since N g = r d ≤ (Γ g · ( α, β )), we infer for s ≤ k small + 30 | d s N g | (cid:46) (cid:15) r − τ − − δ dec +2 δ | d s N g | (cid:46) (cid:15) r − τ − − δ dec +2 δ , | d s e ( N g ) | (cid:46) (cid:15) r − τ − − δ dec +2 δ , | d s e ( N g ) | (cid:46) (cid:15) r − − δB τ − − δ dec +2 δ . (5.2.12)Using these 4 bounds and interpolation, we infer for δ > τ ) q (cid:32) (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r | d ≤ s N g | + (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r δ | d ≤ s e ( N g ) | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) (cid:33) + (1 + τ ) q (cid:18) (cid:90) τ ∗ τ dτ (cid:48) (cid:107) d ≤ s N g (cid:107) L ( ( trap ) Σ( τ (cid:48) )) (cid:19) (cid:46) (cid:15) (1 + τ ) q (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r ( r − τ (cid:48)− − δ dec +2 δ ) δ ( r − τ (cid:48)− − δ dec +2 δ ) − δ + (cid:15) (1 + τ ) q (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r δ ( r − τ (cid:48)− − δ dec +2 δ ) − δ ( r − − δB τ (cid:48)− − δ dec +2 δ ) δ + (cid:15) (1 + τ ) q sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r ( r − τ (cid:48)− − δ dec +2 δ ) + (cid:15) (1 + τ ) q (cid:18) (cid:90) τ ∗ τ τ (cid:48)− − δ dec +2 δ dτ (cid:48) (cid:19) (cid:46) (cid:15) (1 + τ ) q (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r − − δδ B τ (cid:48)− − δ dec + δ +4 δ +2 δδ dec + (cid:15) (1 + τ ) q sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r − τ (cid:48)− − δ dec +4 δ + (cid:15) (1 + τ ) q (cid:18) (cid:90) τ ∗ τ τ (cid:48)− − δ dec +2 δ dτ (cid:48) (cid:19) and since δ >
0, we obtain(1 + τ ) q (cid:32) (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r | d ≤ s N g | + (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r δ | d ≤ s e ( N g ) | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) (cid:33) + (1 + τ ) q (cid:18) (cid:90) τ ∗ τ dτ (cid:48) (cid:107) d ≤ s N g (cid:107) L ( ( trap ) Σ( τ (cid:48) )) (cid:19) (cid:46) (cid:15) (1 + τ ) q − δ dec + δ +4 δ +2 δδ dec . .3. IMPROVED WEIGHTED ESTIMATES q < δ dec − δ , there exists δ > q − δ dec + δ + 4 δ + 2 δδ dec ≤ , and hence (1 + τ ) q (cid:32) (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r | d ≤ s N g | + (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r δ | d ≤ s e ( N g ) | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) (cid:33) + (1 + τ ) q (cid:18) (cid:90) τ ∗ τ dτ (cid:48) (cid:107) d ≤ s N g (cid:107) L ( ( trap ) Σ( τ (cid:48) )) (cid:19) (cid:46) (cid:15) . This is the desired control of the terms with highest power of τ in (5.2.11). Together with(5.2.11) and the above control of the terms with highest power of r , we infer I sq +2 , [ N g ] + I sδ, q − δ [ N g ] (cid:46) sup ≤ τ ≤ τ ∗ (cid:34) (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r q | d ≤ s +1 N g | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r q (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) +(1 + τ ) q (cid:32) (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r | d ≤ s N g | + (cid:90) ( trap (cid:14) ) M ( τ,τ ∗ ) r δ | d ≤ s e ( N g ) | + sup τ (cid:48) ∈ [ τ,τ ∗ ] (cid:90) Σ( τ (cid:48) ) r (cid:12)(cid:12) d ≤ s N g (cid:12)(cid:12) (cid:33) + (1 + τ ) q (cid:18) (cid:90) τ ∗ τ dτ (cid:48) (cid:107) d ≤ s N g (cid:107) L ( ( trap ) Σ( τ (cid:48) )) (cid:19) (cid:35) (cid:46) (cid:15) which is the desired estimate. This concludes the proof of Proposition 5.2.2. The goal of this section is to prove the two following theorems on improved weightedestimates.
Theorem 5.3.1.
Assume q verifies following wave equation, see (5.0.1) , (cid:3) q + κκ q = N with N given, in view of Lemma 5.1.2, by N = N g + e ( rN g ) + N m [ q ] . Then, for any δ ≤ p ≤ − δ , ≤ s ≤ k small + 30 , sup τ ∈ [ τ ,τ ] E sp [ q ]( τ ) + B sp [ q ]( τ , τ ) + F sp [ q ]( τ , τ ) (cid:46) E sp [ q ]( τ ) + I s +1 p [ N g ]( τ , τ ) . (5.3.1)58 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) The next result deals with weighted estimates for the quantityˇ q = f ˇ e q , (5.3.2)where f is a fixed smooth function of r defined as follows, f ( r ) = (cid:40) r for r ≥ m , r ≤ m . (5.3.3) Theorem 5.3.2.
Assume q verifies equation, see (5.0.1) , (cid:3) q + κκ q = N with, N = N g + e ( rN g ) + N m [ q ] as in Lemma 5.1.2. Then, for any − δ < q ≤ − δ , ≤ s ≤ k small + 29 , sup τ ∈ [ τ ,τ ] E sq [ˇ q ]( τ ) + B sq [ˇ q ]( τ , τ ) (cid:46) E sq [ˇ q ]( τ ) + E s +1 q +1 [ q ]( τ ) + I s +2 q +2 [ N g ]( τ , τ ) . (5.3.4) Remark 5.3.3.
Note that in (5.3.1) and (5.3.4) , the term N m [ q ] does not appear in theright-hand side since it turns out that it can be absorbed by the left hand side. The proof of Theorem 5.3.1 is postponed to section 5.3.2, and the proof of Theorem 5.3.2is postponed to section 5.3.3. These proofs will rely on weighted energy flux estimatesintroduced in the next section.
Assume given a spacetime M verifying the bootstrap assumptions with small constant (cid:15) >
0. The proof of Theorem 5.3.1 and Theorem 5.3.2 will rely on estimates stated belowfor solutions ψ ∈ s ( M ) of the equation, (cid:3) ψ + V ψ = N, V = κκ. (5.3.5) Basic weighted estimatesTheorem 5.3.4.
Recall the definitions in (5.1.21) , (5.1.15) . The following holds for any ≤ s ≤ k small + 30 . For all δ ≤ p ≤ − δ , we have, sup τ ∈ [ τ ,τ ] E sp [ ψ ]( τ ) + B sp [ ψ ]( τ , τ ) + F sp [ ψ ]( τ , τ ) (cid:46) E sp [ ψ ]( τ ) + J sp [ ψ, N ]( τ , τ ) , (5.3.6) .3. IMPROVED WEIGHTED ESTIMATES where, for p ≥ δ , we have introduced the notation J p,R [ ψ, N ]( τ , τ ) : = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) M ≥ R ( τ ,τ ) r p ˇ e ψN (cid:12)(cid:12)(cid:12)(cid:12) ,J p [ ψ, N ]( τ , τ ) : = (cid:18) (cid:90) τ τ dτ (cid:107) N (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | N | + J p, m [ ψ, N ]( τ , τ ) , (5.3.7) and J sp [ ψ, N ]( τ , τ ) := (cid:88) k ≤ s J p [ d k ψ, d k N ]( τ , τ ) . The proof of Theorem 5.3.4 is postponed to section 10.4.5.
Higher weighted estimates
The next result deals with weighted estimates for the quantityˇ ψ = f ˇ e ψ, (5.3.8)where f is a fixed smooth function of r defined as follows, f ( r ) = (cid:40) r for r ≥ m , r ≤ m . (5.3.9) Theorem 5.3.5.
The following holds for any − δ < q ≤ − δ , ≤ s ≤ k small + 29 , sup τ ∈ [ τ ,τ ] E sq [ ˇ ψ ]( τ ) + B sq [ ˇ ψ ]( τ , τ ) (cid:46) E sq [ ˇ ψ ]( τ ) + ˇ J sq [ ˇ ψ, N ]( τ , τ )+ E s +1max( q,δ ) [ ψ ]( τ ) + J s +1max( q,δ ) [ ψ, N ] , (5.3.10) where we have introduced the notation ˇ J q [ ˇ ψ, N ]( τ , τ ) := J q, m (cid:20) ˇ ψ, r (cid:18) e N + 3 r N (cid:19)(cid:21) ( τ , τ )= (cid:90) M ≥ m ( τ ,τ ) r q +2 (cid:0) ˇ e ˇ ψ (cid:1) · (cid:18) e N + 3 r N (cid:19) , and ˇ J sq [ ˇ ψ, N ]( τ , τ ) := (cid:88) k ≤ s ˇ J q [ d k ˇ ψ, d k N ]( τ , τ ) . CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) The proof of Theorem 5.3.5 is postponed to section 10.4.6.We now proceed to the proof of Theorem 5.3.1 and Theorem 5.3.2 in the next 2 sections.The proofs will follow from the structure of the nonlinear term N of q provided by Lemma5.1.2 and the use of Theorem 5.3.4 and Theorem 5.3.5. Applying Theorem 5.3.4 to the equation for q , with N given by Lemma 5.1.2, we derivecorresponding estimates with the norm J sp [ q , N ]( τ , τ ) on the right hand side, i.e. for0 ≤ s ≤ k small + 30, and for δ ≤ p ≤ − δ ,sup τ ∈ [ τ ,τ ] E sp [ q ]( τ ) + B sp [ q ]( τ , τ ) + F sp [ q ]( τ , τ ) (cid:46) E sp [ q ]( τ ) + J sp [ q , N ]( τ , τ ) . (5.3.11)To prove Theorem 5.3.1, it suffices, in view of (5.3.11), to estimate J sp [ q , N ]( τ , τ ). Recallthat, see (5.3.7) and (5.1.25) I p [ N ]( τ , τ ) = (cid:18) (cid:90) τ τ dτ (cid:107) N (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p | N | + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p | N g || e ( N g ) | + sup τ ∈ [ τ ,τ ] (cid:90) Σ( τ ) r p +2 (cid:12)(cid:12) N (cid:12)(cid:12) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | e ( N g ) | and, J p,R [ q , N ] = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) M ≥ R ( τ ,τ ) r p ˇ e ( q ) N (cid:12)(cid:12)(cid:12)(cid:12) ,J p [ q , N ]( τ , τ ) = (cid:18) (cid:90) τ τ dτ (cid:107) N (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | N | + J sp, m [ q , N ]( τ , τ ) ,J sp [ q , N ]( τ , τ ) = (cid:88) k ≤ s J p [ d k q , d k N ] , Recall also from (5.1.9) d k N = d ≤ k N g + e ( d k ( rN g )) + d k N m [ q ] (5.3.12) .3. IMPROVED WEIGHTED ESTIMATES Case of N m [ q ] . Recall that N m [ q ] = d ≤ (Γ g · q ). We have, schematically, d k N m [ q ] = d k (Γ g · q ) = (cid:88) i + j = k +1 d ≤ i Γ g d ≤ j q . We make use of the following consequence of the bootstrap assumptions for k ≤ k large − (cid:12)(cid:12) d ≤ k Γ g (cid:12)(cid:12) ≤ (cid:15)r − to deduce, (cid:12)(cid:12) d k N m [ q ] (cid:12)(cid:12) (cid:46) (cid:15)r − (cid:12)(cid:12) d ≤ k +1 q (cid:12)(cid:12) . (5.3.13)We deduce, J sp, m [ q , N m [ q ]]( τ , τ ) (cid:46) (cid:88) k ≤ s (cid:90) M ≥ m ( τ ,τ ) r p (cid:12)(cid:12) ˇ e q ( k ) (cid:12)(cid:12) (cid:12)(cid:12) d k N m [ q ] (cid:12)(cid:12) (cid:46) (cid:15) (cid:88) k ≤ s (cid:90) M ≥ m ( τ ,τ ) r p − (cid:12)(cid:12) d k q (cid:12)(cid:12) . Thus, recalling Remark 5.1.8, we infer J sp, m [ q , N m [ q ]]( τ , τ ) (cid:46) (cid:15)B sp [ q ]( τ , τ ) . (5.3.14)Next, we estimate in view of (5.3.13) (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | d k N m [ q ] | (cid:46) (cid:15) (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ − | d ≤ k +1 q | which yields, using again Remark 5.1.8, (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | d k N m [ q ] | (cid:46) (cid:15)B sδ [ q ]( τ , τ ) . (5.3.15)We next estimate the integral (cid:90) τ τ dτ (cid:107) d k N m [ q ] (cid:107) L ( ( trap ) Σ( τ )) . In view of the definition of N m [ q ] = d ≤ (Γ g · q ), d k N m [ q ] = d ≤ k +1 (Γ g · q ) = (cid:88) i + j = k +1 d ≤ i Γ g d ≤ j q = d j ≤ ( k +1) / Γ g d ≤ k +1 q + d j ≤ ( k +1) / q d ≤ k +1 Γ g = J + J . CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) Now, since k +12 ≤ k small we have (cid:12)(cid:12)(cid:12) d j ≤ ( k +1) / Γ g (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) (1 + τ ) − − δ dec Hence, (cid:107) J (cid:107) L ( ( trap ) Σ( τ )) = (cid:90) ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d j ≤ ( k +1) / Γ g (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) d ≤ k +1 q (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) (1 + τ ) − − δ dec E s [ q ]( τ )i.e., (cid:107) J (cid:107) L ( ( trap ) Σ( τ )) (cid:46) (cid:15) (1 + τ ) − − δ dec ( E s [ q ]( τ )) / . For J we write, (cid:107) J (cid:107) L ( ( trap ) Σ( τ )) = (cid:90) ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d j ≤ ( k +1) / q (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) d ≤ k +1 Γ g (cid:12)(cid:12)(cid:12) (cid:46) (cid:32) sup ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d j ≤ ( k +1) / q (cid:12)(cid:12)(cid:12)(cid:33) (cid:90) ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d ≤ k +1 Γ g (cid:12)(cid:12)(cid:12) (cid:46) (cid:90) ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d ≤ ( k +1) / q (cid:12)(cid:12)(cid:12) (cid:90) ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d ≤ k +1 Γ g (cid:12)(cid:12)(cid:12) or, since ( k + 1) / ≤ s , (cid:107) J (cid:107) L ( ( trap ) Σ( τ )) (cid:46) (cid:20)(cid:90) ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d ≤ s q (cid:12)(cid:12)(cid:12) (cid:21) / (cid:20)(cid:90) ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d ≤ k +1 Γ g (cid:12)(cid:12)(cid:12) (cid:21) / . In view of the above estimates for J and J , we deduce, for all k ≤ s ≤ k large − (cid:90) τ τ dτ (cid:107) d k N m [ q ] (cid:107) L ( ( trap ) Σ( τ )) (cid:46) (cid:15) sup τ ≤ τ ≤ τ ( E s [ q ]( τ )) / + (cid:90) τ τ dτ (cid:20)(cid:90) ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d ≤ s q (cid:12)(cid:12)(cid:12) (cid:21) / (cid:20)(cid:90) ( trap ) Σ( τ ) (cid:12)(cid:12)(cid:12) d ≤ s Γ g (cid:12)(cid:12)(cid:12) (cid:21) / (cid:46) (cid:15) sup τ ≤ τ ≤ τ ( E s [ q ]( τ )) / + (cid:32)(cid:90) ( trap (cid:14) ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12) d ≤ s q (cid:12)(cid:12)(cid:12) (cid:33) (cid:32)(cid:90) M r ≤ m (cid:12)(cid:12)(cid:12) d ≤ s Γ g (cid:12)(cid:12)(cid:12) (cid:33) / Making use of the following consequence of the bootstrap assumptions (cid:32)(cid:90) M r ≤ m (cid:12)(cid:12)(cid:12) d ≤ s Γ g (cid:12)(cid:12)(cid:12) (cid:33) / (cid:46) (cid:15), .3. IMPROVED WEIGHTED ESTIMATES (cid:90) ( trap (cid:14) ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12) d ≤ s q (cid:12)(cid:12)(cid:12) (cid:46) Morr s [ q ]( τ , τ ) , we deduce, (cid:18) (cid:90) τ τ dτ (cid:107) d k N m [ q ] (cid:107) L ( ( trap ) Σ( τ )) (cid:19) (cid:46) (cid:15) sup τ ≤ τ ≤ τ E s [ q ]( τ ) + (cid:15) Morr s [ q ]( τ , τ ) (5.3.16)which together with (5.3.15) and (5.3.14) yields for any p ≥ δJ sp [ q , N m [ q ]]( τ , τ ) (cid:46) (cid:15) sup τ ≤ τ ≤ τ E s [ q ]( τ ) + (cid:15)B sp [ q ]( τ , τ ) . (5.3.17) Case of N g . We write, as before, J sp, m [ q , N g ]( τ , τ ) (cid:46) (cid:88) k ≤ s (cid:90) M ≥ m ( τ ,τ ) r p (cid:12)(cid:12) ˇ e q ( k ) d k N g (cid:12)(cid:12) (cid:46) (cid:88) k ≤ s (cid:16) (cid:90) M ≥ m ( τ ,τ ) r p − (cid:12)(cid:12) ˇ e q ( k ) (cid:12)(cid:12) (cid:17) / (cid:16) (cid:90) M ≥ m ( τ ,τ ) r p +1 (cid:12)(cid:12) d k N g (cid:12)(cid:12) (cid:17) / . Therefore, J sp, m [ q , N g ]( τ , τ ) (cid:46) (cid:0) B sp [ q ]( τ , τ ) (cid:1) / (cid:0) I sp [ N g ]( τ , τ ) (cid:1) / (cid:46) δ B sp [ q ]( τ , τ ) + δ − I sp [ N g ]( τ , τ )where δ > δ B sp [ q ]( τ , τ )by the left hand side of our main estimate.Also, we have in view of the definition of I sp [ N ]( τ , τ ) and the fact that p ≥ δ (cid:18) (cid:90) τ τ dτ (cid:107) d ≤ s N g (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | d ≤ s N g | (cid:46) I sp [ N g ]( τ , τ ) . Therefore, J sp [ q , N g ]( τ , τ ) = (cid:18) (cid:90) τ τ dτ (cid:107) d ≤ s N g (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | d ≤ s N g | + J sp, m [ q , N g ]( τ , τ ) (cid:46) I sδ [ N g ]( τ , τ ) + δ − I sp [ N g ]( τ , τ ) + δ B sp [ q ]( τ , τ ) , CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) i.e., J sp [ q , N g ]( τ , τ ) (cid:46) δ − I sp [ N g ]( τ , τ ) + δ B sp [ q ]( τ , τ ) . (5.3.18) Case of e ( rN g ) . First, note that we have (cid:18) (cid:90) τ τ dτ (cid:107) d ≤ s e ( rN g ) (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | d ≤ s e ( rN g ) | (cid:46) (cid:18) (cid:90) τ τ dτ (cid:107) d ≤ s +1 N g (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | d ≤ s N g | + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | d ≤ s e ( N g ) | where we used the fact that | d ≤ s e ( r ) | (cid:46) | d ≤ s r | (cid:46) r . Hence, we infer in view of thedefinition of I sp [ N ]( τ , τ ) and the fact that p ≥ δ (cid:18) (cid:90) τ τ dτ (cid:107) d ≤ s e ( rN g ) (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | d ≤ s e ( rN g ) | (cid:46) I s +1 p [ N g ]( τ , τ ) . (5.3.19)We then estimate J p, m [ q ( k ) , e ( d k ( rN g ))]( τ , τ ) , k ≤ s. To this end, we introduce a smooth cut-off function φ vanishing for r ≤ m and equalto 1 for r ≥ m . Then, we have J p, m [ q ( k ) , d k ( rN g )]( τ , τ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) r p ˇ e q ( k ) e d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) J p, m [ q ( k ) , φ d k ( rN g )]( τ , τ )+ J p, m [ q ( k ) , (1 − φ ) rN g ]( τ , τ ) . (5.3.20)In view of the fact that 1 − φ is supported in r ≤ m , we easily obtain J p, m [ q ( k ) , (1 − φ ) rN g ]( τ , τ ) (cid:46) (cid:18) sup τ ≤ τ ≤ τ E s [ q ]( τ ) + B sp [ q ]( τ , τ ) (cid:19) / (cid:0) I s +1 p [ N g ]( τ , τ ) (cid:1) and hence J p, m [ q ( k ) , (1 − φ ) rN g ]( τ , τ ) (cid:46) δ (cid:16) sup τ ≤ τ ≤ τ E s [ q ]( τ ) + B sp [ q ]( τ , τ ) (cid:17) + δ − I s +1 p [ N g ]( τ , τ ) (5.3.21) .3. IMPROVED WEIGHTED ESTIMATES δ > δ sup τ ≤ τ ≤ τ E s [ q ]( τ )and δ B sp [ q ]( τ , τ ) by the left hand side of our main estimate.It remains to estimate the terms J p, m [ q ( k ) , φ e ( d k ( rN g ))]( τ , τ ) , k ≤ s which is supported for r ≥ m . Note that e ( rN g ) behaves like rN g and therefore thesame sequence of estimates as for N g would lead to a loss of r − . For this reason we needto integrate by parts by parts in e . Proposition 5.3.6.
The following estimate holds true, for all k ≤ s ≤ k large − , (cid:88) k ≤ s J p, m [ q ( k ) , φ e ( d k ( rN g ))]( τ , τ ) (cid:46) δ B sp [ q ]( τ , τ ) + δ − I s +1 p [ N g ]( τ , τ ) (5.3.22) for a sufficiently small δ > . We postponed the proof of Proposition 5.3.6 to the end of the section. We are now inposition to conclude the proof of Theorem 5.3.1.
Proof of Theorem 5.3.1. (5.3.21) and (5.3.22) yield (cid:88) k ≤ s J p, m [ q ( k ) , e ( d k ( rN g ))]( τ , τ ) (cid:46) δ B sp [ q ]( τ , τ ) + δ − I s +1 p [ N g ]( τ , τ ) . Together with (5.3.17), (5.3.18) and (5.3.19), we infer J sp [ q , N ]( τ , τ ) (cid:46) ( δ + (cid:15) ) B sp [ q ]( τ , τ ) + δ − I s +1 p [ N g ]( τ , τ ) + (cid:15) sup τ ≤ τ ≤ τ E s [ q ]( τ ) . In view of (5.3.11), this concludes the proof of Theorem 5.3.1.The proof of Proposition 5.3.6 will rely in particular on the following identity.
Lemma 5.3.7.
The following hold true for any ψ ∈ s • We have, schematically, e e ( rψ ) = − r (cid:3) ψ + r (cid:52) / ψ + r − d ψ. (5.3.23) • The following identity holds true, schematically, e e ( r d k ψ ) = − d ≤ k ( r (cid:3) ψ ) + r (cid:52) / ( d ≤ k ψ ) + r − d ≤ k +1 ψ. (5.3.24)66 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) Proof.
We start with the following identity for ψ ∈ s , see Definition 2.4.2, (cid:3) ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ from which we deduce, r (cid:3) ψ = − re e ψ + r (cid:18) (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ (cid:19) . On the other hand, re e ψ = e ( re ψ ) − ( e r ) e ψ = e ( e ( rψ ) − e ( r ) ψ ) − ( e r ) e ψ = e e ( rψ ) − e ( r ) e ψ − ( e r ) e ψ − ( e e r ) ψ. Hence, r (cid:3) ψ = − e e ( rψ ) + e ( r ) e ψ + ( e r ) e ψ + ( e e r ) ψ + r (cid:52) / ψ + r (cid:18) ω − κ (cid:19) e ψ − rκe ψ + 2 rηe θ ψ = − e e ( rψ ) + r (cid:52) / ψ + (cid:18) e r − rκ (cid:19) e ψ + (cid:18) e r − rκ + 2 rω (cid:19) e ψ + 2 rηe θ ψ = − e e ( rψ ) + r (cid:52) / ψ + r Ae ψ + r A + 4 ω ) e ψ + 2 rηe θ ψ i.e., e e ( rψ ) = − r (cid:3) ψ + r (cid:52) / ψ + r Ae ψ + r A + 4 ω ) e ψ + 2 rηe θ ψ. or, schematically, in view of the definition of d ψ and the estimate | ω | + r | Γ g | + | Γ b | (cid:46) r − , e e ( rψ ) = − r (cid:3) ψ + r (cid:52) / ψ + (cid:16) r Γ g + Γ b + r − (cid:17) e ψ = − r (cid:3) ψ + r (cid:52) / ψ + r − d ψ which is (5.3.23).To derive the identity for higher derivatives we write, schematically, d k e e ( rψ ) = − d k ( r (cid:3) ψ ) + d k ( r (cid:52) / ψ ) + d k ( r Γ g d ψ ) . We write, d k e e ( rψ ) = e e ( r d k ψ ) + [ d k , e e r ] ψ = e e ( r d k ψ ) + [ d k , e ] d ψ + e [ d k , e r ] ψ, d k ( r (cid:52) / ψ ) = r (cid:52) / d k ψ + [ d k , r (cid:52) / ] ψ = r (cid:52) / d k ψ + [ d k , r − ] d ψ + r − [ d k , r (cid:52) / ] ψ. .3. IMPROVED WEIGHTED ESTIMATES e , d / ] and [ e , d / ] of Lemma 2.2.13, the identities of Proposition2.1.25 for commutation formulas involving d/ k and d (cid:63) / k derivatives, and the commutatorformula for [ e , e ], we have schematically[ e , e ] = 0 , [ d /, r (cid:52) / ] = d / + 1 , [ e , e r ] = ( r − + Γ g ) d [ e , d / ] = Γ b d + Γ b , [ e r, d / ] = ( r ξ + r Γ g ) d + r Γ g In view of the estimates for Γ g , Γ b , and the fact that ξ = 0 for r ≥ m , we infer[ d k , e ] = r − d ≤ k , [ d k , r (cid:52) / ] = d ≤ k +1 , [ d k , r − ] = r − d ≤ k − and hence d k e e ( rψ ) = e e ( r d k ψ ) + e [ d k , e r ] ψ + r − d ≤ k +1 ψ, d k ( r (cid:52) / ψ ) = r (cid:52) / d k ψ + r − d ≤ k +1 ψ. Also, we have[ re , e r ] = [ re , e ] r + e [ re , r ] = − e ( r ) e r − e re ( r ) = − e r + r − d and we infer by induction, schematically,[( re ) j , e r ] = e r ( re ) ≤ j − + r − d ≤ j so that, together with [ d k − j (cid:38) , e r ] = r − d ≤ k − j , we infer [ d k , e r ] = [( re ) j d k − j (cid:38) , e r ] = e r ( re ) ≤ j − d k − j (cid:38) + r − d ≤ k . We deduce e e ( r ( re ) j d k − j (cid:38) ψ ) = − ( re ) j d k − j (cid:38) ( r (cid:3) ψ ) + r (cid:52) / ( d k ψ ) + r − d ≤ k +1 ψ + e r ( re ) ≤ j − d k − j (cid:38) ψ. We infer by induction on je e ( r ( re ) j d k − j (cid:38) ψ ) = − ( re ) ≤ j d k − j (cid:38) ( r (cid:3) ψ ) + r (cid:52) / ( d ≤ k ψ ) + r − d ≤ k +1 ψ and hence e e ( r d k ψ ) = − d ≤ k ( r (cid:3) ψ ) + r (cid:52) / ( d ≤ k ψ ) + r − d ≤ k +1 ψ which is (5.3.24). This concludes the proof of Lemma 5.3.7.68 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) We now are in position to prove Proposition 5.3.6.
Proof of Proposition 5.3.6.
We integrate by parts, J p, m [ q ( k ) , φ d k ( rN g )]( τ , τ ) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) e (cid:0) φ ( r ) r p ˇ e q ( k ) (cid:1) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) + | B kp ( τ ) | + | B kp ( τ ) | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) Div ( e ) φ ( r ) r p ˇ e q ( k ) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) (5.3.25)where Div ( e ) denotes the spacetime divergence of e , and where the with boundaryterms are bounded by | B kp ( τ ) | (cid:46) (cid:90) Σ( τ ) r p | ˇ e q ( k ) | | d k ( rN g ) | , | B kp ( τ ) | (cid:46) (cid:90) Σ( τ ) r p | ˇ e q ( k ) | | d k ( rN g ) | . We estimate, | B kp ( τ ) | (cid:46) (cid:90) Σ( τ ) r p | ˇ e q ( k ) | | d k ( rN g ) | (cid:46) (cid:16) (cid:90) Σ( τ ) r p | ˇ e q ( k ) | (cid:17) / (cid:16) (cid:90) Σ( τ ) r p | d k ( rN g ) | (cid:17) / (cid:46) (cid:16) E kp [ q ]( τ ) (cid:17) / (cid:16) (cid:90) Σ( τ ) r p +2 | d k N g | (cid:17) / . We deduce, with δ > τ ∈ [ τ , τ ], (cid:12)(cid:12) B kp ( τ ) (cid:12)(cid:12) (cid:46) δ sup τ ≤ τ ≤ τ E kp [ q ]( τ ) + δ − sup τ ≤ τ ≤ τ (cid:90) Σ( τ ) r p +2 | N ≤ kg | , (cid:12)(cid:12) B kp ( τ ) (cid:12)(cid:12) (cid:46) δ sup τ ≤ τ ≤ τ E kp [ q ]( τ ) + δ − sup τ ≤ τ ≤ τ (cid:90) Σ( τ ) r p +2 | N ≤ kg | . (5.3.26)Next, notice that Div ( e ) = κ − ω so that | Div ( e ) | (cid:46) r − . Together with the fact that e (Φ ( r )) is supported in 4 m ≤ r ≤ m , the fact that | e ( r ) | (cid:46) r ˇ e q ( k ) = e ( r q ( k ) ) + O ( r − ) e ( q ( k ) ) , .3. IMPROVED WEIGHTED ESTIMATES (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) e (cid:0) φ ( r ) r p ˇ e q ( k ) (cid:1) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) Div ( e ) φ ( r ) r p ˇ e q ( k ) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) φ ( r ) r p − e e ( r q ( k ) ) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:90) M ≥ m ( τ ,τ ) r p − | ˇ e ( q ( k ) ) || d k ( rN g ) | + (cid:90) M m ≤ r ≤ m ( τ ,τ ) | ˇ e ( q ( k ) ) || d k ( rN g ) | (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) φ ( r ) r p − e e ( r q ( k ) ) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:32)(cid:90) M ≥ m ( τ ,τ ) r p − | ˇ e ( q ( k ) ) | (cid:33) (cid:32)(cid:90) M ≥ m ( τ ,τ ) r p +1 | d ≤ k N g | (cid:33) and hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) e (cid:0) φ ( r ) r p ˇ e q ( k ) (cid:1) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) Div ( e ) φ ( r ) r p ˇ e q ( k ) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) φ ( r ) r p − e e ( r q ( k ) ) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:0) B sp [ q ]( τ , τ ) (cid:1) / (cid:0) I sp [ N g ]( τ , τ ) (cid:1) which yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) e (cid:0) φ ( r ) r p ˇ e q ( k ) (cid:1) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) Div ( e ) φ ( r ) r p ˇ e q ( k ) d k ( rN g ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12) L k (cid:12)(cid:12) + δ B sp [ q ]( τ , τ ) + δ − I sp [ N g ]( τ , τ ) (5.3.27)where δ > δ B sp [ q ]( τ , τ )by the left hand side of our main estimate, and where we have introduced the notation L k : = (cid:90) M ( τ ,τ ) φ ( r ) r p − e e (cid:0) r q ( k ) (cid:1) d k ( rN g ) . (5.3.28)70 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) It remains to estimate the term L k . Making use of Lemma 5.3.7, we deduce L k = (cid:90) M ( τ ,τ ) φ ( r ) r p − e e ( r q ( k ) ) d k ( rN g )= − (cid:90) M ( τ ,τ ) φ ( r ) r p − d ≤ k ( r (cid:3) q ) d k ( rN g )+ (cid:90) M ( τ ,τ ) φ ( r ) r p (cid:52) / ( d ≤ k q ) d k ( rN g )+ (cid:90) M ( τ ,τ ) φ ( r ) r p − d ≤ k +1 q d k ( rN g )= L k + L k + L k . We first estimate L k as follows (cid:12)(cid:12) L k (cid:12)(cid:12) (cid:46) (cid:90) M ≥ m ( τ ,τ ) r p − | d ≤ k +1 q | | d k ( rN g ) | (cid:46) (cid:16) (cid:90) M ≥ m ( τ ,τ ) r p − (cid:12)(cid:12) d ≤ k +1 q (cid:12)(cid:12) (cid:17) / (cid:16) (cid:90) M ≥ m ( τ ,τ ) r p +1 | d ≤ k N g | (cid:17) / In view of Remark 5.1.8 we thus deduce, (cid:12)(cid:12) L k (cid:12)(cid:12) (cid:46) (cid:0) B kp [ q ] (cid:1) / (cid:16) (cid:90) M ≥ m ( τ ,τ ) r p +1 | d ≤ k N g | (cid:17) / (cid:46) (cid:0) B kp [ q ]( τ , τ ) (cid:1) / (cid:0) I kp [ N g ]( τ , τ ) (cid:1) / and hence (cid:12)(cid:12) L k (cid:12)(cid:12) (cid:46) δ B sp [ q ]( τ , τ ) + δ − I sp [ N g ]( τ , τ ) (5.3.29)where δ > δ B sp [ q ]( τ , τ )by the left hand side of our main estimate.We now estimate the term L k = (cid:90) M ( τ ,τ ) φ ( r ) r p (cid:52) / ( d k q ) d k ( rN g )by first performing another integration by parts in the angular directions (cid:12)(cid:12) L k (cid:12)(cid:12) (cid:46) (cid:90) M ≥ m ( τ ,τ ) r p − (cid:12)(cid:12) d k +1 q (cid:12)(cid:12)(cid:12)(cid:12) d k +1 ( rN g ) (cid:12)(cid:12) (cid:46) (cid:32)(cid:90) M ≥ m ( τ ,τ ) r p − (cid:12)(cid:12) d k +1 q (cid:12)(cid:12) (cid:33) / (cid:32)(cid:90) M ≥ m ( τ ,τ ) r p +1 (cid:12)(cid:12) d ≤ k +1 N g (cid:12)(cid:12) (cid:33) / (cid:46) (cid:0) B kp [ q ]( τ , τ ) (cid:1) / (cid:0) I k +1 p [ N g ]( τ , τ ) (cid:1) / . .3. IMPROVED WEIGHTED ESTIMATES (cid:12)(cid:12) L k (cid:12)(cid:12) (cid:46) δ B sp [ q ]( τ , τ ) + δ − I s +1 p [ N g ]( τ , τ ) (5.3.30)where δ > δ B sp [ q ]( τ , τ )by the left hand side of our main estimate.It remains to estimate the term, L k = − (cid:90) M ( τ ,τ ) φ ( r ) r p − d ≤ k ( r (cid:3) q ) d k ( rN g ) . Making use of the equation verified by q , i.e., (cid:3) q = − κκ q + N , we deduce, d k ( r (cid:3) q ) = − d k ( rκκ q ) + d k ( rN ) . Recall (5.1.9) d k N = d ≤ k N g + e ( d k ( rN g )) + d k N m [ q ] . We infer d ≤ k ( rN ) = r d ≤ k N + d ≤ k − N = r d ≤ k N g + re ( d ≤ k ( rN g )) + r d ≤ k N m [ q ]and hence | d k ( r (cid:3) q ) | (cid:46) r − (cid:12)(cid:12) d ≤ k q (cid:12)(cid:12) + r (cid:12)(cid:12) d ≤ k N g | + r (cid:12)(cid:12) d ≤ k e ( N g ) | + r (cid:12)(cid:12) d k N m [ q ] | (cid:46) r − (cid:12)(cid:12) d ≤ k +1 q (cid:12)(cid:12) + r (cid:12)(cid:12) d ≤ k N g | + r (cid:12)(cid:12) d ≤ k e ( N g ) | . (5.3.31)Note that we have used in the last inequality the form of N m [ q ] = d ≤ (Γ g q ) and the factthat | Γ g | ≤ (cid:15)r − . We deduce, using (5.3.31), (cid:12)(cid:12) L k (cid:12)(cid:12) (cid:46) (cid:90) M ≥ m ( τ ,τ ) r p − | d ≤ k +1 q (cid:12)(cid:12) | d ≤ k N g | + (cid:90) M ≥ m ( τ ,τ ) r p +1 | d ≤ k N g | + (cid:90) M ≥ m ( τ ,τ ) r p +2 | d ≤ k e ( N g ) || d ≤ k N g | (cid:46) (cid:0) B kp [ q ]( τ , τ ) (cid:1) / (cid:0) I kp [ N g ]( τ , τ ) (cid:1) / + I kp [ N g ]( τ , τ ) . We deduce (cid:12)(cid:12) L k (cid:12)(cid:12) (cid:46) δ B sp [ q ]( τ , τ ) + δ − I sp [ N g ]( τ , τ ) (5.3.32)72 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) where δ > δ B sp [ q ]( τ , τ )by the left hand side of our main estimate.Together with (5.3.29) and (5.3.30) we deduce, (cid:12)(cid:12) L k (cid:12)(cid:12) (cid:46) δ B kp [ q ]( τ , τ ) + δ − I kp [ N g ]( τ , τ ) . (5.3.33)Together with (5.3.25), (5.3.26) and (5.3.28), we infer, (cid:88) k ≤ s J p, m [ q ( k ) , φ d k ( rN g )]( τ , τ ) (cid:46) δ B sp [ q ]( τ , τ ) + δ − I s +1 p [ N g ]( τ , τ )which concludes the proof of Proposition 5.3.6. We apply Theorem 5.3.5 to the case when ψ = q . Hence, E sq [ˇ q ]( τ ) + B sq [ˇ q ]( τ , τ ) (cid:46) E sq [ˇ q ]( τ ) + ˇ J sq [ˇ q , N ]( τ , τ )+ E s +1max( q,δ ) [ q ]( τ ) + J s +1max( q,δ ) [ q , N ]( τ , τ ) . (5.3.34)Also, recall that ˇ q = f ˇ e q , where f is a fixed smooth function of r defined as follows, f ( r ) = (cid:40) r for r ≥ m , r ≤ m . (5.3.35)In particular, ˇ q is supported in r ≥ m , and hence, in view of Remark 5.1.5, B q [ˇ q ]( τ , τ ) (cid:39) (cid:90) M ≥ m ( τ ,τ ) r q − | d ˇ q | , (5.3.36)where we have used the fact that − δ ≤ q ≤ − δ .First, notice that the proof of Theorem 5.3.1 yields J s +1max( q,δ ) [ q , N ]( τ , τ ) (cid:46) sup τ ≤ τ ≤ τ E s +1 [ q ]( τ ) + B s +1max( q,δ ) [ q ]( τ , τ ) + I s +2max( q,δ ) [ N g ]( τ , τ ) . .3. IMPROVED WEIGHTED ESTIMATES q, δ ) ≤ − δ , we infer J s +1max( q,δ ) [ q , N ]( τ , τ ) (cid:46) E s +1max( q,δ ) [ q ]( τ ) + I s +2max( q,δ ) [ N g ]( τ , τ ) . Since q ≥ − δ , we have max( q, δ ) ≤ δ ≤ q + 1 and thus J s +1max( q,δ ) [ q , N ]( τ , τ ) (cid:46) E s +1 q +1 [ q ]( τ ) + I s +2 q +1 [ N g ]( τ , τ ) . (5.3.37)It only remains to estimate the termˇ J sq [ˇ q , N ]( τ , τ ) = (cid:88) k ≤ s ˇ J q [ d k ˇ q , d k N ]( τ , τ )with, ˇ J q [ˇ q , N ]( τ , τ ) = J q, m (cid:20) ˇ q , r (cid:18) e N + 3 r N (cid:19)(cid:21) ( τ , τ )= (cid:90) M ≥ m ( τ ,τ ) r q +2 (cid:0) ˇ e ˇ q (cid:1) · (cid:18) e N + 3 r N (cid:19) . We rewrite in the equivalent form,ˇ J q [ d k ˇ q , d k N ]( τ , τ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ≥ m ( τ ,τ ) r q (cid:0) r ˇ e d k ˇ q (cid:1) (cid:0) d k +1 N (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.3.38)Using the identity (5.1.9), we have d k +1 N = d ≤ k +1 N g + e ( d ≤ k +1 rN g ) + d k +1 N m [ q ] . The integral due to d ≤ k +1 N g is treated as followsˇ J q [ d k ˇ q , d k N g ]( τ , τ ) (cid:46) (cid:90) M ≥ m ( τ ,τ ) r q (cid:12)(cid:12) r ˇ e d k ˇ q (cid:12)(cid:12) (cid:12)(cid:12) d ≤ k +1 N g (cid:12)(cid:12) (cid:46) (cid:16) (cid:90) M ≥ m ( τ ,τ ) r q − (cid:12)(cid:12) r ˇ e d k ˇ q (cid:12)(cid:12) (cid:17) / (cid:16) (cid:90) M ≥ m ( τ ,τ ) r q +3 (cid:12)(cid:12) d ≤ k +1 N g (cid:12)(cid:12) (cid:17) / Therefore, ˇ J sq [ˇ q , N g ]( τ , τ ) (cid:46) (cid:0) B sq [ˇ q ]( τ , τ ) (cid:1) / (cid:0) I s +1 q +2 [ N g ]( τ , τ ) (cid:1) / (cid:46) δ B sq [ˇ q ]( τ , τ ) + δ − I s +1 q +2 [ N g ]( τ , τ ) (5.3.39)74 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) where δ > δ B sq [ q ]( τ , τ )by the left hand side of our main estimate.The integral due to d k +1 N m [ q ] is treated as followsˇ J q [ d k ˇ q , d k N m [ q ]]( τ , τ ) (cid:46) (cid:90) M ≥ m ( τ ,τ ) r q (cid:12)(cid:12) r ˇ e d k ˇ q (cid:12)(cid:12) (cid:12)(cid:12) d k +1 N m [ q ] (cid:12)(cid:12) (cid:46) (cid:90) M ≥ m ( τ ,τ ) r q +1 (cid:12)(cid:12) ˇ e d k ˇ q (cid:12)(cid:12) (cid:12)(cid:12) d ≤ k +2 q (cid:12)(cid:12) (cid:12)(cid:12) d ≤ k +2 Γ g (cid:12)(cid:12) (cid:46) (cid:15) (cid:90) M ≥ m ( τ ,τ ) r q − τ − − δ dec +2 δ (cid:12)(cid:12) ˇ e d k ˇ q (cid:12)(cid:12) (cid:12)(cid:12) d ≤ k +2 q (cid:12)(cid:12) (cid:46) (cid:15) (cid:32)(cid:90) M ≥ m ( τ ,τ ) r q τ − − δ dec +4 δ (cid:12)(cid:12) ˇ e d k ˇ q (cid:12)(cid:12) (cid:33) (cid:32)(cid:90) M ≥ m τ ,τ ) r q − (cid:12)(cid:12) d ≤ k +2 q (cid:12)(cid:12) (cid:33) (cid:46) (cid:15) (cid:18) sup τ ≤ τ ≤ τ E sq [ˇ q ]( τ ) (cid:19) (cid:16) B s +1 q +1 [ q ]( τ , τ ) (cid:17) where we have used | Γ g | (cid:46) (cid:15)r − τ − / − δ dec +2 δ and 2 δ < δ dec . Since δ ≤ q + 1 ≤ − δ and s ≤ k small + 29, we have in view of Theorem 5.3.1 B s +1 q +1 [ q ]( τ , τ ) (cid:46) E s +1 q +1 [ q ]( τ ) + I s +2 q +1 [ N g ]( τ , τ ) . We infer (cid:88) k ≤ s ˇ J q [ d k ˇ q , d k N m [ q ]]( τ , τ ) (cid:46) (cid:15) sup τ ≤ τ ≤ τ E sq [ˇ q ]( τ ) + E s +1 q +1 [ q ]( τ ) + I s +2 q +1 [ N g ]( τ , τ ) . (5.3.40)It remains to estimate the integral due to e ( d ≤ k +1 rN g ). We proceed as in Proposition5.3.6 by integration by parts, and obtain in particular the following analog of (5.3.27)ˇ J q [ d k ˇ q , d k e ( rN g )]( τ , τ ) (cid:46) (cid:12)(cid:12) P k (cid:12)(cid:12) + δ B sq [ˇ q ]( τ , τ ) + δ − I s +1 q +2 [ N g ]( τ , τ ) (5.3.41)where δ > δ B sq [ˇ q ]( τ , τ )by the left hand side of our main estimate, and where we have introduced the notation P k for the analog of L k in (5.3.28), i.e. P k := (cid:90) M ( τ ,τ ) r q e e (cid:0) r d k ˇ q (cid:1) d ≤ k +1 ( rN g ) . (5.3.42) Recall that ˇ q is localized in r ≥ m so that we don’t need in (5.3.42) the cutoff function φ ( r )introduced in Proposition 5.3.6. .3. IMPROVED WEIGHTED ESTIMATES e e ( r d k ˇ q ) = − d ≤ k ( r (cid:3) ˇ q ) + r (cid:52) / ( d ≤ k ˇ q ) + r − d ≤ k +1 ˇ q . (5.3.43)We infer P k = (cid:90) M ( τ ,τ ) r q e e (cid:0) r d k ˇ q (cid:1) d ≤ k +1 ( rN g )= − (cid:90) M ( τ ,τ ) r q d ≤ k ( r (cid:3) ˇ q ) d ≤ k +1 ( rN g )+ (cid:90) M ( τ ,τ ) r q +1 (cid:52) / ( d ≤ k ˇ q ) d ≤ k +1 ( rN g )+ (cid:90) M ( τ ,τ ) r q − d ≤ k +1 ˇ q d ≤ k +1 ( rN g )= P k + P k + P k . The last two terms on the right can be treated exactly as the the corresponding terms inthe treatment of L k . This yields to the following analog of (5.3.29) and (5.3.30) (cid:12)(cid:12) P k (cid:12)(cid:12) (cid:46) δ B sq [ˇ q ]( τ , τ ) + δ − I s +1 q +2 [ N g ]( τ , τ ) , (cid:12)(cid:12) P k (cid:12)(cid:12) (cid:46) δ B sq [ˇ q ]( τ , τ ) + δ − I s +2 q +2 [ N g ]( τ , τ ) , (5.3.44)where δ > δ B sp [ˇ q ]( τ , τ )by the left hand side of our main estimate.It thus only remains to consider the analogous of the term L k , i.e. P k = (cid:90) M ( τ ,τ ) r q d k ( r (cid:3) ˇ q ) d ≤ k +1 ( rN g ) . Now, in view of Proposition 10.3.1, q verifies, schematically, (cid:3) ˇ q = r − d ≤ ˇ q + r − d ≤ q + r d ≤ N so that d k ( r (cid:3) ˇ q ) = r − d ≤ k +1 ˇ q + r − d ≤ k +2 q + r d ≤ k +1 N = r − d ≤ k +1 ˇ q + r − d ≤ k +2 q + r d ≤ k +1 N g + r d ≤ k +1 N m [ q ] + r d ≤ k +1 e ( rN g ) . CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) We infer the following decomposition of P k P k = (cid:90) M ( τ ,τ ) r q − (cid:16) d ≤ k +1 ˇ q + d ≤ k +2 q (cid:17) d ≤ k +1 ( rN g )+ (cid:90) M ( τ ,τ ) r q +2 d ≤ k +1 N m [ q ] d ≤ k +1 ( rN g )+ (cid:90) M ( τ ,τ ) r q +2 (cid:16) d ≤ k +1 N g + d ≤ k +1 e ( rN g ) (cid:17) d ≤ k +1 ( rN g )= P k + P k + P k .P k is estimated as ˇ J sq [ˇ q , N g ]( τ , τ ), see (5.3.39), and hence | P k | (cid:46) (cid:0) B sq [ˇ q ]( τ , τ ) (cid:1) / (cid:0) I s +1 q +2 [ N g ]( τ , τ ) (cid:1) / (cid:46) δ B sq [ˇ q ]( τ , τ ) + B s max( q,δ ) [ q ]( τ , τ ) + δ − I s +1 q +2 [ N g ]( τ , τ )which in view of Theorem 5.3.1 yields | P k | (cid:46) δ B sq [ˇ q ]( τ , τ ) + E s +1max( q,δ ) [ q ]( τ ) + δ − I s +1 q +2 [ N g ]( τ , τ ) . (5.3.45)Next, P k is estimated as follows | P k | (cid:46) (cid:90) M ≥ m ( τ ,τ ) r q +3 | d ≤ k +1 N m [ q ] || d ≤ k +1 N g | (cid:46) (cid:90) M ≥ m ( τ ,τ ) r q +3 | d ≤ k +2 Γ g || d ≤ k +2 q || d ≤ k +1 N g | (cid:46) (cid:15) (cid:90) M ≥ m ( τ ,τ ) r q +1 τ − − δ dec +2 δ | d ≤ k +2 q || d ≤ k +1 N g | (cid:46) (cid:15) (cid:32)(cid:90) M ≥ m ( τ ,τ ) r q − | d ≤ k +2 q | (cid:33) (cid:32)(cid:90) M ≥ m ( τ ,τ ) r q +4 τ − − δ dec +4 δ | d ≤ k +1 N g | (cid:33) (cid:46) (cid:15) (cid:16) B s +1 q +1 [ q ]( τ , τ ) (cid:17) (cid:32) sup τ ∈ [ τ ,τ ] (cid:90) Σ( τ ) r q +4 | d ≤ k +1 N g | (cid:33) where we have used | Γ g | (cid:46) (cid:15)r − τ − / − δ dec +2 δ and 2 δ < δ dec . We infer | P k | (cid:46) B s +1 q +1 [ q ]( τ , τ ) + I s +1 q +2 [ N g ]( τ , τ ) . (5.3.46) .4. DECAY ESTIMATES P k is estimated as follows | P k | (cid:46) (cid:90) M ≥ m ( τ ,τ ) r q +3 (cid:16) | d ≤ k +1 N g | + | d ≤ k +1 e ( rN g ) | (cid:17) | d ≤ k +1 N g | (cid:46) (cid:90) M ≥ m ( τ ,τ ) r q +3 | d ≤ k +1 N g | + (cid:90) M ≥ m ( τ ,τ ) r q +4 | d ≤ k +1 e ( N g ) || d ≤ k +1 N g | (cid:46) I s +1 q +2 [ N g ]( τ , τ ) . Together with (5.3.45) and (5.3.46), we infer | P k | ≤ | P k | + | P k | + | P k | (cid:46) δ B sq [ˇ q ]( τ , τ ) + E s +1max( q,δ ) [ q ]( τ ) + δ − I s +1 q +2 [ N g ]( τ , τ ) + B s +1 q +1 [ q ]( τ , τ ) . Together with (5.3.44), we deduce | P k | ≤ | P k | + | P k | + | P k | (cid:46) δ B sq [ˇ q ]( τ , τ ) + E s +1max( q,δ ) [ q ]( τ ) + δ − I s +2 q +2 [ N g ]( τ , τ ) + B s +1 q +1 [ q ]( τ , τ ) . Together with (5.3.34), (5.3.37), (5.3.39), (5.3.40) and (5.3.41), this concludes the proofof Theorem 5.3.2.
In this section we prove the decay estimates. In particular • In section 5.4.1, we prove first flux decay estimates for q . • In section 5.4.2, we prove flux decay estimates for ˇ q . • In section 5.4.3, we prove Theorem 5.2.1. • In section 5.4.4, we prove Proposition 5.2.4 on pointwise decay estimates for q . • In section 5.4.5, we prove Proposition 5.2.5 on flux estimates on Σ ∗ and on improvedpointwise estimates for e ( q ).78 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) The decay estimates rely on the norms (5.1.27) which we recall below. E sp,d [ ψ ] = sup ≤ τ ≤ τ ∗ (1 + τ ) d E sp [ ψ ]( τ ) , B sp,d [ ψ ] = sup ≤ τ ≤ τ ∗ (1 + τ ) d (cid:90) τ ∗ τ M sp − [ ψ ]( τ ) dτ, F sp,d [ ψ ] = sup ≤ τ ≤ τ ∗ (1 + τ ) d F sp [ ψ ]( τ ) , I sp,d [ N g ] = sup ≤ τ ≤ τ ∗ (1 + τ ) d I sp [ N g ]( τ, τ ∗ ) . The goal of this section is to prove the following flux decay estimates for q . Theorem 5.4.1.
Assume q verifies all the estimates of Theorem 5.3.1. Then the followingestimates hold true for all s ≤ k small + 30 and for all δ ≤ p ≤ − δ E s − [2 − δ − p ] p, − δ − p [ q ] + B s − [2 − δ − p ] p, − δ − p [ q ] + F s − [2 − δ − p ] p, − δ − p [ q ] (cid:46) E s − δ [ q ](0) + I s +12 − δ, [ N g ] + I s +1 δ, − δ [ N g ] . (5.4.1) Here [ x ] denotes the least integer greater or equal to x .Proof. We make use of Theorem 5.3.1 according to which we have, for δ ≤ p ≤ − δ , and0 ≤ k ≤ k small + 30, E sp [ q ]( τ ) + B sp [ q ]( τ , τ ) + F sp [ q ]( τ , τ ) (cid:46) E sp [ q ]( τ ) + I s +1 p [ N g ]( τ , τ )which we write in the form, E sp ( τ ) + (cid:90) τ τ M sp − ( τ ) dτ (cid:46) E sp ( τ ) + I s +1 p [ N g ]( τ , τ ) , δ ≤ p ≤ − δ. (5.4.2)In particular, E s − δ ( τ ) + (cid:90) ττ/ M s − δ ( λ ) dλ (cid:46) E s − δ ( τ /
2) + I s +12 − δ, [ N g ] . By the mean value theorem there exists τ ∈ [ τ / , τ ] such that, M s − δ ( τ ) (cid:46) τ (cid:0) E s − δ ( τ /
2) + I s +12 − δ, [ N g ] (cid:1) . .4. DECAY ESTIMATES E s − − δ ( τ ) (cid:46) M s − δ ( τ ) , we deduce, E s − − δ ( τ ) (cid:46) τ (cid:0) E s − δ ( τ /
2) + I s +12 − δ, [ N g ] (cid:1) . Moreover, applying (5.4.2) again for p = 1 − δ , we deduce, E s − − δ ( τ ) + (cid:90) ττ M s − − δ ( λ ) dλ (cid:46) E s − − δ ( τ ) + (1 + τ ) − I s − δ, [ N g ] (cid:46) (1 + τ ) − (cid:0) E s − δ ( τ /
2) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] (cid:1) . In particular, E s − − δ ( τ ) (cid:46) (1 + τ ) − (cid:0) E s − δ ( τ /
2) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] (cid:1) . (5.4.3)Interpolating with E s − δ ( τ ) (cid:46) E s − δ ( τ /
2) + I s +12 − δ, [ N g ]by using, E sp (cid:46) ( E sp ) p − pp − p ( E sp ) p − p p − p , p ≤ p ≤ p , we deduce E s − ( τ ) (cid:46) ( E s − − δ ( τ )) − δ ( E s − − δ ( τ )) δ (cid:46) (1 + τ ) − δ (cid:0) E s − δ ( τ /
2) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] (cid:1) . The same inequality hods for τ replaced by τ / E s − ( τ / (cid:46) (1 + τ ) − δ (cid:0) E s − δ ( τ /
4) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] (cid:1) . (5.4.4)We now repeat the procedure starting this time with the inequality (5.4.2) for p = 1, E s − ( τ ) + (cid:90) ττ/ M s − ( λ ) dλ (cid:46) E s − ( τ /
2) + I s [ N g ]( τ / , τ ) (cid:46) E s − ( τ /
2) + (1 + τ ) − δ I s , − δ [ N g ] . Note that the loss of derivative is due to the degeneracy of the bulk integral in the trapping region. CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) Thus, in view of (5.4.4), (cid:90) ττ/ M s − ( λ ) dλ (cid:46) (1 + τ ) − δ (cid:0) E s − δ ( τ /
4) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] + I s , − δ [ N g ] (cid:1) or, since E s − δ ( τ / (cid:46) E s − δ (0) + I s +12 − δ, [ N g ] , we infer that, (cid:90) ττ/ M s − ( λ ) dλ (cid:46) B (1 + τ ) − δ where, B : = E s − δ (0) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] + I s , − δ [ N g ] . (5.4.5)Repeating the mean value argument, we can find τ ∈ [ τ / , τ ] such that, M s − ( τ ) (cid:46) τ (cid:90) ττ/ M s − ( λ ) dλ (cid:46) B (1 + τ ) − δ . We now make use of the fact that the energy norm E s − is comparable with M s − every-where except in the trapping region where we lose a derivative. Thus E s − ( τ ) (cid:46) M s − ( τ )and therefore, E s − ( τ ) (cid:46) B (1 + τ ) − δ . (5.4.6)We would like now to compare E s − ( τ ) with E s − ( τ ) using the usual version of the energyinequality and thus derive a similar estimate for the former. Unfortunately , we don’thave a closed energy inequality for E and we therefore have instead to rely on E δ forwhich we have the inequality, E s − δ ( τ ) (cid:46) E s − δ ( τ ) + I s − δ [ N g ]( τ , τ ) . (5.4.7)We also have in view of (5.4.3) E s − − δ ( τ ) (cid:46) (1 + τ ) − (cid:0) E s − δ (0) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] (cid:1) . The loss of δ is due to the fact that we are on a perturbation of Schwarzschild rather than onSchwarzschild. .4. DECAY ESTIMATES δ > E s − δ ( τ ) (cid:46) (cid:0) E s − ( τ ) (cid:1) − δ − δ (cid:0) E s − − δ ( τ ) (cid:1) δ − δ (cid:46) (1 + τ ) − δ ( B + E s − − δ (0) + I s − − δ, [ N g ]) (cid:46) (1 + τ ) − δ B. Thus, in view of (5.4.7), E s − δ ( τ ) (cid:46) E s − δ ( τ ) + I s − δ [ N g ]( τ , τ ) (cid:46) (1 + τ ) − δ ( B + I s − δ, − δ [ N g ])i.e., E s − δ ( τ ) (cid:46) (1 + τ ) − δ (cid:0) E s − δ (0) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] + I s , − δ [ N g ] + I s − δ, − δ [ N g ] (cid:1) . We infer E s − δ, − δ (cid:46) (cid:0) E s − δ (0) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] + I s , − δ [ N g ] + I s − δ, − δ [ N g ] (cid:1) which can be written in the shorter form (by interpolation of the middle terms), E s − δ, − δ (cid:46) E s − δ (0) + I s +12 − δ, [ N g ] + I s +1 δ, − δ [ N g ] . (5.4.8)Also, (5.4.3) yields E s − − δ, (cid:46) E s − δ (0) + I s +12 − δ, [ N g ] + I s − δ, [ N g ] (cid:46) E s − δ (0) + I s +12 − δ, [ N g ] + I s +1 δ, − δ [ N g ] . (5.4.9)while from Theorem 5.3.1, we have E s − δ, (cid:46) E s − δ (0) + I s +12 − δ, [ N g ] . (5.4.10)Interpolating (5.4.8) and (5.4.9), as well as (5.4.9) and (5.4.10), we infer for all s ≤ k small + 30 and for all δ ≤ p ≤ − δ E s − [2 − δ − p ] p, − δ − p [ q ] (cid:46) E s − δ [ q ](0) + I s +12 − δ, [ N g ] + I s +1 δ, − δ [ N g ] . (5.4.11)Finally, making use of Theorem 5.3.1 between τ and τ ∗ , we have in particular B s − [2 − δ − p ] p [ q ]( τ, τ ∗ ) + F s − [2 − δ − p ] p [ q ]( τ, τ ∗ ) (cid:46) E s − [2 − δ − p ] p [ q ]( τ ) + I s +1 − [2 − δ − p ] p [ N g ]( τ, τ ∗ ) (cid:46) (1 + τ ) − (2 − δ − p ) (cid:16) E s − [2 − δ − p ] p, − δ − p [ q ] + I s +1 p, − δ − p [ N g ] (cid:17) and hence, we infer for all s ≤ k small + 30 and for all δ ≤ p ≤ − δ B s − [2 − δ − p ] p, − δ − p [ q ] + F s − [2 − δ − p ] p, − δ − p [ q ] (cid:46) E s − [2 − δ − p ] p, − δ − p [ q ] + I s +12 − δ, [ N g ] + I s +1 δ, − δ [ N g ] . Together with (5.4.11), this concludes the proof of Theorem 5.4.1.82
CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) ˇ q The goal of this section is to prove the following flux decay estimates for ˇ q . Theorem 5.4.2.
The following estimates hold for all q − ≤ q ≤ q , where q is a fixednumber δ < q ≤ − δ , and s ≤ k small + 28 E sq,q − q [ˇ q ] + B sq,q − q [ˇ q ] (cid:46) E sq [ˇ q ](0) + E s +22 − δ [ q ](0) + I s +3 q +2 , [ N g ] + I s +3 δ, q − δ [ N g ] . Proof.
Since δ < q ≤ − δ , according to Theorem 5.3.2, ˇ q = f ˇ e q verifies, for any q − ≤ q ≤ q and any s ≤ k small + 29, E sq [ˇ q ]( τ ) + B sq [ˇ q ]( τ , τ ) (cid:46) E sq [ˇ q ]( τ ) + E s +1 q +1 [ q ]( τ ) + I s +2 q +2 [ N g ]( τ , τ ) . According to the definition of our decay norms above we have, I s +2 q +2 [ N g ]( τ , τ ) (cid:46) (1 + τ ) q − q I s +2 q +2 ,q − q [ N g ] . (5.4.12)Also, according to the definition 5.1.27 for the decay norms for q we also have E s +1 q +1 [ q ]( τ ) (cid:46) (1 + τ ) q − q E s +2 q +1 ,q − q [ q ] . We deduce , for all q − ≤ q ≤ q , E sq [ˇ q ]( τ ) + (cid:90) τ τ M sq [ˇ q ]( τ ) (cid:46) E sq [ˇ q ]( τ ) + (1 + τ ) q − q ˜ E sq,q − q (5.4.13)where, ˜ E sq,q − q := E s +1 q +1 ,q − q [ q ] + I s +2 q +2 ,q − q [ N g ] . (5.4.14)In particular, E sq [ˇ q ]( τ ) + (cid:90) τ τ M sq − [ˇ q ]( τ ) dτ (cid:46) E sq [ˇ q ]( τ ) + ˜ E sq , . (5.4.15)By the mean value theorem we deduce that there exists τ ∈ [ τ , τ ] such that, M sq − [ˇ q ]( τ ) (cid:46) τ − τ (cid:16) E sq [ˇ q ]( τ ) + ˜ E sq , (cid:17) (cid:46) τ − τ (cid:16) E sq , [ˇ q ] + ˜ E sq , (cid:17) . Note that it is important in what follows that the r q weighted estimates hold also for negative valuesof q . .4. DECAY ESTIMATES E sq − [ˇ q ]( τ ) (cid:46) τ − τ (cid:16) E sq , [ˇ q ] + ˜ E sq , (cid:17) . (5.4.16)We now make use of (5.4.13) to compare the quantities E q [ˇ q ] for negative weights ( q = q −
1) at different values of τ . E sq − [ˇ q ]( τ ) (cid:46) E sq − [ˇ q ]( τ ) + (1 + τ ) − ˜ E sq − , . Combining this with (5.4.16) we deduce, E sq − [ˇ q ]( τ ) (cid:46) τ − τ (cid:16) E sq , [ˇ q ] + ˜ E sq , (cid:17) + (1 + τ ) − ˜ E sq − , . Applying this inequality for τ = τ ≤ τ ∗ , τ = τ , τ ∈ [ τ , τ ] we deduce, E sq − [ˇ q ]( τ ) (cid:46) (1 + τ ) − (cid:16) E sq , [ˇ q ] + ˜ E sq , + ˜ E sq − , (cid:17) . (5.4.17)We now interpolate this last inequality with the following immediate consequence of(5.4.15) E sq [ˇ q ]( τ ) (cid:46) E sq , [ˇ q ] + ˜ E sq , to deduce, for all q − ≤ q ≤ q , E sq [ˇ q ]( τ ) (cid:46) (1 + τ ) q − q (cid:16) E sq , [ˇ q ] + ˜ E sq , + ˜ E sq − , (cid:17) i.e., E sq,q − q [ˇ q ] (cid:46) E sq , [ˇ q ] + ˜ E sq , + ˜ E sq − , . In view of the definition of ˜ E sq,q − q , this yields for all q − ≤ q ≤ q , E sq,q − q [ˇ q ] (cid:46) E sq , [ˇ q ] + E s +1 q +1 , [ q ] + E s +1 q , [ q ] + I s +2 q +2 , [ N g ] + I s +2 q +1 , [ N g ] . On the other hand, we have in view of Theorem 5.3.2, E sq , [ˇ q ] (cid:46) E sq [ˇ q ](0) + E s +1 q +1 , [ q ] + I s +2 q +2 , [ N g ]and hence E sq,q − q [ˇ q ] (cid:46) E sq [ˇ q ](0) + E s +1 q +1 , [ q ] + E s +1 q , [ q ] + I s +2 q +2 , [ N g ] + I s +2 q +1 , [ N g ] . CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) Now, since δ < q ≤ − δ , we have δ < q < q + 1 ≤ − δ and thus, we may applyTheorem 5.4.1 to obtain for all q − ≤ q ≤ q E s +1 q +1 ,q − q [ q ] (cid:46) E s +22 − δ [ q ](0) + I s +32 − δ, [ N g ] + I s +3 δ, − δ [ N g ] . (5.4.18)We thus infer E sq,q − q [ˇ q ] (cid:46) E sq [ˇ q ](0) + E s +22 − δ [ q ](0) + I s +3 q +2 , [ N g ] + I s +3 q +1 , [ N g ] + I s +32 − δ, [ N g ] + I s +3 δ, − δ [ N g ]and hence, for all q − ≤ q ≤ q , E sq,q − q [ˇ q ] (cid:46) E sq [ˇ q ](0) + E s +22 − δ [ q ](0) + I s +3 q +2 , [ N g ] + I s +3 δ, − δ [ N g ] . (5.4.19)Finally, making use of Theorem 5.3.2 between τ and τ ∗ , we have in particular B sq [ˇ q ]( τ, τ ∗ ) (cid:46) E sq [ˇ q ]( τ ) + E s +1 q +1 [ q ]( τ ) + I s +2 q +2 [ N g ]( τ, τ ∗ ) (cid:46) (1 + τ ) − ( q − q ) (cid:16) E sq,q − q [ˇ q ] + E s +1 q +1 ,q − q [ q ] + I s +2 q +2 ,q − q [ N g ] (cid:17) (cid:46) (1 + τ ) − ( q − q ) (cid:16) E sq,q − q [ˇ q ] + E s +22 − δ [ q ](0) + I s +3 q +2 , [ N g ] + I s +3 δ, − δ [ N g ] (cid:17) where we used (5.4.18) in the last inequality. Hence, we infer for all s ≤ k small + 28 andfor all q − ≤ q ≤ q B sq,q − q [ˇ q ] (cid:46) E sq,q − q [ˇ q ] + E s +22 − δ [ q ](0) + I s +3 q +2 , [ N g ] + I s +3 δ, − δ [ N g ] . Together with (5.4.19), this concludes the proof of Theorem 5.4.2.
In this section, we prove Theorem 5.2.1 by making use of Theorem 5.4.1 and Theorem5.4.2. We start with the main estimate of Theorem 5.4.2 with q = − δ which we write inthe form, E s − δ [ˇ q ] (cid:46) (1 + τ ) − q − δ C sq where, C sq := E sq [ˇ q ](0) + E s +22 − δ [ q ](0) + I s +3 q , [ N g ] + I s +3 δ,q +2 − δ [ N g ] . In view of the definition (5.1.21) of E s − δ [ˇ q ] and since ˇ q = f ˇ e q , (cid:90) Σ ≥ m ( τ ) r − δ (cid:0) | ˇ e ˇ q | + r − | ˇ q | (cid:1) (cid:46) (1 + τ ) − q − δ C sq . .4. DECAY ESTIMATES E s − δ, m [ q ] = (cid:90) Σ ≥ m ( τ ) r − δ | ˇ e q | (cid:46) (1 + τ ) − q − δ C sq . (5.4.20)In view of the decay estimates (5.4.1) for q established in Theorem 5.4.1 we have, E s ( τ ) (cid:46) (1 + τ ) − δ B s − δ ,B s − δ : = E s +22 − δ [ q ](0) + I s +32 − δ, [ N g ] + I s +3 δ, − δ [ N g ] . Thus, the quantity E s − δ = E s − δ [ q ]( τ ) = ˙ E s − δ, m [ q ] + E s [ q ]verifies, E s − δ (cid:46) (1 + τ ) − q − δ (cid:0) C sq + B s − δ (cid:1) . (5.4.21)On the other hand, E s − δ verifies (5.4.2) for p = 2 − δ , i.e. E s − δ ( τ ) + (cid:90) τ τ M s − δ ( τ ) dτ (cid:46) E s − δ ( τ ) + I s +12 − δ [ N g ]( τ , τ ) . Since I s +12 − δ [ N g ]( τ , τ ) (cid:46) (1 + τ ) − q − δ I s +12 − δ,q + δ [ N g ] , we infer E s − δ ( τ ) + (cid:90) ττ/ M s − δ ( τ (cid:48) ) dτ (cid:48) (cid:46) E s − δ ( τ /
2) + I s +12 − δ [ N g ]( τ / , τ ) (cid:46) (1 + τ ) − q − δ (cid:0) C sq + B s − δ + I s +12 − δ,q + δ [ N g ] (cid:1) . (5.4.22)Following the same arguments as in the proof of Theorem 5.4.1 we deduce, for a τ ∈ [ τ / , τ ], E s − − δ ( τ ) (cid:46) (1 + τ ) − q − − δ (cid:0) C sq + B s − δ + I s +12 − δ,q + δ [ N g ] (cid:1) and since, E s − δ ( τ ) (cid:46) E s − δ ( τ ) + I s +11 − δ ( τ , τ )[ N g ] , we infer that, E s − δ ( τ ) (cid:46) (1 + τ ) − q − − δ (cid:0) C s +1 q + B s − δ + I s +22 − δ,q + δ [ N g ] + I s +11 − δ, q + δ [ N g ] (cid:1) . (5.4.23)86 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) Interpolating with (5.4.21), i.e. E s − δ (cid:46) (1 + τ ) − q − δ (cid:0) C sq + B s − δ (cid:1) we deduce, E s (cid:46) ( E s − δ ) − δ ( E s − δ ) δ (cid:46) (1 + τ ) − q − (cid:0) C s +1 q + B s − δ + I s +22 − δ,q + δ [ N g ] + I s +11 − δ, q + δ [ N g ] (cid:1) . Hence, E s (cid:46) (1 + τ ) − q − (cid:0) C s +1 q + B s − δ + I s +22 − δ,q + δ + I s +11 − δ, q + δ (cid:1) . (5.4.24)As in the proof of Theorem 5.4.1 we repeat the procedure starting with the inequality(5.4.2) for p = 1, E s ( τ ) + (cid:90) ττ/ M s ( λ ) dλ (cid:46) E s ( τ /
2) + I s +11 [ N g ]( τ / , τ ) (cid:46) (1 + τ ) − q − (cid:0) C s +1 q + B s − δ + I s +22 − δ,q + δ [ N g ] + I s +11 − δ, q + δ [ N g ] (cid:1) + (1 + τ ) − − q I s +11 , q [ N g ] (cid:46) (1 + τ ) − q − (cid:0) C s +1 q + B s − δ + I s + s − δ,q + δ [ N g ] + I s +11 − δ, q + δ [ N g ] + I s +11 , q [ N g ] (cid:1) from which we infer that, for a τ ∈ [ τ / , τ ], E s ( τ ) (5.4.25) (cid:46) (1 + τ ) − q − (cid:0) C s +2 q + B s +42 − δ + I s +32 − δ,q + δ [ N g ] + I s +21 − δ, q + δ [ N g ] + I s +21 , q [ N g ] (cid:1) . Interpolating (5.4.23) and (5.4.25) we deduce, for δ > E sδ ( τ ) (cid:46) (cid:0) E s ( τ ) (cid:1) − δ − δ (cid:0) E s − δ ( τ ) (cid:1) δ − δ (cid:46) (1 + τ ) − − q + δ (cid:0) C s +2 q + B s +42 − δ + I s +32 − δ,q + δ [ N g ] + I s +21 − δ, q + δ [ N g ] + I s +21 , q [ N g ] (cid:1) . Thus, since we have, as in (5.4.7), E sδ ( τ ) (cid:46) E sδ ( τ ) + I s +1 δ [ N g ]( τ , τ ) , we deduce E sδ ( τ ) (cid:46) (1 + τ ) − − q + δ (cid:0) C s +2 q + B s +42 − δ + I s +32 − δ,q + δ [ N g ] + I s +21 − δ, q + δ [ N g ] + I s +21 , q [ N g ] (cid:1) + (1 + τ ) − − q + δ I s +1 δ, q − δ [ N g ]i.e., E sδ ( τ ) (cid:46) (1 + τ ) − − q + δ (cid:16) C s +2 q + B s +42 − δ + I s +32 − δ,q + δ [ N g ] + I s +21 − δ, q + δ [ N g ]+ I s +21 , q [ N g ] + I s +1 δ, q − δ [ N g ] (cid:17) . .4. DECAY ESTIMATES E sδ ( τ ) (cid:46) (1 + τ ) − − q + δ (cid:0) C s +2 q + B s +42 − δ + I s +32 − δ,q + δ [ N g ] + I s +3 δ, q − δ [ N g ] (cid:1) . We now recall, C sq := E sq [ˇ q ](0) + E s +22 − δ [ q ](0) + I s +3 q +2 , [ N g ] + I s +3 δ,q +2 − δ [ N g ] B s − δ : = E s +22 − δ [ q ](0) + I s +32 − δ, [ N g ] + I s +3 δ, − δ [ N g ] . Hence, C s +2 q + B s +42 − δ + I s +32 − δ,q + δ [ N g ] + I s +3 δ, q − δ [ N g ]= E s +2 q [ˇ q ](0) + E s +42 − δ [ q ](0) + I s +5 q +2 , [ N g ] + I s +5 δ,q +2 − δ [ N g ]+ E s +42 − δ [ q ](0) + I s +52 − δ, [ N g ] + I s +5 δ, − δ [ N g ] + I s +32 − δ,q + δ [ N g ] + I s +3 δ, q − δ [ N g ] . We deduce, E sδ, q − δ [ q ] (cid:46) E s +2 q [ˇ q ](0) + E s +42 − δ [ q ](0) + I s +5 q +2 , [ N g ] + I s +5 δ, q − δ [ N g ] . (5.4.26)We can also simplify the right hand side of (5.4.24), C s +1 q + B s − δ + I s +22 − δ,q + δ [ N g ] + I s +11 − δ, q + δ [ N g ] (cid:46) E s +2 q [ˇ q ](0) + E s +42 − δ [ q ](0) + I s +5 q +2 , [ N g ] + I s +5 δ, q − δ [ N g ] . Thus (5.4.23) becomes, E s − δ, q + δ (cid:46) E s +2 q [ˇ q ](0) + E s +42 − δ [ q ](0) + I s +5 q +2 , [ N g ] + I s +5 δ, q − δ [ N g ] . (5.4.27)Similarly, (5.4.21) yields E s − δ,q − δ (cid:46) E s +2 q [ˇ q ](0) + E s +42 − δ [ q ](0) + I s +5 q +2 , [ N g ] + I s +5 δ, q − δ [ N g ] . (5.4.28)Interpolating (5.4.26) and (5.4.27), as well as (5.4.27) and (5.4.28), we infer for all s ≤ k small + 25 and for all δ ≤ p ≤ − δ E sp, q − p [ q ] (cid:46) E s +2 q [ˇ q ](0) + E s +42 − δ [ q ](0) + I s +5 q +2 , [ N g ] + I s +5 δ, q − δ [ N g ] . (5.4.29)Finally, making use of Theorem 5.3.1 between τ and τ ∗ , we have in particular B sp [ q ]( τ, τ ∗ ) + F sp [ q ]( τ, τ ∗ ) (cid:46) E sp [ q ]( τ ) + I s +1 p [ N g ]( τ, τ ∗ ) (cid:46) (1 + τ ) − (2+ q − p ) (cid:16) E sp, q − p [ q ] + I s +1 p, q − p [ N g ] (cid:17) and hence, we infer for all s ≤ k small + 25 and for all δ ≤ p ≤ − δ B sp, q − p [ q ] + F sp, q − p [ q ] (cid:46) E sp, q − p [ q ] + I s +5 q +2 , [ N g ] + I s +5 δ, q − δ [ N g ] . Together with (5.4.29), this concludes the proof of Theorem 5.2.1.88
CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) Let χ be a smooth cut-off function vanishing for r ≤ m and equal to 1 for r ≥ m . Toprove estimate (5.2.6) we consider the identity, e (cid:18)(cid:90) S r χ ( q ( s ) ) (cid:19) = (cid:90) S r (cid:16) e ( χ ( q ( s ) ) ) + κχ ( q ( s ) ) (cid:17) = (cid:90) S r (cid:16) χ (2 q ( s ) e q ( s ) + 2 r − ( q ( s ) ) ) + χ (cid:48) ( q ( s ) ) + χ ( κ − r − ) | q ( s ) | (cid:17) = (cid:90) S r (cid:16) χ q ( s ) ˇ e q ( s ) + χ (cid:48) ( q ( s ) ) + O ( r − ) | q ( s ) | (cid:17) . Integrating between 4 m and r for a fixed r ≥ m , we deduce, in view of the definitionsof E [ q ( s ) ]( τ ) and of E p [ q ( s ) ]( τ ), (cid:90) S r | q ( s ) | (cid:46) (cid:90) Σ( τ ) ≥ m | q ( s ) || ˇ e q ( s ) | + E [ q ( s ) ]( τ ) (cid:46) (cid:32)(cid:90) Σ( τ ) ≥ m r δ | ˇ e q ( s ) | (cid:33) / (cid:32)(cid:90) Σ( τ ) ≥ m r − − δ | q ( s ) | (cid:33) / + E [ q ( s ) ]( τ ) (cid:46) (cid:0) E δ [ q ( s ) ]( τ ) (cid:1) / (cid:0) E − δ [ q ( s ) ]( τ ) (cid:1) / . Clearly, this estimate also holds for r ≤ m . Together with the definition (5.1.27) of E sp,d [ q ( s ) ], we immediately infer(1 + τ ) q (cid:90) S r | q ( s ) | (cid:46) (cid:0) E s δ, q − δ [ q ] (cid:1) (cid:0) E s − δ, q + δ [ q ] (cid:1) which is the desired estimate (5.2.6).To prove (5.2.7) we start instead with the identity, e (cid:18) r − (cid:90) S r χ ( q ( s ) ) (cid:19) = (cid:90) S r r − (cid:16) e ( χ ( q ( s ) ) ) + κχ ( q ( s ) ) (cid:17) − e ( r ) r (cid:90) S r χ ( q ( s ) ) = (cid:90) S r r − (cid:16) χ (2 q ( s ) e q ( s ) + r − ( q ( s ) ) ) + χ (cid:48) ( q ( s ) ) + χ ( κ − r − ) | q ( s ) | (cid:17) − e ( r ) − r (cid:90) S r χ ( q ( s ) ) = (cid:90) S r (cid:16) r − χe ( q ( s ) ) q ( s ) + r − χ (cid:48) ( q ( s ) ) + O ( r − ) | q ( s ) | (cid:17) . .4. DECAY ESTIMATES m and r for a fixed r ≥ m , we deduce, in view of the definitionsof E [ q ( s ) ]( τ ) and of E p [ q ( s ) ]( τ ), r − (cid:90) S r | ψ | (cid:46) (cid:90) Σ( τ ) ≥ m r − | q ( s ) || e ( q ( s ) ) | + E [ q ( s ) ]( τ ) (cid:46) (cid:32)(cid:90) Σ( τ ) ≥ m | e ( q ( s ) ) | (cid:33) / (cid:32)(cid:90) Σ( τ ) ≥ m r − | q ( s ) | (cid:33) / + E [ q ( s ) ]( τ ) (cid:46) E [ q ( s ) ]( τ ) (cid:46) E δ [ q ( s ) ]( τ ) . Clearly, this estimate also holds for r ≤ m . Together with the definition (5.1.27) of E sp,d [ q ( s ) ], we immediately infer r − (1 + τ ) q − δ (cid:90) S r | q ( s ) | (cid:46) E sδ, q − δ [ q ]which is the desired estimate (5.2.7). This concludes the proof of Proposition 5.2.4. Recall the following definitions F [ ψ ]( τ , τ ) = (cid:90) A ( τ ,τ ) (cid:16) δ − H | e Ψ | + δ H | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) + (cid:90) Σ ∗ ( τ ,τ ) (cid:16) | e Ψ | + | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) , ˙ F p [ ψ ]( τ , τ ) = (cid:90) Σ ∗ ( τ ,τ ) r p (cid:16) | e ψ | + |∇ / ψ | + r − | ψ | (cid:17) ,F p [ ψ ]( τ , τ ) = F [ ψ ]( τ , τ ) + ˙ F p [ ψ ]( τ , τ ) ,F s [ ψ ]( τ , τ ) = (cid:88) k ≤ s F [ d k ψ ]( τ , τ ) ,F sp [ ψ ]( τ , τ ) = (cid:88) k ≤ s F p [ d k ψ ]( τ , τ ) , F sp,d [ ψ ] = sup ≤ τ ≤ τ ∗ (1 + τ ) d F sp [ ψ ]( τ, τ ∗ ) . We deduce F s [ q ]( τ, τ ∗ ) ≤ F sδ [ q ]( τ, τ ∗ ) ≤ (1 + τ ) − − q + δ F sδ, q − δ [ q ]90 CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) and hence in particular(1 + τ ) q − δ (cid:90) Σ ∗ ( τ,τ ∗ ) (cid:16) | e d ≤ s q | + r − | d ≤ s q | (cid:17) (cid:46) F sδ, q − δ [ q ] (5.4.30)which yields the desired estimate (5.2.8).Next, we focus on the proof of (5.2.9). We start with the following trace estimatesup Σ ∗ ( τ,τ ∗ ) (cid:107) e d ≤ s q (cid:107) L ( S ) (cid:46) (cid:107) νe d ≤ s q (cid:107) L (Σ ∗ ( τ,τ ∗ )) + (cid:107) e d ≤ s q (cid:107) L (Σ ∗ ( τ,τ ∗ )) where we recall that ν is tangent to Σ ∗ , orthogonal to e θ and given by ν = e + ae , − ≤ a ≤ − . We infer sup Σ ∗ ( τ,τ ∗ ) (cid:107) e d ≤ s q (cid:107) L ( S ) (cid:46) (cid:107) e e d ≤ s q (cid:107) L (Σ ∗ ( τ,τ ∗ )) + (cid:107) e e d ≤ s q (cid:107) L (Σ ∗ ( τ,τ ∗ )) + (cid:107) e d ≤ s q (cid:107) L (Σ ∗ ( τ,τ ∗ )) (cid:46) (cid:107) e d ≤ s +1 q (cid:107) L (Σ ∗ ( τ,τ ∗ )) + (cid:107) r − d ≤ s +1 q (cid:107) L (Σ ∗ ( τ,τ ∗ )) + (cid:107) [ e , e ] d ≤ s q (cid:107) L (Σ ∗ ( τ,τ ∗ )) (cid:46) (cid:107) e d ≤ s +1 q (cid:107) L (Σ ∗ ( τ,τ ∗ )) + (cid:107) r − d ≤ s +1 q (cid:107) L (Σ ∗ ( τ,τ ∗ )) . In view of (5.4.30), we deducesup Σ ∗ (cid:110) (1 + τ ) q − δ (cid:107) e d ≤ s q (cid:107) L ( S ) (cid:111) (cid:46) F s +1 δ, q − δ [ q ] . (5.4.31)Next, we extend (5.4.31) to r ≥ m . In view of (5.3.24), we have schematically e e ( r d k q ) = − d ≤ k ( r (cid:3) q ) + r (cid:52) / ( d ≤ k q ) + r − d ≤ k +1 q = − d ≤ k ( r (cid:3) q ) + r − d ≤ k +2 q . Also, we have e ( re ( d k q )) = e e ( r d k q ) + [ e , e ]( r d k q ) − e ( e ( r ) d k q )and hence, we infer schematically e ( re ( d k q )) = − d ≤ k ( r (cid:3) q ) + r − d ≤ k +2 q . .4. DECAY ESTIMATES | d k ( r (cid:3) q ) | (cid:46) r − (cid:12)(cid:12) d ≤ k +1 q (cid:12)(cid:12) + r (cid:12)(cid:12) d ≤ k N g | + r (cid:12)(cid:12) d ≤ k e ( N g ) | . We deduce | e ( re ( d k q )) | (cid:46) r − (cid:12)(cid:12) d ≤ k +2 q (cid:12)(cid:12) + r (cid:12)(cid:12) d ≤ k N g | + r (cid:12)(cid:12) d ≤ k e ( N g ) | Now, we have e (cid:18) r − (cid:90) S ( re ( d ≤ s q )) (cid:19) = r − (cid:90) S (cid:16) e ( re ( d ≤ s q )) re ( d ≤ s q ) + κ ( re ( d ≤ s q )) (cid:17) − e ( r ) r r − (cid:90) S ( re ( d ≤ s q )) = r − (cid:90) S (cid:16) e ( re ( d ≤ s q )) re ( d ≤ s q ) + ( κ − r − )( re ( d ≤ s q )) (cid:17) − e ( r ) − r r − (cid:90) S ( re ( d ≤ s q )) and hence (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18) r − (cid:90) S ( re ( d ≤ s q )) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:90) S (cid:40)(cid:16) r − (cid:12)(cid:12) d ≤ s +2 q (cid:12)(cid:12) + r (cid:12)(cid:12) d ≤ s N g | + r (cid:12)(cid:12) d ≤ s e ( N g ) | (cid:17) | re ( d ≤ s q ) | + r − ( re ( d ≤ s q )) (cid:41) (cid:46) r − (cid:90) S (cid:16) r − (cid:12)(cid:12) d ≤ s +2 q (cid:12)(cid:12) + r (cid:12)(cid:12) d ≤ s N g | + r (cid:12)(cid:12) d ≤ s e ( N g ) | (cid:17) + r − (cid:90) S ( re ( d ≤ s q )) Together with (5.2.7), this yields (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18) r − (cid:90) S ( re ( d ≤ s q )) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:90) S (cid:16) r (cid:12)(cid:12) d ≤ s N g | + r (cid:12)(cid:12) d ≤ s e ( N g ) | (cid:17) + r − (cid:90) S ( re ( d ≤ s q )) + r − (1 + τ ) − − q + δ E s +2 δ, q − δ [ q ] . CHAPTER 5. DECAY ESTIMATES FOR Q (THEOREM M1) Now, recall from (5.2.12) that we have for s ≤ k small + 30 | d s N g | (cid:46) (cid:15) r − τ − − δ dec +2 δ | d s N g | (cid:46) (cid:15) r − τ − − δ dec +2 δ , | d s e ( N g ) | (cid:46) (cid:15) r − τ − − δ dec +2 δ , | d s e ( N g ) | (cid:46) (cid:15) r − − δB τ − − δ dec +2 δ . By interpolation, we infer r − (cid:90) S (cid:16) r (cid:12)(cid:12) d ≤ s N g | + r (cid:12)(cid:12) d ≤ s e ( N g ) | (cid:17) (cid:46) (cid:15) r − τ − − δ dec +4 δ + (cid:15) r − − δB τ − − δ dec +4 δ (cid:46) (cid:15) r − − δB τ − − δ dec +4 δ and hence (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18) r − (cid:90) S ( re ( d ≤ s q )) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:90) S ( re ( d ≤ s q )) + (cid:15) r − − δB τ − − δ dec +4 δ + r − (1 + τ ) − − q + δ E s +2 δ, q − δ [ q ] . We integrate from Σ ∗ . By Gronwall, and in view of (5.4.31), we deduce for r ≥ m (1 + τ ) q − δ (cid:90) S r ( e d ≤ s q ) (cid:46) (cid:15) + F s +1 δ, q − δ [ q ] + E s +2 δ, q − δ [ q ] . On the other hand, we have by a trace estimate for r ≤ m (1 + τ ) q − δ (cid:90) S r ( e d ≤ s q ) (cid:46) E s +20 , q − δ [ q ] . We finally deduce on M (1 + τ ) q − δ (cid:90) S r ( e d ≤ s q ) (cid:46) (cid:15) + F s +1 δ, q − δ [ q ] + E s +2 δ, q − δ [ q ]which is the desired estimate (5.2.9). This concludes the proof of Proposition 5.2.5. hapter 6DECAY ESTIMATES FOR α AND α (Theorems M2, M3) In this section, we rely on the decay of q to prove the decay estimates for α and α . Moreprecisely, we rely on the results of Theorem M1 to prove Theorem M2 and M3. ( ext ) M In Theorem M1, decay estimates are derived for q defined with respect to the global frameconstructed in Proposition 3.5.5. We have the following control for the Ricci coefficientsin that frame. Lemma 6.1.1.
Consider the global null frame ( e , e , e θ ) constructed in Proposition 3.5.5.Then, the Ricci coefficients satisfy the following estimates max ≤ k ≤ k small +20 sup ( ext ) M u (cid:32) r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) ω + mr , κ − r , ϑ, ζ, η, η (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) ξ, ω, κ + 2 r , ϑ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , + (cid:12)(cid:12) d k ( e ( r ) − Υ , e ( r ) + 1) (cid:12)(cid:12) (cid:33) (cid:46) (cid:15). Proof.
This follows immediately from the stronger estimates of Lemma 5.1.1 with the29394
CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) choice k loss = 20. α To recover α from q , we derive below a transport equation for α where q is on the RHS.We are careful to avoid terms of the type e ( ω ) as they are anomalous w.r.t. decay in r .Indeed, they only decay linearly in r − while all comparable term decay like r − in r . Lemma 6.1.2.
We have κ e (cid:26) e (cid:18) ακ (cid:19) − (cid:18) ω − κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) ακ (cid:27) = q r + (cid:26) ω + 4 κ (cid:18) − d/ ξ − η − ζ ) ξ + 14 ϑ (cid:19)(cid:27) e α + (cid:40) − d/ ξ + (cid:18) κ − ω + 8 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) ω + 12 ϑ − κ e (( η − ζ ) ξ )+ (cid:18)
16 + 48 κ ω − κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) ζξ (cid:41) α. Proof.
We compute e e (cid:18) ακ (cid:19) = e (cid:18) e ακ − e ( κ ) ακ (cid:19) = 1 κ (cid:18) e e α − e ( κ ) κ e α − κ e (cid:18) e ( κ ) κ (cid:19) α (cid:19) . Now, recall the following null structure equation e ( κ ) + 12 κ + 2 ω κ = 2 d/ ξ + 2( η − ζ ) ξ − ϑ . We infer e ( κ ) κ = − κ − ω + 1 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) .1. PROOF OF THEOREM M2 e (cid:18) e ( κ ) κ (cid:19) = e (cid:18) − κ − ωκ + 1 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) = e ( κ )2 κ + e (cid:18) − ωκ + 1 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) = −
14 + 12 κ (cid:18) − ω + 1 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) + e (cid:18) − ωκ + 1 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) and hence κ e e (cid:18) ακ (cid:19) = e e α − e ( κ ) κ e α − κ e (cid:18) e ( κ ) κ (cid:19) α = e e α + 2 κe α + 12 κ α + (cid:18) ω − κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) e α + (cid:40) − κ (cid:18) − ω + 1 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) − κ e (cid:18) − ωκ + 1 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) (cid:41) α. Next, recall from section 2.3.3 that q is defined with respect to a general null frame asfollows q = r (cid:18) e ( e ( α )) + (2 κ − ω ) e ( α ) + (cid:18) − e ( ω ) + 8 ω − ω κ + 12 κ (cid:19) α (cid:19) . We infer κ e e (cid:18) ακ (cid:19) = q r + (cid:18) ω − κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) e α + (cid:40) e ( ω ) − ω + 10 ω κ − (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) − κ e (cid:18) − ωκ + 1 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) (cid:41) α. CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) We rewrite the following terms (cid:40) e ( ω ) − κ e (cid:18) − ωκ + 1 κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) (cid:41) α = κ e (cid:26)(cid:18) ωκ − κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) α (cid:27) − κ ωe (cid:18) ακ (cid:19) + 2 κ (cid:18) d/ ξ − ϑ (cid:19) e ( α )= κ e (cid:26)(cid:18) ωκ − κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) α (cid:27) + (cid:18) − ω + 2 κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) e α − κ ωe (cid:18) κ (cid:19) α so that we obtain κ e (cid:26) e (cid:18) ακ (cid:19) − (cid:18) ωκ − κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) α (cid:27) = q r + (cid:26) ω − κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:27) e α + (cid:40) − ω + 10 ω κ − (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) − κ e (cid:18) κ ( η − ζ ) ξ (cid:19) − κ ωe (cid:18) κ (cid:19) (cid:41) α which we rewrite as κ e (cid:26) e (cid:18) ακ (cid:19) − (cid:18) ω − κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) ακ (cid:27) = q r + (cid:26) ω − κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:27) e α + (cid:40) − ω + 10 ω κ − (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) − κ e (( η − ζ ) ξ ) + 12 e ( κ ) κ ( η − ζ ) ξ + 8 e ( κ ) κ ω (cid:41) α. Now, recall from above that we have e ( κ ) κ = − κ − ω + 1 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) . .1. PROOF OF THEOREM M2 κ e (cid:26) e (cid:18) ακ (cid:19) − (cid:18) ω − κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) ακ (cid:27) = q r + (cid:26) ω + 4 κ (cid:18) − d/ ξ − η − ζ ) ξ + 14 ϑ (cid:19)(cid:27) e α + (cid:40) − d/ ξ + (cid:18) κ − ω + 8 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) ω + 12 ϑ − κ e (( η − ζ ) ξ )+ (cid:18)
16 + 48 κ ω − κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) ζξ (cid:41) α. This concludes the proof of the lemma. e The following lemma will be useful to integrate the transport equations in e . Lemma 6.1.3.
Let p ∈ ( ext ) M . Let γ [ p ] the unique integral curve of e starting from apoint on C terminating at p . Then, we have for l ≥ (cid:90) γ [ p ] r (cid:48) l u (cid:48) + δ extra + r (cid:48) l u (cid:48) δ extra (cid:46) r l (2 r + u ) + δ extra + r l (2 r + u ) δ extra and (cid:90) γ [ p ] r (cid:48) u (cid:48) + δ extra + r (cid:48) u (cid:48) δ extra (cid:46) r (2 r + u ) + δ extra + u ) (2 r + u ) δ extra where ( u, r ) correspond to p and ( r (cid:48) , u (cid:48) ) to a point on γ [ p ] , and where the integration along γ [ p ] relies on a parametrization of γ [ p ] normalized with respect to e .Proof. Note first from the construction of ( ext ) M that γ [ p ] exists for any p ∈ ( ext ) M (i.e.any point p can be joined to C by an integral curve of e ), and γ [ p ] is included in ( ext ) M .Next, recall that the integration along γ [ p ] relies on a parametrization of γ [ p ] normalizedwith respect to e . To parametrize the integration by u or r , we will thus have to derivean upper bound for the corresponding Jacobian of the change of variable, i.e. for1 | e ( u ) | , | e ( r ) | . CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) To this end, note that we have on ( ext ) M e ( u ) = 2 ς Υ ≥ O ( (cid:15) )Υ ≥ ≥ ≤ ( ext ) M in view of Lemma 6.1.1 | e ( r ) | ≥ − | e ( r ) + 1 | = 1 + O ( (cid:15) ) ≥ . Hence, we have obtained on ( ext ) M | e ( u ) | ≤ , | e ( r ) | ≤ . (6.1.1)Next, since e ( u ) > e ( r ) < ( ext ) M , we have r (cid:48) ≥ r and 1 ≤ u (cid:48) ≤ u . We startwith the proof of the first inequality. We consider two cases • If r ≥ u , we have (cid:90) γ [ p ] r (cid:48) l u (cid:48) + δ extra + r (cid:48) l u (cid:48) δ extra ≤ r l (cid:90) u | e ( u (cid:48) ) | du (cid:48) u (cid:48) + δ extra (cid:46) u − δ extra r l (cid:46) r l (2 r + u ) + δ extra + r l (2 r + u ) δ extra , where we used (6.1.1). • If r ≤ u , we separate the integral in r (cid:48) ≥ u , which coincides with 1 ≤ u (cid:48) ≤ u , and .1. PROOF OF THEOREM M2 r ≤ r (cid:48) ≤ u and compute (cid:90) γ [ p ] r (cid:48) l u (cid:48) + δ extra + r (cid:48) l u (cid:48) δ extra = (cid:90) u r (cid:48) l u (cid:48) + δ extra + r (cid:48) l u (cid:48) δ extra du (cid:48) | e ( u (cid:48) ) | + (cid:90) ur r (cid:48) l u (cid:48) + δ extra + r (cid:48) l u (cid:48) δ extra dr (cid:48) | e ( r (cid:48) ) | (cid:46) u l (cid:90) u | e ( u (cid:48) ) | du (cid:48) u (cid:48) + δ extra + min (cid:18) u + δ extra (cid:90) ur | e ( r (cid:48) ) | dr (cid:48) r (cid:48) l , u δ extra (cid:90) ur | e ( r (cid:48) ) | dr (cid:48) r (cid:48) l (cid:19) (cid:46) u + δ extra + min (cid:18) u + δ extra r l , u δ extra r l (cid:19) (cid:46) r l (2 r + u ) + δ extra + r l (2 r + u ) δ extra , where we used (6.1.1).This proves the first inequality.The second inequality is obtained similarly as follows • If r ≥ u , we have (cid:90) γ [ p ] r (cid:48) u (cid:48) + δ extra + r (cid:48) u (cid:48) δ extra ≤ r (cid:90) u | e ( u (cid:48) ) | du (cid:48) u (cid:48) + δ extra (cid:46) u − δ extra r (cid:46) r (2 r + u ) + δ extra + (2 r + u ) δ extra , where we used (6.1.1). • If r ≤ u , we separate the integral in r (cid:48) ≥ u , which coincides with 1 ≤ u (cid:48) ≤ u , and00 CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) r ≤ r (cid:48) ≤ u and compute (cid:90) γ [ p ] r (cid:48) u (cid:48) + δ extra + r (cid:48) u (cid:48) δ extra = (cid:90) u r (cid:48) u (cid:48) + δ extra + r (cid:48) u (cid:48) δ extra du (cid:48) | e ( u (cid:48) ) | + (cid:90) ur r (cid:48) u (cid:48) + δ extra + r (cid:48) u (cid:48) δ extra dr (cid:48) | e ( r (cid:48) ) | (cid:46) u (cid:90) u | e ( u (cid:48) ) | du (cid:48) u (cid:48) + δ extra + min (cid:18) u + δ extra (cid:90) ur | e ( r (cid:48) ) | dr (cid:48) r (cid:48) , u δ extra (cid:90) ur | e ( r (cid:48) ) | dr (cid:48) r (cid:48) (cid:19) (cid:46) u + δ extra + min (cid:32) u + δ extra r , u δ extra (cid:90) ru dr (cid:48)(cid:48) r (cid:48)(cid:48) (cid:33) (cid:46) r (2 r + u ) + δ extra + (2 r + u ) δextra log(1+ u ) , where we used (6.1.1).This concludes the proof of the lemma. Corollary 6.1.4.
Let ψ a solution of the following transport equation e ( ψ ) = h on ( ext ) M . Let also < u ≤ u ∗ . Then • If h and ψ satisfy for l ≥ | h | (cid:46) (cid:15) r l u + δ extra + r l u δ extra on ( ext ) M ( u ≤ u ) and | ψ | (cid:46) (cid:15) r l + + δ extra on C , we have sup ( ext ) M ( u ≤ u ) (cid:16) r l (2 r + u ) + δ extra + r l (2 r + u ) δ extra (cid:17) | ψ | (cid:46) (cid:15) . • If h and ψ satisfy | h | (cid:46) (cid:15) r u + δ extra + ru δ extra on ( ext ) M ( u ≤ u ) and | ψ | (cid:46) (cid:15) r + δ extra on C , we have sup ( ext ) M ( u ≤ u ) (cid:18) r (2 r + u ) + δ extra + (2 r + u ) δ extra log(1 + u ) (cid:19) | ψ | (cid:46) (cid:15) . Proof.
This follows immediately from Lemma 6.1.3. .1. PROOF OF THEOREM M2 α We start with an estimate for α on C . Lemma 6.1.5.
We have max ≤ k ≤ k small +22 sup C r + δ extra | d k α | + max ≤ k ≤ k small +21 sup C r + δ extra | d k e α | (cid:46) (cid:15) . Proof.
Recall that on C , we have obtained in Theorem M0max ≤ k ≤ k large (cid:40) sup C (cid:104) r + δ B (cid:0) | d k ( ext ) α | + | d k ( ext ) β | (cid:1) + r + δ B | d k − e ( ( ext ) α ) | (cid:105) + sup C (cid:20) r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) ( ext ) ρ + 2 m r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r | d k ( ext ) β | + r | d k ( ext ) α | (cid:21) (cid:41) (cid:46) (cid:15) . Since we have chosen δ B ≥ δ extra , we deducemax ≤ k ≤ k large sup C (cid:104) r + δ extra | d k ( ext ) α | + r + δ extra | d k − e ( ( ext ) α ) | (cid:105) (cid:46) (cid:15) . Next, recall that q is defined with respect to the global frame constructed in Proposition3.5.5. In view of Proposition 3.5.5 and Proposition 3.4.6, and the change of frame formulafor α in Proposition 2.3.4, we have α = ( ( ext ) Υ) (cid:18) ( ext ) α + 2 f ( ext ) β + 32 f ext ) ρ + l.o.t. (cid:19) (6.1.2)where f satisfies , see (3.4.11), | d k f | (cid:46) (cid:15)ru + u , for k ≤ k small + 22 on ( ext ) M , | d k − e f | (cid:46) (cid:15)ru for k ≤ k small + 22 on ( ext ) M . (6.1.3)We easily infermax ≤ k ≤ k small +22 sup C r + δ extra | d k α | + max ≤ k ≤ k small +21 sup C r + δ extra | d k e α | (cid:46) (cid:15) . This concludes the proof of the lemma. Here we use (3.4.11) with k loss = 20. Note also that the estimates we claim here for f are slightlyweaker that those in (3.4.11). CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) Next, let 0 < u ≤ u ∗ . We introduce the following bootstrap assumption for α on ( ext ) M ( u ≤ u )max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17)(cid:16) | d k α | + r | d k e α | (cid:17) ≤ (cid:15). (6.1.4)The goal of this section will be the following proposition, i.e. the improvement of thesebootstrap assumptions. Proposition 6.1.6.
We have max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17)(cid:16) | d k α | + r | d k e α | (cid:17) (cid:46) (cid:15) . Proposition 6.1.6 will be proved at the end of this section.Based on the bootstrap assumptions (6.1.4), we estimate the RHS of the transport equa-tion for α . Lemma 6.1.7.
We have e (cid:26) e (cid:18) ακ (cid:19) − F (cid:27) = F where F and F satisfy max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra + r (2 r + u ) + δ extra (cid:17) | d k F | + max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r u δ extra + r u + δ extra (cid:17) | d k F | (cid:46) (cid:15) . Proof.
Recall that we have κ e (cid:26) e (cid:18) ακ (cid:19) − (cid:18) ω − κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) ακ (cid:27) = q r + (cid:26) ω + 4 κ (cid:18) − d/ ξ − η − ζ ) ξ + 14 ϑ (cid:19)(cid:27) e α + (cid:40) − d/ ξ + (cid:18) κ − ω + 8 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) ω + 12 ϑ − κ e (( η − ζ ) ξ )+ (cid:18)
16 + 48 κ ω − κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) ζξ (cid:41) α .1. PROOF OF THEOREM M2 e (cid:26) e (cid:18) ακ (cid:19) − F (cid:27) = F where F and F are defined by F := (cid:18) ω − κ (cid:18) d/ ξ − ϑ (cid:19)(cid:19) ακ and F := q r κ + 1 κ (cid:26) ω + 4 κ (cid:18) − d/ ξ − η − ζ ) ξ + 14 ϑ (cid:19)(cid:27) e α + 1 κ (cid:40) − d/ ξ + (cid:18) κ − ω + 8 κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) ω + 12 ϑ − κ e (( η − ζ ) ξ )+ (cid:18)
16 + 48 κ ω − κ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19)(cid:19) ζξ (cid:41) α. In view of the bootstrap assumptions (6.1.4) for α , the estimates of Lemma 6.1.1 for theRicci coefficients, and using Theorem M2 to estimate q , we easily infermax ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra + r (2 r + u ) + δ extra (cid:17) | d k F | (cid:46) (cid:15) max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:18) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:19) (cid:16) | d k α | + r | d k e α | (cid:17) (cid:46) (cid:15) (cid:46) (cid:15) . and max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r u δ extra + r u + δ extra (cid:17) | d k F | (cid:46) max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) u δ extra + ru + δ extra (cid:17) | d k q | + (cid:15) max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:18) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:19) (cid:16) | d k α | + r | d k e α | (cid:17) (cid:46) (cid:15) + (cid:15) (cid:46) (cid:15) . This concludes the proof of the lemma.04
CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) Lemma 6.1.8.
For ≤ k + j ≤ k small + 20 , we have e (cid:26) e d / k e j (cid:18) ακ (cid:19) − F , d / k ,e j (cid:27) = F , d / k ,e j where max ≤ l ≤ k small +20 − k sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra + r (2 r + u ) + δ extra (cid:17) | d l F , d / k | + max ≤ l ≤ k small +20 − k sup ( ext ) M ( u ≤ u ) (cid:16) r u δ extra + r u + δ extra (cid:17) | d l F , d / k | (cid:46) (cid:15) , and for j ≥ ≤ l ≤ k small +20 − k − j sup ( ext ) M ( u ≤ u ) (cid:16) r j (2 r + u ) δ extra + r j (2 r + u ) + δ extra (cid:17) | d l F , d / k ,e j | + max ≤ l ≤ k small +20 − k − j sup ( ext ) M ( u ≤ u ) (cid:16) r j u δ extra + r j u + δ extra (cid:17) | d l F , d / k ,e j | (cid:46) (cid:15) + max ≤ j + k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17) r × (cid:16) | d / k ( re ) j − e α | + | d / k ( re ) j − e α | (cid:17) . Proof.
Recall from Lemma 6.1.7 that we have e (cid:26) e (cid:18) ακ (cid:19) − F (cid:27) = F where F and F satisfymax ≤ k ≤ k small +20 sup ( ext ) M (cid:16) r (2 r + u ) δ extra + r (2 r + u ) + δ extra (cid:17) | d k F | + max ≤ k ≤ k small +20 sup ( ext ) M (cid:16) r u δ extra + r u + δ extra (cid:17) | d k F | (cid:46) (cid:15) . Differentiating with d / k , this yields e (cid:26) e d / k (cid:18) ακ (cid:19) + [ d / k , e ] (cid:18) ακ (cid:19) − d / k F (cid:27) = d / k F − [ d / k , e ] (cid:26) e (cid:18) ακ (cid:19) − F (cid:27) and hence e (cid:26) e d / k (cid:18) ακ (cid:19) − F , d / k (cid:27) = F , d / k .1. PROOF OF THEOREM M2 F , d / k := d / k F − [ d / k , e ] (cid:18) ακ (cid:19) , F , d / k := d / k F − [ d / k , e ] (cid:26) e (cid:18) ακ (cid:19) − F (cid:27) . In view of Lemma 2.2.13, we have schematically[ d /, e ] = ˇΓ g d + ˇΓ g + rβ, [ d /, e ] = ˇΓ b d + ˇΓ b + rβ. Together with the estimates of Lemma 6.1.1 for the Ricci coefficients and curvature com-ponents as well as the bootstrap assumptions (6.1.4) for α on ( ext ) M , we infermax ≤ j ≤ k small +20 − k sup ( ext ) M (cid:16) r (2 r + u ) δ extra + r (2 r + u ) + δ extra (cid:17) | d j F , d / k | + max ≤ j ≤ k small +20 − k sup ( ext ) M (cid:16) r u δ extra + r u + δ extra (cid:17) | d j F , d / k | (cid:46) (cid:15) . Next, we consider the case j ≥
1. We have the commutator[ e , e ] = 2 ωe − ωe − ζe θ . In view of the estimates of Lemma 6.1.1 for the Ricci coefficients, and in view of thebootstrap assumptions (6.1.4) for α , we infer after commutation by e j for 0 ≤ k + j ≤ k small + 20 e (cid:26) e d / k e j (cid:18) ακ (cid:19) − F , d / k ,e j (cid:27) = F , d / k ,e j where max ≤ l ≤ k small +20 − k − j sup ( ext ) M ( u ≤ u ) (cid:16) r j (2 r + u ) δ extra + r j (2 r + u ) + δ extra (cid:17) | d l F , d / k ,e j | + max ≤ l ≤ k small +20 − k − j sup ( ext ) M ( u ≤ u ) (cid:16) r j u δ extra + r j u + δ extra (cid:17) | d l F , d / k ,e j | (cid:46) (cid:15) + max ≤ j + k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17) r × (cid:16) | d / k ( re ) j − e α | + | d / k ( re ) j − e α | (cid:17) (cid:46) (cid:15) . This concludes the proof of the lemma.06
CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) We are now ready to prove Proposition 6.1.6.
Step 1.
For 0 ≤ k ≤ k small + 20, recall from the above lemma with j = 0 that we have e (cid:26) e d / k (cid:18) ακ (cid:19) − F , d / k (cid:27) = F , d / k where max ≤ j ≤ k small +20 − k sup ( ext ) M ( u ≤ u ) (cid:16) r u δ extra + r u + δ extra (cid:17) | d j F , d / k | (cid:46) (cid:15) . Also, we have in view of Lemma 6.1.5max ≤ k ≤ k large − sup C r + δ extra (cid:12)(cid:12)(cid:12)(cid:12) e d / k (cid:18) ακ (cid:19) − F , d / k (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . In view of Corollary 6.1.4, we immediately infer for any 0 ≤ k ≤ k small + 20max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) + δ extra + r (2 r + u ) δ extra (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) e d / k (cid:18) ακ (cid:19) − F , d / k (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Since we have from the above lemma thatmax ≤ j ≤ k small +20 − k sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra + r (2 r + u ) + δ extra (cid:17) | d j F , d / k | (cid:46) (cid:15) , we deduce that we have for any 0 ≤ k ≤ k small + 20max ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) + δ extra + r (2 r + u ) δ extra (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) e d / k (cid:18) ακ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . (6.1.5) Step 2.
Next, note that we have in view of Lemma 6.1.5max ≤ k ≤ k large − sup C r + δ extra (cid:12)(cid:12)(cid:12)(cid:12) d / k (cid:18) ακ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . Together with the transport equation (6.1.5), and in view of Corollary 6.1.4, we infermax ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:18) r (2 r + u ) + δ extra + (2 r + u ) δ extra log(1 + u ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) d / k (cid:18) ακ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . In view of the control of κ provided by Lemma 6.1.1, we easily deducemax ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:18) r (2 r + u ) + δ extra + r (2 r + u ) δ extra log(1 + u ) (cid:19) (cid:12)(cid:12) d / k α (cid:12)(cid:12) (cid:46) (cid:15) . .1. PROOF OF THEOREM M2 ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:18) r (2 r + u ) + δ extra + r (2 r + u ) δ extra log(1 + u ) (cid:19) (cid:16) | d / k α | + r | d / k e α | (cid:17) (cid:46) (cid:15) . Step 3.
Next, recall from section 2.3.3 that q is defined with respect to a general nullframe as follows q = r (cid:18) e ( e ( α )) + (2 κ − ω ) e ( α ) + (cid:18) − e ( ω ) + 8 ω − ω κ + 12 κ (cid:19) α (cid:19) . We infer e ( e ( α )) = q r − (2 κ − ω ) e ( α ) − (cid:18) − e ( ω ) + 8 ω − ω κ + 12 κ (cid:19) α. Together with the above estimate for α and e α , we infer by iterationmax ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17) × (cid:16) | ( d /, e ) k α | + r | ( d /, e ) k e α | (cid:17) (cid:46) (cid:15) . Step 4.
Arguing as for Step 1, but with j ≥
1, we infer the following analog of (6.1.5)max ≤ j + k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) + δ extra + r (2 r + u ) δ extra (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) e d / k ( re ) j (cid:18) ακ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) + max ≤ j + k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17) r × (cid:16) | d / k ( re ) j − e α | + | d / k ( re ) j − e α | (cid:17) . Step 5.
Arguing as for Step 2, but with j ≥
1, we infer the following analog of the lastestimate of Step 2max ≤ j ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:18) r (2 r + u ) + δ extra + r (2 r + u ) δ extra log(1 + u ) (cid:19) (cid:16) | d / k ( re ) j α | + r | d / k ( re ) j e α | (cid:17) (cid:46) (cid:15) + max ≤ j + k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17) r × (cid:16) | d / k ( re ) j − e α | + | d / k ( re ) j − e α | (cid:17) . CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) Step 6.
Arguing as for Step 3, but with j ≥
1, we infer the following analog of the lastestimate of Step 3max ≤ j + k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17) × (cid:16) | ( d /, e ) k ( re ) j α | + r | ( d /, e ) k ( re ) j e α | (cid:17) (cid:46) (cid:15) + max ≤ j + k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17) r × (cid:16) | ( d /, e ) k ( re ) j − e α | + r | ( d /, e ) k ( re ) j − e α | (cid:17) . Step 7.
Arguing by iteration on j , noticing that the last estimate of Step 3 correspondsto desired estimate for j = 0, and in view of the estimate derived in Step 6, we finallyobtain max ≤ j + k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17) × (cid:16) | ( d /, e ) k ( re ) j α | + r | ( d /, e ) k ( re ) j e α | (cid:17) (cid:46) (cid:15) and hencemax ≤ k ≤ k small +20 sup ( ext ) M ( u ≤ u ) (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17)(cid:16) | d k α | + r | d k e α | (cid:17) (cid:46) (cid:15) . This concludes the proof of Proposition 6.1.6.
First, note in view of the estimates for α on C provided by Lemma 6.1.5 that the bootstrapassumptions (6.1.4) for α hold by continuity for some sufficiently small u >
0. Then, wemay in view of Proposition 6.1.6 choose u = u ∗ . We deduce thereforemax ≤ k ≤ k small +20 sup ( ext ) M (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17)(cid:16) | d k α | + r | d k e α | (cid:17) (cid:46) (cid:15) . Next, recall from (6.1.2) and (6.1.3) that we have α = ( ( ext ) Υ) (cid:18) ( ext ) α + 2 f ( ext ) β + 32 f ext ) ρ + l.o.t. (cid:19) .2. PROOF OF THEOREM M3 f satisfies | d k f | (cid:46) (cid:15)ru + u , for k ≤ k small + 22 on ( ext ) M , | d k − e f | (cid:46) (cid:15)ru for k ≤ k small + 22 on ( ext ) M . Together with bootstrap assumptions for ( ext ) β and ( ext ) ρ , we easily infermax ≤ k ≤ k small +20 sup ( ext ) M (cid:16) r (2 r + u ) δ extra log(1 + u ) + r (2 r + u ) + δ extra (cid:17)(cid:16) | d k ( ext ) α | + r | d k e ext ) α | (cid:17) (cid:46) (cid:15) . This concludes the proof of Theorem M2.
Theorem M3 contains decay estimates for α in ( int ) M and on Σ ∗ . We first proceed withthe estimate on ( int ) M before moving to ( ext ) M . α in ( int ) M Recall that q , controlled in Theorem M1, is defined with respect to the global frameof Proposition 3.5.5. Recall also that we may choose the global null frame to coincidewith the ingoing geodesic null frame of ( int ) M in ( int ) M (see property (b) in Proposition3.5.5 together with property (d) ii. in Proposition 3.5.2). Thus, in this section, as weonly work on ( int ) M , the null frame ( e , e , e θ ) denotes both the frame of ( int ) M and theglobal frame with respect to which q is defined. We start with the following definition. Definition 6.2.1. In ( int ) M , we define with respect to the ingoing geodesic frame of ( int ) M (cid:101) T := e − κ (cid:16) κ + A (cid:17) e . (6.2.1)The estimate for α in ( int ) M relies on the following proposition. Proposition 6.2.2.
Let ≤ k ≤ k small + 17 . Then, α satisfies in ( int ) M m (cid:101) T ( d k α ) + r d (cid:63) / d (cid:63) / d/ d/ ( d k α ) = F k CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) where F k satisfies max ≤ k ≤ k small +17 (cid:90) ( int ) M u δ dec | d ≤ F k | (cid:46) (cid:15) . Remark 6.2.3.
In view of the definition of (cid:101) T , we have (cid:101) T ( r ) = e ( r ) − κ (cid:16) κ + A (cid:17) e ( r ) = 0 so that (cid:101) T is tangent to the hypersurfaces of constant r . In particular, ( (cid:101) T , e θ ) spans thetangent space of hypersurfaces of constant r . Therefore, in view of Proposition 6.2.2, α and its derivatives satisfy on each hyper surface of contant r in ( int ) M , i.e. on { r = r } for m (1 − δ H ) ≤ r ≤ r T , a forward parabolic equation. Furthermore, since we have (cid:101) T ( u ) = 2 /ς = 2 + O ( (cid:15) ) , u plays the role of time in this forward parabolic equation. We also derive estimates for the control of the parabolic equation appearing in the state-ment of Proposition 6.2.2.
Lemma 6.2.4.
Let f and h reduced 2-scalars such that (cid:16) m (cid:101) T + r d (cid:63) / d (cid:63) / d/ d/ (cid:17) f = h. Then, for any real number n ≥ and any r such that m (1 − δ H ) ≤ r ≤ r T , we have sup ≤ u ≤ u ∗ (cid:90) S ( r = r ,u ) (1 + u n ) f (cid:46) n (cid:90) S ( r = r , f + (cid:15) (cid:90) u ∗ (cid:90) S ( r = r ,u ) (1 + u n − )( d f ) + (cid:90) ( int ) M (1 + u n )( d ≤ h ) . We are now in position to control α in ( int ) M . Recall from Proposition 6.2.2 that α satisfies in ( int ) M for 0 ≤ k ≤ k small + 176 m (cid:101) T ( d k α ) + r d (cid:63) / d (cid:63) / d/ d/ ( d k α ) = F k . Applying Lemma 6.2.4 with n = 2 + 2 δ dec , f = d k α and h = F k , we infer for any r suchthat 2 m (1 − δ H ) ≤ r ≤ r T sup ≤ u ≤ u ∗ (cid:90) S ( r = r ,u ) (1 + u δ dec )( d k α ) (cid:46) (cid:90) S ( r = r , ( d k α ) + (cid:15) (cid:90) u ∗ (cid:90) S ( r = r ,u ) (1 + u δ dec )( d k +1 α ) + (cid:90) ( int ) M (1 + u δ dec )( d ≤ F k ) . .2. PROOF OF THEOREM M3 α on C provided by Theorem M0, the bootstrap assump-tions on decay and energy for α in ( int ) M , and the bound for F k provided by Proposition6.2.2, we infer for 0 ≤ k ≤ k small + 17 in ( int ) M sup ≤ u ≤ u ∗ (cid:90) S ( r = r ,u ) (1 + u δ dec )( d k α ) (cid:46) (cid:15) . In particular, we have obtainedmax ≤ k ≤ k small +17 sup ( int ) M u δ dec (cid:107) d k α (cid:107) L ( S ) (cid:46) (cid:15) . Using the Sobolev embedding on 2-surface and the fact that r is bounded on ( int ) M , weinfer max ≤ k ≤ k small +15 sup ( int ) M u δ dec | d k α | (cid:46) (cid:15) and hence ( int ) D k small +15 [ α ] (cid:46) (cid:15) (6.2.2)which is the desired estimate for α in ( int ) M .The proof of Proposition 6.2.2 will be given in section 6.2.3, and to the proof of Lemma6.2.4 which will be given in section 6.2.4. But first, we conclude in the next section theproof of Theorem M3 by controlling α on Σ ∗ . α on Σ ∗ Recall that q , controlled in Theorem M1, is defined with respect to the global frame ofProposition 3.5.5. We will first control α in this frame, before coming back to ( ext ) M atthe end of the argument. We start with the following definition. Definition 6.2.5. In Σ ∗ , we define, with respect to the the global frame of Proposition3.5.5, (cid:101) ν := e + ae , (6.2.3) where the scalar function a is uniquely defined so that (cid:101) ν is tangent to Σ ∗ . The estimate for α on Σ ∗ relies on the following proposition.12 CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) Proposition 6.2.6.
Let ≤ k ≤ k small + 18 . Then, α satisfies on Σ ∗ m (cid:101) ν ( d k α ) + r d (cid:63) / d (cid:63) / d/ d/ ( d k α ) = F k where F k satisfies max ≤ k ≤ k small +18 (cid:90) Σ ∗ u δ dec | F k | (cid:46) (cid:15) . Remark 6.2.7.
Since (cid:101) ν is tangent to Σ ∗ , and since ( (cid:101) ν, e θ ) spans the tangent space of Σ ∗ , in view of Proposition 6.2.6, α and its derivatives satisfy on Σ ∗ a forward parabolicequation. Furthermore, since we have (cid:101) ν ( u ) = 2 + O ( (cid:15) ) , u plays the role of time in thisforward parabolic equation. We also derive estimates for the control of the parabolic equation appearing in the state-ment of Proposition 6.2.2.
Lemma 6.2.8.
Let f and h reduced 2-scalars such that (cid:16) m (cid:101) ν + r d (cid:63) / d (cid:63) / d/ d/ (cid:17) f = h. Then, for any real number n ≥ , we have (cid:90) Σ ∗ (1 + u n ) f (cid:46) n (cid:90) Σ ∗ ∩C f + (cid:15) (cid:90) Σ ∗ (1 + u n − )( d f ) + (cid:90) Σ ∗ (1 + u n ) h . We are now in position to control α on Σ ∗ . Recall from Proposition 6.2.6 that α satisfiesin Σ ∗ for 0 ≤ k ≤ k small + 186 m (cid:101) ν ( d k α ) + r d (cid:63) / d (cid:63) / d/ d/ ( d k α ) = F k . Applying Lemma 6.2.8 with n = 2 + 2 δ dec , f = d k α and h = F k , we infer (cid:90) Σ ∗ (1 + u δ dec )( d k α ) (cid:46) (cid:90) Σ ∗ ∩C ( d k α ) + (cid:15) (cid:90) Σ ∗ (1 + u δ dec )( d k +1 α ) + (cid:90) Σ ∗ (1 + u δ dec )( F k ) . Together with the bounds for α on C provided by Theorem M0, the bootstrap assump-tions on decay and energy for α in ( ext ) M , and the bound for F k provided by Proposition6.2.6, we infer (cid:90) Σ ∗ (1 + u δ dec )( d k α ) (cid:46) (cid:15) . .2. PROOF OF THEOREM M3 ≤ k ≤ k small +18 (cid:90) Σ ∗ (1 + u δ dec )( d k α ) (cid:46) (cid:15) . Now, recall that α in the above estimate is defined with respect to the global frameconstructed in Proposition 3.5.5. In view of Proposition 3.5.5 and Proposition 3.4.6, andthe change of frame formula for α in Proposition 2.3.4, we have α = ( ( ext ) Υ) − ext ) α. Hence, we immediately infermax ≤ k ≤ k small +18 (cid:90) Σ ∗ (1 + u δ dec )( d k ( ext ) α ) (cid:46) (cid:15) . which is the desired estimate in Σ ∗ . Together with (6.2.2), this concludes the proof ofTheorem M3.The proof of Proposition 6.2.6 will be given in section 6.2.5, and to the proof of Lemma6.2.8 which will be given in section 6.2.6. In this section, we infer from the Teukolsky-Starobinski identity, see Proposition 2.3.15,a parabolic equation for α . Corollary 6.2.9. α satisfies in ( int ) M the following equation m (cid:101) T α + r d (cid:63) / d (cid:63) / d/ d/ α = 1 r (cid:16) e ( r e ( r q )) + 2 ωr e ( r q ) (cid:17) − r − Err [ T S ] − (cid:26) r (cid:18) ρ + 2 mr (cid:19) κ − mr (cid:18) κ + 2 r (cid:19)(cid:27) e α − (cid:26) − r (cid:18) ρ + 2 mr (cid:19) κ + 3 mrκ (cid:18) κ + 2 r (cid:19) κ + 3 mr ˇ κ + 6 mκ A (cid:27) e α where the vectorfield (cid:101) T is defined by (6.2.1) .Proof. Recall from (2.3.15) that we have e ( r e ( r q )) + 2 ωr e ( r q ) = r (cid:26) d (cid:63) / d (cid:63) / d/ d/ α + 32 ρ (cid:16) κe − κe (cid:17) α (cid:27) + Err[ T S ] . CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) This yields32 r ρ (cid:16) κe − κe (cid:17) α + r d (cid:63) / d (cid:63) / d/ d/ α = 1 r (cid:16) e ( r e ( r q )) + 2 ωr e ( r q ) (cid:17) − r − Err[
T S ] . Now, we have in view of the definition of (cid:101) T r ρ (cid:16) κe − κe (cid:17) − m (cid:101) T = (cid:26) r (cid:18) ρ + 2 mr (cid:19) κ − mr (cid:18) κ + 2 r (cid:19)(cid:27) e + (cid:26) − r (cid:18) ρ + 2 mr (cid:19) κ + 3 mrκ (cid:18) κ + 2 r (cid:19) κ + 3 mr ˇ κ + 6 mκ A (cid:27) e . We infer6 m (cid:101) T α + r d (cid:63) / d (cid:63) / d/ d/ α = 1 r (cid:16) e ( r e ( r q )) + 2 ωr e ( r q ) (cid:17) − r − Err[
T S ] − (cid:26) r (cid:18) ρ + 2 mr (cid:19) κ − mr (cid:18) κ + 2 r (cid:19)(cid:27) e α − (cid:26) − r (cid:18) ρ + 2 mr (cid:19) κ + 3 mrκ (cid:18) κ + 2 r (cid:19) κ + 3 mr ˇ κ + 6 mκ A (cid:27) e α. This concludes the proof of the corollary.
Corollary 6.2.10. α satisfies in ( int ) M m (cid:101) T α + r d (cid:63) / d (cid:63) / d/ d/ α = F where F satisfies max ≤ k ≤ k small +18 (cid:90) ( int ) M u δ dec | d k F | (cid:46) (cid:15) . Proof.
In view of Corollary 6.2.9, α satisfies6 m (cid:101) T α + r d (cid:63) / d (cid:63) / d/ d/ α = F with F := 1 r (cid:16) e ( r e ( r q )) + 2 ωr e ( r q ) (cid:17) + F ,F := − r − Err[
T S ] − (cid:26) r (cid:18) ρ + 2 mr (cid:19) κ − mr (cid:18) κ + 2 r (cid:19)(cid:27) e α − (cid:26) − r (cid:18) ρ + 2 mr (cid:19) κ + 3 mrκ (cid:18) κ + 2 r (cid:19) κ + 3 mr ˇ κ + 6 mκ A (cid:27) e α. .2. PROOF OF THEOREM M3 ( int ) M for decay and energies, and in view of the factthat F contains only quadratic or higher order terms, we easily derivemax ≤ k ≤ k small +18 sup ( int ) M u + δ dec | d k F | (cid:46) (cid:15) (cid:46) (cid:15) . In view of the definition of F , this yieldsmax ≤ k ≤ k small +18 (cid:90) ( int ) M u δ dec | d k F | (cid:46) (cid:15) + max ≤ k ≤ k small +20 (cid:90) ( int ) M u δ dec | d k q | . Together with Theorem M1, and the fact that δ extra > δ dec , we infermax ≤ k ≤ k small +18 (cid:90) ( int ) M u δ dec | d k F | (cid:46) (cid:15) . This concludes the proof of the corollary.We are now ready to prove Proposition 6.2.2. In view of Corollary 6.2.10, α satisfies6 m (cid:101) T α + r d (cid:63) / d (cid:63) / d/ d/ α = F. Commuting with d k , we infer6 m (cid:101) T ( d k α ) + r d (cid:63) / d (cid:63) / d/ d/ ( d k α ) = F k where F k is defined by F k := − m [ d k , (cid:101) T ] α − k (cid:88) j =1 d j ( m ) d k − j (cid:101) T α − [ d k , r d (cid:63) / ] r d (cid:63) / r d/ r d/ α − r d (cid:63) / [ d k , r d (cid:63) / ] r d/ r d/ α − r d (cid:63) / r d (cid:63) / [ d k , r d/ ] r d/ α − r d (cid:63) / r d (cid:63) / r d/ [ d k , r d/ ] α + d k F. Note that we have schematically[ d , d / ] = ˇΓ d , [ (cid:101) T , d / ] = (cid:16) d ˇΓ + ˇΓ (cid:17) d , CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) as well as[ (cid:101) T , re ] = e ( r ) e − κ (cid:16) κ + A (cid:17) [ e , re ] + re (cid:18) κ (cid:16) κ + A (cid:17)(cid:19) e = r (cid:16) κ + A (cid:17) e − κ (cid:16) κ + A (cid:17) (cid:16) r κe + r (cid:16) − ωe + 4 ζe θ (cid:17)(cid:17) + re (cid:18) κκ (cid:19) e + re (cid:18) κ A (cid:19) e = (cid:26) rκ (cid:16) κ + A (cid:17) ω + re (cid:18) κκ (cid:19) + re (cid:18) κ A (cid:19)(cid:27) e − rκ (cid:16) κ + A (cid:17) ζe θ = (cid:40) − mr (cid:18) κκ + Υ (cid:19) + 2 rκ κ (cid:16) ω + mr (cid:17) − m ( e ( r ) − Υ) r + 2 e ( m ) + re (cid:18) κκ + Υ (cid:19) + 2 rκ Aω + re (cid:18) κ A (cid:19) (cid:41) e − rκ (cid:16) κ + A (cid:17) ζe θ = (cid:16) d ˇΓ + ˇΓ (cid:17) d , and[ (cid:101) T , e ] = [ e , e ] + e (cid:18) κ (cid:16) κ + A (cid:17)(cid:19) e = (cid:26) ω + e (cid:18) κκ (cid:19) + e (cid:18) κ A (cid:19)(cid:27) e − ζe θ = (cid:26) (cid:16) ω + mr (cid:17) − m ( e ( r ) + 1) r + 2 e ( m ) r + e (cid:18) κκ + Υ (cid:19) + e (cid:18) κ A (cid:19)(cid:27) e − ζe θ = (cid:16) d ˇΓ + ˇΓ (cid:17) d . Together with the bootstrap assumptions in ( int ) M for decay and energies, and in viewof the fact that F contains only quadratic or higher order terms, we easily derivemax ≤ k ≤ k small +18 sup ( int ) M u + δ dec (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − m [ d k , (cid:101) T ] α − k (cid:88) j =1 d j ( m ) d k − j (cid:101) T α − [ d k , r d (cid:63) / ] r d (cid:63) / r d/ r d/ α − r d (cid:63) / [ d k , r d (cid:63) / ] r d/ r d/ α − r d (cid:63) / r d (cid:63) / [ d k , r d/ ] r d/ α − r d (cid:63) / r d (cid:63) / r d/ [ d k , r d/ ] α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) . .2. PROOF OF THEOREM M3 F k , this yieldsmax ≤ k ≤ k small +18 (cid:90) ( int ) M u δ dec | F k | (cid:46) (cid:15) + max ≤ k ≤ k small +18 (cid:90) ( int ) M u δ dec | d k F | . Together with the estimate for F of Corollary 6.2.10, we infermax ≤ k ≤ k small +18 (cid:90) ( int ) M u δ dec | F k | (cid:46) (cid:15) + (cid:15) (cid:46) (cid:15) . This concludes the proof of Proposition 6.2.2.
In this section we prove Lemma 6.2.4, i.e. we derive estimates for the control of theparabolic equation appearing in the statement of Proposition 6.2.2. To this end, we firststart with a Poincar´e inequality.
Lemma 6.2.11.
We have (cid:90) S f d (cid:63) / d (cid:63) / d/ d/ f ≥ (cid:90) S (1 + O ( (cid:15) )) K f . Proof.
We have d (cid:63) / d (cid:63) / d/ d/ = d (cid:63) / ( −(cid:52) / + K ) d/ = − d (cid:63) / (cid:52) / d/ + K d (cid:63) / d/ + d (cid:63) / ( K ) d/ = −(cid:52) / d (cid:63) / d/ + (cid:16) (cid:52) / d (cid:63) / − d (cid:63) / (cid:52) / (cid:17) d/ + K d (cid:63) / d/ + d (cid:63) / ( K ) d/ = ( d (cid:63) / d/ − K ) d (cid:63) / d/ + (cid:16) K d (cid:63) / − d (cid:63) / ( K ) (cid:17) d/ + K d (cid:63) / d/ + d (cid:63) / ( K ) d/ = ( d (cid:63) / d/ ) + 2 K d (cid:63) / d/ . Recall also the Poincar´e inequality for d/ which holds for any reduced 2-scalar f (cid:90) S | d/ f | ≥ (cid:90) S Kf . Then, we easily infer (cid:90) S f d (cid:63) / d (cid:63) / d/ d/ f = (cid:90) S f ( d (cid:63) / d/ ) f + (cid:90) S Kf d (cid:63) / d/ f ≥ (cid:90) S (1 + O ( (cid:15) )) K f + 8 (cid:90) S (1 + O ( (cid:15) )) K f ≥ (cid:90) S (1 + O ( (cid:15) )) K f CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) where we also used the estimates for the Gauss curvature K = 1 r + O (cid:16) (cid:15)r (cid:17) , re θ ( K ) = O (cid:16) (cid:15)r (cid:17) , which follow from the bootstrap assumptions.The following identity will be useful. Lemma 6.2.12.
We have for any reduced scalar f (cid:101) T (cid:18)(cid:90) S f (cid:19) = 2 (cid:90) S f (cid:101) T f + (cid:90) S (cid:18) κ Af e ( f ) + ˇ κf (cid:19) − κκ (cid:18)(cid:90) S ˇ κf (cid:19) − κ A (cid:18)(cid:90) S (2 f e ( f ) + κf ) (cid:19) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) . Proof.
Recall from the definition of (cid:101) T that (cid:101) T = e − κ (cid:16) κ + A (cid:17) e . We infer, in view of the analog of Proposition 2.2.9 for an ingoing geodesic foliation, (cid:101) T (cid:18)(cid:90) S f (cid:19) = e (cid:18)(cid:90) S f (cid:19) − κ (cid:16) κ + A (cid:17) e (cid:18)(cid:90) S f (cid:19) = (cid:90) S (2 f e ( f ) + κf ) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) − κ (cid:16) κ + A (cid:17) (cid:18)(cid:90) S (2 f e ( f ) + κf ) (cid:19) = (cid:90) S (cid:18) f (cid:101) T f + 2 κ (cid:16) κ + A (cid:17) f e ( f ) + κf (cid:19) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) − κ (cid:16) κ + A (cid:17) (cid:18)(cid:90) S (2 f e ( f ) + κf ) (cid:19) = 2 (cid:90) S f (cid:101) T f + (cid:90) S (cid:18) κ Af e ( f ) + ˇ κf (cid:19) − κκ (cid:18)(cid:90) S ˇ κf (cid:19) − κ A (cid:18)(cid:90) S (2 f e ( f ) + κf ) (cid:19) +Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) . This concludes the proof of the lemma. .2. PROOF OF THEOREM M3 (cid:101) T (cid:18)(cid:90) S f (cid:19) = 2 (cid:90) S f (cid:101) T f + (cid:90) S (cid:18) κ Af e ( f ) + ˇ κf (cid:19) − κκ (cid:18)(cid:90) S ˇ κf (cid:19) − κ A (cid:18)(cid:90) S (2 f e ( f ) + κf ) (cid:19) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) . In view of the equation satisfied by f , we infer (cid:101) T (cid:18)(cid:90) S f (cid:19) = − m (cid:18)(cid:90) S r f d (cid:63) / d (cid:63) / d/ d/ f (cid:19) + 13 m (cid:18)(cid:90) S hf (cid:19) + (cid:90) S (cid:18) κ Af e ( f ) + ˇ κf (cid:19) − κκ (cid:18)(cid:90) S ˇ κf (cid:19) − κ A (cid:18)(cid:90) S (2 f e ( f ) + κf ) (cid:19) +Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) . Now, from the definition of (cid:101) T , we have (cid:101) T ( u ) = 2 /ς . We deduce (cid:101) T (cid:18) u n (cid:90) S f (cid:19) + u n m (cid:18)(cid:90) S r f d (cid:63) / d (cid:63) / d/ d/ f (cid:19) = u n m (cid:18)(cid:90) S hf (cid:19) + u n (cid:90) S (cid:18) κ Af e ( f ) + ˇ κf (cid:19) − u n κκ (cid:18)(cid:90) S ˇ κf (cid:19) − u n κ A (cid:18)(cid:90) S (2 f e ( f ) + κf ) (cid:19) + u n Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) + 2 ς nu n − (cid:90) S f . This yields in view of the bootstrap assumptions (cid:101) T (cid:18) u n (cid:90) S f (cid:19) + u n m (cid:18)(cid:90) S r f d (cid:63) / d (cid:63) / d/ d/ f (cid:19) (cid:46) u n m (cid:107) h (cid:107) L ( S ) (cid:107) f (cid:107) L ( S ) + (cid:15)u n − (cid:90) S | f || d ≤ f | + nu n − (cid:90) S f . Next, we rely on the Poincar´e inequality of Lemma 6.2.11 to deduce (cid:101) T (cid:18) u n (cid:90) S f (cid:19) + u n (cid:90) S f (cid:46) u n (cid:90) S h + (cid:15) u n − (cid:90) S ( d f ) + nu n − (cid:90) S f . Integrating in u between 1 and u ∗ , and recalling that (cid:101) T ( u ) = 2 /ς , we infer for any r suchthat 2 m (1 − δ H ) ≤ r ≤ r T (cid:90) S ( r = r ,u ) u n f + (cid:90) u (cid:18)(cid:90) S ( r = r ,u (cid:48) ) u (cid:48) n f (cid:19) du (cid:48) (cid:46) (cid:90) S ( r = r , f + (cid:90) u (cid:18)(cid:90) S ( r = r ,u (cid:48) ) u (cid:48) n h (cid:19) du (cid:48) + (cid:15) (cid:90) u (cid:18)(cid:90) S ( r = r ,u (cid:48) ) u (cid:48) n − ( d f ) (cid:19) du (cid:48) + n (cid:90) u (cid:18)(cid:90) S ( r = r ,u (cid:48) ) u (cid:48) n − f (cid:19) du (cid:48) . CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) In particular, we have for n = 0sup ≤ u ≤ u ∗ (cid:90) S ( r = r ,u ) f + (cid:90) u ∗ (cid:18)(cid:90) S ( r = r ,u ) f (cid:19) du (cid:46) (cid:90) S ( r = r , f + (cid:90) u ∗ (cid:18)(cid:90) S ( r = r ,u ) h (cid:19) du + (cid:15) (cid:90) u ∗ (cid:18)(cid:90) S ( r = r ,u ) u − ( d f ) (cid:19) du. Then, starting from the case n = 0 and arguing by iteration on the largest integer below n , one immediately deduces for any real n ≥ ≤ u ≤ u ∗ (cid:90) S ( r = r ,u ) (1 + u n ) f + (cid:90) u ∗ (cid:18)(cid:90) S ( r = r ,u ) (1 + u n ) f (cid:19) du (cid:46) (cid:90) S ( r = r , f + (cid:90) u ∗ (cid:18)(cid:90) S ( r = r ,u ) (1 + u n ) h (cid:19) du + (cid:15) (cid:90) u ∗ (cid:18)(cid:90) S ( r = r ,u ) (1 + u n − )( d f ) (cid:19) du. Now, a simple trace estimate yields (cid:90) S ( r = r ,u ) (1 + u n ) h (cid:46) (cid:90) C u (1 + u n ) (cid:16) | h | + | e ( h ) | (cid:17) so that (cid:90) u ∗ (cid:18)(cid:90) S ( r = r ,u ) (1 + u n ) h (cid:19) du (cid:46) (cid:90) u ∗ (cid:90) C u (1 + u n ) (cid:16) | h | + | e ( h ) | (cid:17) du (cid:46) (cid:90) ( int ) M (1 + u n )( d ≤ h ) . We deducesup ≤ u ≤ u ∗ (cid:90) S ( r = r ,u ) (1 + u n ) f + (cid:90) u ∗ (cid:18)(cid:90) S ( r = r ,u ) (1 + u n ) f (cid:19) du (cid:46) (cid:90) S ( r = r , f + (cid:90) ( int ) M (1 + u n )( d ≤ h ) + (cid:15) (cid:90) u ∗ (cid:18)(cid:90) S ( r = r ,u ) (1 + u n − )( d f ) (cid:19) du which concludes the proof of Lemma 6.2.4. In this section, we infer from the Teukolsky-Starobinski identity, see Proposition 2.3.15,a parabolic equation for α . .2. PROOF OF THEOREM M3 Corollary 6.2.13. α satisfies in ( int ) M the following equation m (cid:101) να + r d (cid:63) / d (cid:63) / d/ d/ α = 1 r (cid:16) e ( r e ( r q )) + 2 ωr e ( r q ) (cid:17) − r − Err [ T S ] − (cid:26) r ρκ − am (cid:27) e α − (cid:26) − r (cid:18) ρ + 2 mr (cid:19) κ + 3 mr (cid:18) κ − r (cid:19) − mr (cid:27) e α. where the vectorfield (cid:101) ν is defined by (6.2.3) .Proof. Recall from (2.3.15) that we have e ( r e ( r q )) + 2 ωr e ( r q ) = r (cid:26) d (cid:63) / d (cid:63) / d/ d/ α + 32 ρ (cid:16) κe − κe (cid:17) α (cid:27) + Err[ T S ] . This yields32 r ρ (cid:16) κe − κe (cid:17) α + r d (cid:63) / d (cid:63) / d/ d/ α = 1 r (cid:16) e ( r e ( r q )) + 2 ωr e ( r q ) (cid:17) − r − Err[
T S ] . Now, we have in view of the definition of (cid:101) ν r ρ (cid:16) κe − κe (cid:17) − m (cid:101) ν = (cid:26) r ρκ − am (cid:27) e + (cid:26) − r (cid:18) ρ + 2 mr (cid:19) κ + 3 mr (cid:18) κ − r (cid:19) − mr (cid:27) e . We infer 6 m (cid:101) να + r d (cid:63) / d (cid:63) / d/ d/ α = 1 r (cid:16) e ( r e ( r q )) + 2 ωr e ( r q ) (cid:17) − r − Err[
T S ] − (cid:26) r ρκ − am (cid:27) e α − (cid:26) − r (cid:18) ρ + 2 mr (cid:19) κ + 3 mr (cid:18) κ − r (cid:19) − mr (cid:27) e α. This concludes the proof of the corollary.
Corollary 6.2.14. α satisfies in ( int ) M m (cid:101) να + r d (cid:63) / d (cid:63) / d/ d/ α = F where F satisfies max ≤ k ≤ k small +18 (cid:90) Σ ∗ u δ dec | d k F | (cid:46) (cid:15) . CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) Proof.
In view of Corollary 6.2.9, α satisfies6 m (cid:101) να + r d (cid:63) / d (cid:63) / d/ d/ α = F with F := e ( e ( q )) + F ,F := 1 r (cid:16) e ( r e ( r ) q ) + e ( r ) e ( q ) + 2 ωr e ( r q ) (cid:17) − r − Err[
T S ] − (cid:26) r ρκ − am (cid:27) e α − (cid:26) − r (cid:18) ρ + 2 mr (cid:19) κ + 3 mr (cid:18) κ − r (cid:19) − mr (cid:27) e α. Recall also that Err[
T S ] is given schematically by, see Proposition 2.3.15,Err[
T S ] = r (cid:0) d / Γ b + r Γ b · Γ b ) · α + r (cid:0) Γ b e ( r q ) + ( d ≤ Γ b ) r q (cid:17) + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) . We infer that F is given schematically by F = r (cid:0) d / Γ b + r Γ b · Γ b ) · α + r − (cid:0) Γ b e ( r q ) + ( d ≤ Γ b ) r q (cid:17) + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) + r − Γ b = r − Γ b + r d ≤ (cid:0) Γ b · Γ g (cid:1) + r d ≤ (cid:0) Γ g · β (cid:1) where we have used • The fact we are working here with the global frame of Proposition 3.5.5 which hasthe property that η ∈ Γ g . • The fact that Γ b behave better that r Γ g . • The fact that α and q behaves at least as good as Γ b . • The fact that ρ + mr behaves as good as r − Γ g . • The fact that e ( r ) + 1 belongs to r Γ b .Now, recall from Lemma 5.1.1 that the global frame of Proposition satisfies in particular max ≤ k ≤ k small +22 sup M (cid:110) r + δ dec − δ | d k β | + r u + δ dec − δ | d k Γ g | + ru δ dec − δ | d k Γ b | (cid:111) (cid:46) (cid:15). (6.2.4) Here we use (3.4.11) with k loss = 22. .2. PROOF OF THEOREM M3 F and the behavior (3.3.4) of r on Σ ∗ , and the factthat δ can be chosen to satisfy δ ≤ δ dec , we infermax ≤ k ≤ k small +18 sup Σ ∗ ru + δ dec | d k F | (cid:46) u + δ dec ∗ sup Σ ∗ ( r − ) (cid:15) + (cid:15) (cid:46) (cid:15) . In view of the definition of F , this yieldsmax ≤ k ≤ k small +18 (cid:90) Σ ∗ u δ dec | d k F | (cid:46) (cid:15) + max ≤ k ≤ k small +19 (cid:90) Σ ∗ u δ dec | d k e ( q ) | . Together with Theorem M1, and the fact that δ extra > δ dec , we infermax ≤ k ≤ k small +18 (cid:90) Σ ∗ u δ dec | d k F | (cid:46) (cid:15) . This concludes the proof of the corollary.We are now ready to prove Proposition 6.2.6. In view of Corollary 6.2.14, α satisfies6 m (cid:101) να + r d (cid:63) / d (cid:63) / d/ d/ α = F. Commuting with d k , we infer6 m (cid:101) ν ( d k α ) + r d (cid:63) / d (cid:63) / d/ d/ ( d k α ) = F k where F k is defined by F k := − m [ d k , (cid:101) ν ] α − k (cid:88) j =1 d j ( m ) d k − j (cid:101) να − [ d k , r d (cid:63) / ] r d (cid:63) / r d/ r d/ α − r d (cid:63) / [ d k , r d (cid:63) / ] r d/ r d/ α − r d (cid:63) / r d (cid:63) / [ d k , r d/ ] r d/ α − r d (cid:63) / r d (cid:63) / r d/ [ d k , r d/ ] α + d k F. Note that we have schematically[ d , d / ] = r Γ b d , [ (cid:101) ν, d / ] = (cid:16) O ( r − ) + r Γ b (cid:17) d , [ (cid:101) ν, re ] = O ( r − ) d , [ (cid:101) ν, e ] = O ( r − ) d . Recall from Lemma 5.1.1 that we have δ = k loss k large − k small . Since we have here k loss = 22, and since we have 2 k small ≤ k large + 1 and k large δ dec (cid:29)
1, we deduce δ (cid:28) δ dec and we have indeed 8 δ ≤ δ dec . CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) Together with the fact that α behaves at least as good as Γ b , we infer, schematically, F k = d k F + r − d ≤ k +4 Γ b + r d ≤ k +4 (Γ b ) . In view of (6.2.4) and the behavior (3.3.4) of r on Σ ∗ , we havemax ≤ k ≤ k small +18 sup Σ ∗ ru + δ dec | r − d ≤ k +4 Γ b + r d ≤ k +4 (Γ b ) | (cid:46) u + δ dec ∗ sup Σ ∗ ( r − ) (cid:15) + (cid:15) (cid:46) (cid:15) . This yields max ≤ k ≤ k small +18 (cid:90) Σ ∗ u δ dec | F k | (cid:46) (cid:15) + max ≤ k ≤ k small +18 sup Σ ∗ u δ dec | d k F | . Together with the estimate for F of Corollary 6.2.14, we infermax ≤ k ≤ k small +18 (cid:90) Σ ∗ u δ dec | F k | (cid:46) (cid:15) + (cid:15) (cid:46) (cid:15) . This concludes the proof of Proposition 6.2.6.
In this section we prove Lemma 6.2.8, i.e. we derive estimates for the control of theparabolic equation appearing in the statement of Proposition 6.2.6. The following identitywill be useful.
Lemma 6.2.15.
We have for any reduced scalar f (cid:101) ν (cid:18)(cid:90) S f (cid:19) = 2 (cid:90) S f (cid:101) ν ( f ) + (cid:90) S ( − af e ( f ) + κf ) + a (cid:90) S (2 f e ( f ) + κf ) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) . Proof.
Recall from the definition of (cid:101) ν that (cid:101) ν = e + ae . .2. PROOF OF THEOREM M3 (cid:101) ν (cid:18)(cid:90) S f (cid:19) = e (cid:18)(cid:90) S f (cid:19) + ae (cid:18)(cid:90) S f (cid:19) = (cid:90) S (2 f e ( f ) + κf ) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) + a (cid:90) S (2 f e ( f ) + κf )= 2 (cid:90) S f (cid:101) ν ( f ) + (cid:90) S ( − af e ( f ) + κf ) + a (cid:90) S (2 f e ( f ) + κf ) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) . This concludes the proof of the lemma.We are now ready to prove Lemma 6.2.4. Recall from Lemma 6.2.15 that we have (cid:101) ν (cid:18)(cid:90) S f (cid:19) = 2 (cid:90) S f (cid:101) ν ( f ) + (cid:90) S ( − af e ( f ) + κf ) + a (cid:90) S (2 f e ( f ) + κf ) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) . In view of the equation satisfied by f , we infer (cid:101) ν (cid:18)(cid:90) S f (cid:19) = − m (cid:18)(cid:90) S r f d (cid:63) / d (cid:63) / d/ d/ f (cid:19) + 13 m (cid:18)(cid:90) S hf (cid:19) + (cid:90) S ( − af e ( f ) + κf ) + a (cid:90) S (2 f e ( f ) + κf ) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) . Now, from the definition of (cid:101) ν , we have (cid:101) ν ( u ) = 2 /ς . We deduce (cid:101) ν (cid:18) u n (cid:90) S f (cid:19) + u n m (cid:18)(cid:90) S r f d (cid:63) / d (cid:63) / d/ d/ f (cid:19) = u n m (cid:18)(cid:90) S hf (cid:19) + u n (cid:90) S ( − af e ( f ) + κf ) + au n (cid:90) S (2 f e ( f ) + κf )+ u n Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) + 2 ς nu n − (cid:90) S f . This yields in view of the bootstrap assumptions (cid:101) ν (cid:18) u n (cid:90) S f (cid:19) + u n m (cid:18)(cid:90) S r f d (cid:63) / d (cid:63) / d/ d/ f (cid:19) (cid:46) u n m (cid:107) h (cid:107) L ( S ) (cid:107) f (cid:107) L ( S ) + (cid:18) r + (cid:15)u − (cid:19) u n (cid:90) S | f || d ≤ f | + nu n − (cid:90) S f (cid:46) u n m (cid:107) h (cid:107) L ( S ) (cid:107) f (cid:107) L ( S ) + (cid:15)u n − (cid:90) S | f || d ≤ f | + nu n − (cid:90) S f CHAPTER 6. DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3) where we have used in the last inequality the behavior (3.3.4) of r on Σ ∗ . Next, we relyon the Poincar´e inequality of Lemma 6.2.11 to deduce (cid:101) ν (cid:18) u n (cid:90) S f (cid:19) + u n (cid:90) S f (cid:46) u n (cid:90) S h + (cid:15) u n − (cid:90) S ( d f ) + nu n − (cid:90) S f . Integrating in u between 1 and u ∗ , and recalling that (cid:101) ν ( u ) = 2 /ς , we infer (cid:90) Σ ∗ u n f (cid:46) (cid:90) Σ ∗ ∩C f + (cid:90) Σ ∗ u n h + (cid:15) (cid:90) Σ ∗ u n − ( d f ) + n (cid:90) Σ ∗ u n − f . In particular, we have for n = 0 (cid:90) Σ ∗ f (cid:46) (cid:90) Σ ∗ ∩C f + (cid:90) Σ ∗ h + (cid:15) (cid:90) Σ ∗ u − ( d f ) . Then, starting from the case n = 0 and arguing by iteration on the largest integer below n , one immediately deduces for any real n ≥ (cid:90) Σ ∗ (1 + u n ) f (cid:46) (cid:90) Σ ∗ ∩C f + (cid:90) Σ ∗ (1 + u n ) h + (cid:15) (cid:90) Σ ∗ (1 + u n − )( d f ) which concludes the proof of Lemma 6.2.8. hapter 7DECAY ESTIMATES (TheoremsM4, M5) In this chapter, we rely on the decay of q , α and α to prove the decay estimates for allthe other quantities. More precisely, we rely on the results of Theorem M1, M2 and M3to prove Theorem M4 and M5. In what follows we give a detailed proof of Theorem M4 , which, we recall, provides themain decay estimates in ( ext ) M . The proof makes use of the bootstrap assumptions BA-D , BA-E , the results of Theorems
M1, M2, M3 and Lemmas 3.4.1, 3.4.2. In thissection, we start with some preliminaries. Σ ∗ The proof of Theorem M4 depends in a fundamental way on the geometric properties ofthe GCM hypersuface Σ ∗ , the spacelike future boundary of ( ext ) M introduced in section3.1.2. For the convenience of the reader, we recall below its main features.1. The affine parameter s is initialized on Σ ∗ such that s = r .32728 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
2. There exists a constant c ∗ such thatΣ ∗ := { u + r = c ∗ } .
3. Let ν ∗ = e + a ∗ e be the unique vectorfield tangent to the hypersurface Σ ∗ , per-pendicular to the foliation S ( u ) induced on Σ ∗ and normalized by the condition g ( ν ∗ , e ) = −
2. The following normalization condition holds true at the South Pole SP of every sphere S , a ∗ (cid:12)(cid:12)(cid:12) SP = − − m S r S . (7.1.1)4. We have r ∗ ≥ (cid:15) − u on Σ ∗ . (7.1.2)5. The following GCM conditions hold on Σ ∗ κ = 2 r , d (cid:63) / d (cid:63) / κ = 0 , d (cid:63) / d (cid:63) / µ = 0 , (7.1.3) (cid:90) S ηe Φ = (cid:90) S ξe Φ = 0 . (7.1.4)Moreover on S ∗ = Σ ∗ ∩ C ∗ , (cid:90) S ∗ βe Φ = 0 , (cid:90) S ∗ e θ ( κ ) e Φ = 0 . (7.1.5)6. According to the definition of the Hawking mass, i.e. 1 − mr = − r κκ , and theGCM assumption for κ we also have, κ = − r (cid:18) − mr (cid:19) . (7.1.6)Thus on Σ ∗ , e ( r ) = r κ + A ) = − Υ + r A, e ( r ) = 1 . (7.1.7)7. In view of the definition of ν ∗ and and that of ς we we easily deduce the followingrelation between a ∗ and ς on Σ ∗ . a ∗ = − ς + Υ − r A. (7.1.8)8. Since on Σ ∗ we have r = s we deduce,Ω = e ( r ) = − Υ + r A on Σ ∗ . (7.1.9) Indeed, since ν ∗ is tangent to Σ ∗ along which u = − r + c ∗ , using also (7.1.7), ς = e ( u ) = ν ∗ ( u ) = − ν ∗ ( r ) = − e ( r ) − a ∗ e ( r ) = − a ∗ + Υ − r A. .1. PRELIMINARIES TO THE PROOF OF THEOREM M4 We reformulate below the main bootstrap assumption in the form needed in the proofof Theorem M4. Definition 7.1.1.
We make use of the following norms on S = S ( u, r ) ⊂ ( ext ) M , (cid:107) f (cid:107) ∞ ( u, r ) : = (cid:107) f (cid:107) L ∞ (cid:0) S ( u,r ) (cid:1) , (cid:107) f (cid:107) ( u, r ) := (cid:107) f (cid:107) L (cid:0) S ( u,r ) (cid:1) , (cid:107) f (cid:107) ∞ ,k ( u, r ) := k (cid:88) i =0 (cid:107) d i f (cid:107) ∞ ( u, r ) , (cid:107) f (cid:107) ,k ( u, r ) := k (cid:88) i =0 (cid:107) d i f (cid:107) ( u, r ) . (7.1.10)To simplify the exposition it also helps to introduce the following schematic notation forthe connection coefficients (recall ω, ξ = 0 and ζ = − η ),Γ g = (cid:110) ˇ κ, ϑ, η, ζ, ˇ κ (cid:111) ∪ (cid:110) κ − r , κ + 2Υ r (cid:111) , Γ b = (cid:110) ϑ, η, ˇ ω, ξ, A, r − ˇ ς, r − ˇΩ , (cid:111) ∪ (cid:110) ω − mr , r − ( ς − , r − (Ω + Υ) (cid:111) . (7.1.11) Remark 7.1.2.
It is important to note that η belongs to Γ b rather than Γ g as it mayhave been expected. Note also that A ∈ Γ b in view of Proposition 2.2.9 and the factthat (ˇ ς, ˇΩ) ∈ r Γ b . We also note that the averaged quantities (cid:110) κ − r , κ + r (cid:111) and (cid:110) ω − mr , r − ( ς − , r − (Ω + Υ) (cid:111) are actually better behaved in view of Lemmas 3.4.1, 3.4.2. Ref 1.
According to our bootstrap assumptions
BA-D , and the pointwise estimates ofProposition 3.4.5, which themselves follow from
BA-E , as well as the control of averagesin Lemma 3.4.1 and the control of the Hawking mass in Lemma 3.4.2, we have on ( ext ) M ,1. For 0 ≤ k ≤ k small , (cid:107) Γ g (cid:107) ∞ ,k (cid:46) (cid:15) min (cid:110) r − u − − δ dec , r − u − − δ dec (cid:111) , (cid:107) e Γ g (cid:107) ∞ ,k − (cid:46) (cid:15)r − u − − δ dec , (cid:107) Γ b (cid:107) ∞ ,k (cid:46) (cid:15)r − u − − δ dec . (7.1.12)2. For k ≤ k large − (cid:107) Γ g (cid:107) ∞ ,k (cid:46) (cid:15)r − , (cid:107) Γ b (cid:107) ∞ ,k (cid:46) (cid:15)r − . (7.1.13) Based on bootstrap assumptions
BA-D , BA-E , Theorems
M1, M2, M3 and Lemmas 3.4.1, 3.4.2. CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Ref 2.
The quantity q satisfies on ( ext ) M , for all 0 ≤ k ≤ k small + 20, (cid:107) q (cid:107) ∞ ,k (cid:46) (cid:15) min (cid:110) u − − δ extra , r − u − − δ extra (cid:111) , (cid:107) e q (cid:107) ∞ ,k − (cid:46) (cid:15) r − u − − δ extra . (7.1.14)In addition, on the last slice Σ ∗ , for all k ≤ k small + 20, (cid:90) Σ ∗ ( τ,τ ∗ ) | e d k q | + | e d k q | + r − | q | (cid:46) (cid:15) (1 + τ ) − − δ dec . (7.1.15)According to Theorem M2 we have on ( ext ) M , for all 0 ≤ k ≤ k small + 20, (cid:107) α (cid:107) ∞ ,k (cid:46) (cid:15) min (cid:110) r − ( u + 2 r ) − − δ extra , log(1 + u ) r − ( u + 2 r ) − − δ extra (cid:111) , (cid:107) e α (cid:107) ∞ ,k − (cid:46) (cid:15) min (cid:110) r − ( u + 2 r ) − − δ extra , log(1 + u ) r − ( u + 2 r ) − − δ extra (cid:111) . (7.1.16)According to Theorem M3 , the component α verifies the following estimate holds on T ,for 0 ≤ k ≤ k small + 16, sup T u δ dec | d k α | (cid:46) (cid:15) , (7.1.17)and on the last slice Σ ∗ for all k ≤ k small + 16 (cid:90) Σ ∗ ( τ,τ ∗ ) | d k α | (cid:46) (cid:15) (1 + τ ) − − δ dec . (7.1.18) Ref 3.
In view of the bootstrap assumptions
BA-D and the pointwise estimates ofProposition 3.4.5 for the curvature components, which themselves follow from
BA-E , wehave in ( ext ) M , Recall (see Remark 2.4.9) that the quantity q we are working with is defined relative to the globalframe of Proposition 3.5.5. In fact, the corresponding estimate in Theorem M3 holds on ( int ) M , and hence in particular on T since T ⊂ ( int ) M . .1. PRELIMINARIES TO THE PROOF OF THEOREM M4 ≤ k ≤ k small , (cid:107) β (cid:107) ∞ ,k (cid:46) (cid:15) min (cid:110) r − ( u + 2 r ) − − δ dec , r − ( u + 2 r ) − − δ dec (cid:111) , (cid:107) e β (cid:107) ∞ ,k − (cid:46) (cid:15) min (cid:110) r − ( u + 2 r ) − − δ dec , r − ( u + 2 r ) − − δ dec (cid:111) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ˇ ρ, ρ + 2 mr (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) min (cid:110) r − u − − δ dec , r − u − − δ dec (cid:111) , (cid:13)(cid:13)(cid:13)(cid:13) e (cid:18) ˇ ρ, ρ + 2 mr (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k − (cid:46) (cid:15)r − u − − δ dec , (cid:13)(cid:13)(cid:13)(cid:13) ˇ µ, µ − mr (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15)r − u − − δ dec , (cid:107) β (cid:107) ∞ ,k (cid:46) (cid:15)r − u − − δ dec . (7.1.19)Since K = − ρ − κκ + ϑϑ = r − ( ρ − ρ ) − ( κκ − κκ ) + l.o.t. we also deduce forall 0 ≤ k ≤ k small , (cid:13)(cid:13)(cid:13)(cid:13) K − r (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) min (cid:110) r − u − − δ dec , r − u − − δ dec (cid:111) . ii. For all k ≤ k large − r + δB (cid:16) (cid:107) α (cid:107) ∞ ,k + (cid:107) β (cid:107) ∞ ,k (cid:17) (cid:46) (cid:15),r (cid:107) ˇ ρ (cid:107) ∞ ,k + r (cid:107) β (cid:107) ∞ ,k + r (cid:107) α (cid:107) ∞ ,k (cid:46) (cid:15). (7.1.20) Remark 7.1.3.
In view of the control of averages Lemma 3.4.1 we have in fact betterestimates for the scalars, κ − r , κ + 2Υ r , ω − mr , ρ + 2 mr . In particular they can be estimated by (cid:15) replaced by (cid:15) in Ref 1 . Remark 7.1.4.
Note that r ( ˇ ρ, ρ + mr ) , r ( K − r ) behave as Γ g . For convenience we shalljust simply add them to Γ g . Similarly ( rβ, α ) behave as Γ b . Thus, our extended Γ g , Γ b are Γ g = (cid:110) ˇ κ, ϑ, η, ζ, ˇ κ, r ˇ ρ (cid:111) ∪ (cid:26) κ − r , κ + 2Υ r , r (cid:18) ρ + 2 mr (cid:19)(cid:27) , Γ b = (cid:110) ϑ, η, ˇ ω, ξ, A, r − ˇ ς, r − ˇΩ , rβ, α (cid:111) ∪ (cid:110) ω − mr , r − ( ς − , r − (Ω + Υ) (cid:111) . Note also that we can write e (Γ g ) = r − d Γ b . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Commutation identitiesLemma 7.1.5.
We have, schematically, [ d /, e ] ψ = Γ g d (cid:37) ψ + l.o.t. , [ d /, e ] ψ = r Γ b e ψ + Γ ≤ b d (cid:37) ψ + l.o.t. (7.1.21) Proof.
Follows from Lemma 2.2.13 and the symbolic notation introduced in (7.1.11), seealso Remark 7.1.4.
Product Estimates
We estimate quadratic error terms with the help of the following,
Lemma 7.1.6.
Let k small < k large and k loss > three positive integers verifying theconditions, k loss < k small , k loss ≤ k large − − k small . (7.1.22) The following product estimates hold true for all ≤ k ≤ k small + k loss , (cid:107) Γ g · Γ g (cid:107) ∞ ,k + r (cid:13)(cid:13)(cid:13)(cid:16) ˇ ρ, β, α (cid:17) · Γ g (cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:13)(cid:13)(cid:13) Γ g · (cid:16) Γ b , α (cid:17)(cid:13)(cid:13)(cid:13) ∞ ,k + r (cid:107) Γ g · β (cid:107) ∞ ,k + r (cid:107) ˇ ρ · Γ b (cid:107) ∞ ,k + r (cid:13)(cid:13)(cid:13)(cid:16) β, α (cid:17) · Γ b (cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:13)(cid:13)(cid:13)(cid:16) Γ b , α (cid:17) · Γ b (cid:13)(cid:13)(cid:13) ∞ ,k + r (cid:107) β · Γ b (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) e (Γ · β ) (cid:107) ∞ ,k (cid:46) r − u − − δ dec . (7.1.23) Proof.
All estimates are easy to prove in the range 0 ≤ k ≤ k small . We shall thus assumethat k small ≤ k ≤ k small + k loss . Since k loss < k small we have k/ < k small for all k inthat range. We start with the first estimate. Since r ˇ ρ satisfies the same estimates as Γ g ,and as rβ and rα satisfy even better estimates, it suffices to prove the first estimate for .1. PRELIMINARIES TO THE PROOF OF THEOREM M4 g · Γ g . For simplicity of notation we write L := k large − S := k small . By standardinterpolation inequalities, for all S ≤ k ≤ L , (cid:107) Γ g (cid:107) ∞ ,k (cid:46) (cid:107) Γ g (cid:107) k − SL − S ∞ ,L (cid:107) Γ g (cid:107) L − kL − S ∞ ,S (cid:46) (cid:15)r − (cid:104) u − − δ dec (cid:105) L − kL − S . Therefore, for any S ≤ k ≤ L (cid:107) Γ g · Γ g (cid:107) ∞ ,k (cid:46) (cid:107) Γ g (cid:107) ∞ (cid:107) Γ g (cid:107) ∞ ,k (cid:46) (cid:107) Γ g (cid:107) ∞ (cid:15)r − (cid:104) u − − δ dec (cid:105) L − kL − S (cid:46) (cid:15)r − (cid:107) Γ g (cid:107) ∞ (cid:107) Γ g (cid:107) ∞ u − ( + δ dec ) L − kL − S (cid:46) (cid:15) r − r − u − (1+ δ dec ) r − u − ( + δ dec ) u − ( + δ dec ) L − kL − S = (cid:15) r − u − − δ dec − ( + δ dec ) L − kL − S . Now, we have − − L − kL − S = − − (cid:18) − k − SL − S (cid:19) = − − (cid:18) − k loss k large − − k small (cid:19) ≤ − k loss k loss ≤ k large − − k small . We infer (cid:107) Γ g · Γ g (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec . Next, we consider the second estimate. Since r ˇ ρ satisfies the same estimates as Γ g , andas α and rβ satisfy the same estimate as Γ b , it suffices to prove the second estimate forΓ g · Γ b and ( α, β ) · Γ b . Starting with, (cid:107) Γ b (cid:107) ∞ ,k (cid:46) (cid:107) Γ b (cid:107) k − SL − S ∞ ,L (cid:107) Γ b (cid:107) L − kL − S ∞ ,S (cid:46) (cid:15)r − (cid:104) u − − δ dec (cid:105) L − kL − S , we deduce, (cid:107) Γ g · Γ b (cid:107) ∞ ,k (cid:46) (cid:107) Γ g (cid:107) ∞ (cid:107) Γ b (cid:107) ∞ ,k + (cid:107) Γ b (cid:107) ∞ (cid:107) Γ g (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec r − (cid:104) u − − δ dec (cid:105) L − kL − S + (cid:15) r − u − − δ dec r − (cid:104) u − − δ dec (cid:105) L − kL − S (cid:46) (cid:15) r − u − − δ dec . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Also, (cid:13)(cid:13)(cid:13)(cid:16) α, β (cid:17) · Γ b (cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:107) ( α, β ) (cid:107) ∞ (cid:107) Γ b (cid:107) ∞ ,k + (cid:107) Γ b (cid:107) ∞ (cid:107) ( α, β ) (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec r − (cid:104) u − − δ dec (cid:105) L − kL − S + (cid:15) r − u − − δ dec r − (cid:46) (cid:15) r − u − − δ dec . Finally, the estimates for Γ b · Γ b , α · Γ b , β · Γ b as well as e (Γ · β ) are easier and left tothe reader. Elliptic Estimates
We shall often make use of the results of Proposition 2.1.30 and Lemma 2.1.35 whichwe rewrite as follows with respect to the L based h k ( S ) spaces introduced in Definition2.1.36. Lemma 7.1.7.
Under the assumptions
Ref1 − Ref3 the following elliptic estimates holdtrue for the Hodge operators d/ , d/ , d (cid:63) / , d (cid:63) / , for all k ≤ k small + 20 .1. If f ∈ s ( S ) , (cid:107) d /f (cid:107) h k ( S ) + (cid:107) f (cid:107) h k ( S ) (cid:46) r (cid:107) d/ f (cid:107) h k ( S ) .
2. If f ∈ s ( S ) , (cid:107) d /f (cid:107) h k ( S ) + (cid:107) f (cid:107) h k ( S ) (cid:46) r (cid:107) d/ f (cid:107) h k ( S ) .
3. If f ∈ s ( S ) , (cid:107) d /f (cid:107) h k ( S ) (cid:46) r (cid:107) d (cid:63) / f (cid:107) h k ( S ) .
4. If f ∈ s ( S ) , (cid:107) f (cid:107) h k +1 ( S ) (cid:46) r (cid:107) d (cid:63) / f (cid:107) h k ( S ) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e Φ f (cid:12)(cid:12)(cid:12)(cid:12) .
5. If f ∈ s ( S ) , (cid:13)(cid:13)(cid:13)(cid:13) f − (cid:82) S f e Φ (cid:82) S e e Φ (cid:13)(cid:13)(cid:13)(cid:13) h k +1 ( S ) (cid:46) r (cid:107) d (cid:63) / f (cid:107) h k ( S ) . Note also that in view of Remark 7.1.4 we can write β ∈ r − Γ b , α ∈ Γ b . .1. PRELIMINARIES TO THE PROOF OF THEOREM M4 The proof of Theorem M4 relies heavily on the null structure and null Bianchi identitiesderived in section 2.2.4, see Propositions 2.2.8. We also rely on Proposition 2.2.18 forequations verified by the check quantities. We rewrite them below in a schematic form. Proposition 7.1.8 (Transport equations for checked quantities) . We have the followingtransport equations in the e direction, e ˇ κ + κ ˇ κ = Γ g · Γ g ,e ˇ κ + 12 κ ˇ κ + 12 ˇ κκ = − d/ ζ + 2 ˇ ρ + Γ g · Γ b ,e ˇ ω = ˇ ρ + Γ g · Γ b ,e ˇ ρ + 32 κ ˇ ρ + 32 ρ ˇ κ = d/ β + Γ b · α + Γ g · β + ˇ κ · ˇ ρ,e ˇ µ + 32 κ ˇ µ + 32 µ ˇ κ = r − Γ g · d / ≤ Γ g . (7.1.24) Also, we have in the e direction, e ˇ κ = r − d / ≤ Γ b + Γ b · d / ≤ Γ b ,e ˇ ρ = r − d / ≤ Γ b + r − Γ b · d / ≤ Γ b . (7.1.25) Proof.
The statements follow from the precise formulas of Proposition 2.2.18 and thesymbolic notation in (7.1.11). We also use the convention made in Remark 7.1.4 accordingto which we write r ˇ ρ, r ˇ µ ∈ Γ g , ( rβ, α ) ∈ Γ b and e (Γ g ) = r − ( d Γ b ). q Recall that our main quantity q has been introduced in Definition 2.3.12 with respect tothe global frame of Proposition 3.5.5 (see Remark 2.4.9). The passage from the geodesicframe ( e , e θ , e ) of ( ext ) M to the global frame ( e (cid:48) , e (cid:48) θ , e (cid:48) ) is given by e (cid:48) = Υ (cid:18) e + f e θ + 14 f e (cid:19) , e (cid:48) θ = e θ + 12 f e , e (cid:48) = Υ − e . (7.1.26)with a reduced scalar f which was constructed in Proposition 3.4.6. We recall below themain relevant statements of Proposition 3.4.6 in connection to the construction of theglobal frame.36 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Proposition 7.1.9.
Under assumptions
Ref 1-2 on ( ext ) M there exists a frame trans-formation of the form, (7.1.26) verifying the following properties :1. Everywhere in ( ext ) M we have ξ (cid:48) = 0 .2. The transition function f verifies, relative to the background frame ( e , e θ , e ) , theestimates | d k f | (cid:46) (cid:15)ru + δ dec − δ + u δ dec − δ , for k ≤ k small + 20 on ( ext ) M , | d k − e (cid:48) f | (cid:46) (cid:15)ru δ dec − δ for k ≤ k small + 20 on ( ext ) M . (7.1.27)
3. The primed Ricci coefficients and curvature components verify max ≤ k ≤ k small + k loss sup ( ext ) M (cid:40)(cid:16) r u + δ dec − δ + ru δ dec − δ (cid:17) | d k Γ (cid:48) g | + ru δ dec − δ | d k Γ (cid:48) b | + r u δ dec − δ (cid:12)(cid:12)(cid:12)(cid:12) d k − e (cid:48) (cid:18) κ (cid:48) − r , κ (cid:48) + 2 r , ϑ (cid:48) , ζ (cid:48) , η (cid:48) , η (cid:48) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) r ( u + 2 r ) + δ dec − δ + r ( u + 2 r ) δ dec − δ (cid:17)(cid:16) | d k α (cid:48) | + | d k β (cid:48) | (cid:17) + (cid:16) r (2 r + u ) δ dec + r (2 r + u ) + δ dec − δ (cid:17) | d k − e (cid:48) ( α (cid:48) ) | + (cid:16) r u δ dec + r u + δ dec − δ (cid:17) | d k − e (cid:48) ( β (cid:48) ) | + (cid:16) r u + δ dec − δ + r ru δ dec − δ (cid:17) | d k ˇ ρ (cid:48) | + u δ dec − δ (cid:16) r | d k β (cid:48) | + r | d k α (cid:48) | (cid:17)(cid:41) (cid:46) (cid:15). We have the following analog of Proposition 2.3.13.
Proposition 7.1.10.
We have, relative to the background frame of ( ext ) M , r (cid:18) d (cid:63) / d (cid:63) / ρ + 34 κρϑ + 34 κρϑ (cid:19) = q + Err (7.1.28) with error term expressed schematically in the formErr = r d / ≤ (Γ b · Γ g ) . (7.1.29) We denote by primes the Ricci and curvature components w.r.t. to the primed frame. In fact, the estimates hold for k small + k loss , see Proposition 3.4.6, and we choose here k loss = 20. Note that u and r here are the outgoing optical function and area radius of the foliation of ( ext ) M . .1. PRELIMINARIES TO THE PROOF OF THEOREM M4 Proof.
We make use of Proposition 2.3.13. Recall (see Remark 2.4.9) that the quantity q we are working with is defined relative to the global frame of Proposition 3.5.5. We thuswrite , q = r (cid:18) ( d (cid:63) / ) (cid:48) ( d (cid:63) / ) (cid:48) ρ (cid:48) + 34 κ (cid:48) ρ (cid:48) ϑ (cid:48) + 34 κ (cid:48) ρ (cid:48) ϑ (cid:19) + Err (cid:48) , Err (cid:48) = r e (cid:48) η (cid:48) · β (cid:48) + r d ≤ (cid:0) Γ b · Γ g ) , where the primes refer to the global frame in which q was defined. Since in that frame e (cid:48) η (cid:48) ∈ r − d Γ b and β (cid:48) ∈ r − Γ g we can simplify and write,Err (cid:48) = r d ≤ (cid:0) Γ b · Γ g ) . We also have in view of Proposition 2.3.4 ρ (cid:48) = ρ + f β + O ( f α ) ,β (cid:48) = β + 12 f α,α (cid:48) = α,κ (cid:48) = κ + f ξ,κ (cid:48) = κ + d/ (cid:48) ( f ) + f ( ζ + η ) + O ( r − f ) ,ϑ (cid:48) = ϑ − d (cid:63) / (cid:48) ( f ) + f ( ζ + η ) + O ( r − f ) ,ϑ (cid:48) = ϑ + f ξ. Note that ( d (cid:63) / ) (cid:48) ρ = − e (cid:48) θ ( ρ ) = − e θ ρ − f e ρ = d (cid:63) / ρ − f e ρ. We deduce, ( d (cid:63) / ) (cid:48) ( d (cid:63) / ) (cid:48) ρ (cid:48) = ( d (cid:63) / ) (cid:48) ( d (cid:63) / ) (cid:48) ρ + ( d (cid:63) / ) (cid:48) ( d (cid:63) / ) (cid:48) (Γ b · Γ g ) + l.o.t.= ( d (cid:63) / ) (cid:48) (cid:18) d (cid:63) / − f e (cid:19) ρ + r − d / ≤ (Γ b · Γ g )= d (cid:63) / (cid:18) d (cid:63) / − f e (cid:19) ρ + r − d / ≤ (Γ b · Γ g )= d (cid:63) / d (cid:63) / ρ − d (cid:63) / f e ρ + r − d / ≤ (Γ b · Γ g ) . The values of r and r (cid:48) differ only by lower order terms which do not affect the result. CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Similarly, κ (cid:48) ρ (cid:48) ϑ (cid:48) = ρκ (cid:0) ϑ − d (cid:63) / f (cid:1) + r − d / ≤ (Γ g · Γ g ) ,κ (cid:48) ρ (cid:48) ϑ (cid:48) = κρϑ + r − d / ≤ (Γ b · Γ g ) . We deduce,( d (cid:63) / ) (cid:48) ( d (cid:63) / ) (cid:48) ρ (cid:48) + 34 κ (cid:48) ρ (cid:48) ϑ (cid:48) + 34 κ (cid:48) ρ (cid:48) ϑ (cid:48) = d (cid:63) / d (cid:63) / ρ + 34 κρϑ + 34 κρϑ − d (cid:63) / f (cid:18) e ρ + 32 κρ (cid:19) + r − d / ≤ (Γ b · Γ g ) . Note that, d (cid:63) / f (cid:18) e ρ + 32 κρ (cid:19) = d (cid:63) / f (cid:18) d/ β − ϑα + l.o.t. (cid:19) = r − d / ≤ (Γ g · Γ b ) . Hence( d (cid:63) / ) (cid:48) ( d (cid:63) / ) (cid:48) ρ (cid:48) + 34 κ (cid:48) ρ (cid:48) ϑ (cid:48) + 34 κ (cid:48) ρ (cid:48) ϑ (cid:48) = d (cid:63) / d (cid:63) / ρ + 34 κρϑ + 34 κρϑ + r − d / ≤ (Γ b · Γ g ) . This concludes the proof of Proposition 7.1.10.We shall also need the following analogue of Proposition 2.3.14.
Proposition 7.1.11.
The following identity holds true in ( ext ) M , with respect to itsbackground frame e ( r q ) = r (cid:40) d (cid:63) / d (cid:63) / d/ β − ρ d (cid:63) / d (cid:63) / κ − κρ d (cid:63) / ζ − κρα + 34 (2 ρ − κκρ ) ϑ (cid:41) + Err [ e ( r q )] , (7.1.30) where Err [ e ( r q )] = r d ≤ (cid:0) Γ b · Γ g (cid:1) . (7.1.31) Proof.
We start with the result of Proposition 2.3.14 which we write in the form,( r (cid:48) ) − e (cid:48) ( r (cid:48) q ) = ( d (cid:63) / d (cid:63) / d/ ) (cid:48) β (cid:48) − κ (cid:48) ρ (cid:48) α (cid:48) − ρ (cid:48) ( d (cid:63) / d (cid:63) / ) (cid:48) κ (cid:48) − κ (cid:48) ρ ( (cid:48) d (cid:63) / ) (cid:48) ζ (cid:48) + 34 (2( ρ (cid:48) ) − κ (cid:48) κ (cid:48) ρ (cid:48) ) ϑ (cid:48) + ( r (cid:48) ) − Err[ e (cid:48) ( r (cid:48) q )]Err (cid:48) [ e (cid:48) ( r (cid:48) q )] = r (cid:48) Γ b q + r d (cid:48)≤ (cid:0) e (cid:48) η (cid:48) · β (cid:48) (cid:1) + r (cid:48) d ≤ (cid:0) Γ b · Γ g (cid:1) . .1. PRELIMINARIES TO THE PROOF OF THEOREM M4 e (cid:48) η (cid:48) ∈ r − Γ b and q ∈ Γ b , we deduce,Err (cid:48) [ e (cid:48) ( r (cid:48) q )] = r d ≤ (cid:0) Γ b · Γ g (cid:1) . Now, in view of Proposition 2.3.4,( d (cid:63) / d (cid:63) / d/ ) (cid:48) β (cid:48) = ( d (cid:63) / d (cid:63) / d/ ) (cid:48) (cid:18) β + 12 f α (cid:19) = ( d (cid:63) / d (cid:63) / d/ ) (cid:48) β + r − d / (Γ b · Γ g ) . Proceeding in the same manner with all other terms we find,( d (cid:63) / d (cid:63) / d/ ) (cid:48) β (cid:48) − κ (cid:48) ρ (cid:48) α (cid:48) − ρ (cid:48) ( d (cid:63) / d (cid:63) / ) (cid:48) κ (cid:48) − κ (cid:48) ρ ( (cid:48) d (cid:63) / ) (cid:48) ζ (cid:48) + 34 (2( ρ (cid:48) ) − κ (cid:48) κ (cid:48) ρ (cid:48) ) ϑ (cid:48) = d (cid:63) / d (cid:63) / d/ β − κρα − ρ d (cid:63) / d (cid:63) / κ − κρ d (cid:63) / ζ + 34 (2 ρ − κκρ ) ϑ + r − d / ≤ (Γ b · Γ g )from which the result easily follows. The following proposition is an immediate corollary of Proposition 2.2.19.
Proposition 7.1.12.
We have, schematically, d (cid:63) / ω = (cid:18) κ + 2 ω (cid:19) η + e ( ζ ) − β − κξ + r − Γ g + Γ b , d/ d (cid:63) / η = κ (cid:0) − e ( ζ ) + β (cid:1) − e ( e θ ( κ )) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) , d/ d (cid:63) / ξ = κ (cid:0) e ( ζ ) − β (cid:1) − e ( e θ ( κ )) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) . Remark 7.1.13.
Note that in fact Γ g = { ˇ κ, ϑ, ζ, ˇ κ, r ˇ ρ } and Γ b = { ϑ, η, ξ, ˇ ω, rβ, α } inthe derivation of this proposition. It is important to note also that the terms denotedschematically by d / (Γ b · Γ b ) do not contain derivatives of ˇ ω . The following corollary of Proposition 7.1.12 which will be very useful later on.
Proposition 7.1.14.
The following identities hold true on Σ ∗ . d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:16) e ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) − d (cid:63) / d (cid:63) / d/ e ( e θ ( κ ))+ r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t. (7.1.32)2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / ξ = e (cid:16) ( d (cid:63) / d/ + 2 K ) d (cid:63) / d (cid:63) / κ ) (cid:17) − κ (cid:16) e ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t. (7.1.33)40 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Remark 7.1.15.
Here, as in the remark following Proposition 7.1.12, Γ g = { ˇ κ, ϑ, ζ, ˇ κ, r ˇ ρ } and Γ b = { ϑ, η, ξ, ˇ ω, rβ, α } . The quadratic terms denoted l.o.t. are lower order both interms of decay in r, u as well in terms of number of derivatives. They also contain onlyangular derivatives d / and not e nor e .Proof. We make use of Proposition 7.1.12 . We shall also make use of the conventionsmentioned in Remark 7.1.4, i.e. ˇ ρ, ˇ µ ∈ r − Γ g , β ∈ r − Γ b , α ∈ Γ b .We start with,2 d/ d (cid:63) / η = κ (cid:0) − e ( ζ ) + β (cid:1) − e ( e θ ( κ )) + r − d / ≤ Γ g + r − d / (Γ b · Γ b )We apply d (cid:63) / d/ to derive,2 d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:0) − d (cid:63) / d/ e ( ζ ) + d (cid:63) / d/ β (cid:1) − d (cid:63) / d/ e ( e θ ( κ )) + r − d / ≤ Γ g + r − d / (Γ b · Γ b )= κ (cid:0) − e ( d (cid:63) / d/ ( ζ ) + d (cid:63) / d/ β (cid:1) − d (cid:63) / d/ e ( e θ ( κ )) − κ [ d (cid:63) / d/ , e ] ζ + r − d / ≤ Γ g + r − d / (Γ b · Γ b )Making use of the commutation formula, see Lemma 7.1.5, and the null structure equa-tions for e ζ, e ζ ,[ d/ , e ] ζ = − ηe ζ + r − d /ζ + Γ b e ζ + l.o.t. = r − Γ b · Γ b + r − d / Γ g + l.o.t.we deduce, schematically,[ d (cid:63) / d/ , e ] ζ = d (cid:63) / [ d/ , e ] ζ + [ d (cid:63) / , e ] d/ ζ = r − d / (cid:0) r − Γ b · Γ b + r − d /ζ + l.o.t. (cid:1) + Γ b e d/ ζ + r − d/ ζ + l.o.t.= r − d / (Γ b · Γ b ) + r − d / ζ + Γ b (cid:0) d/ e ζ + Γ b e ζ + r − d /ζ (cid:1) + l.o.t.= r − d / (Γ b · Γ b ) + r − Γ b d / ( d Γ b ) + r − Γ b · Γ b · Γ b + r − d / Γ g = r − d / (Γ b d ≤ Γ b ) + r − d / Γ g + l.o.t.Hence, 2 d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:16) − e ( d (cid:63) / d/ ζ ) + d (cid:63) / d/ β (cid:17) − d (cid:63) / d/ e ( e θ ( κ ))+ r − d / ≤ Γ g + r − d / (Γ b · d / Γ b ) (7.1.34)Since µ = − d/ ζ − ρ + ϑϑ , we deduce, d (cid:63) / µ = − d (cid:63) / d/ ζ − d (cid:63) / ρ + 14 d (cid:63) / ( ϑϑ ) ,e d (cid:63) / µ = − e ( d (cid:63) / d/ ζ ) − e d (cid:63) / ρ + 14 e d (cid:63) / ( ϑϑ )= − e ( d (cid:63) / d/ ζ ) − d (cid:63) / e ρ − [ d (cid:63) / , e ] ρ + 14 d (cid:63) / e ( ϑϑ ) + 14 [ e , d (cid:63) / ]( ϑ · ϑ ) . .1. PRELIMINARIES TO THE PROOF OF THEOREM M4 e ρ = d/ β − κρ + Γ g · Γ b and also the equations for e ρ, e ϑ, e ϑ, e ϑ, e ϑ (and writing d/ β = r − d /β = r − d / Γ b )[ e , d (cid:63) / ] ρ = Γ b e ρ + Γ b e ζ + r − d /ρ = r − Γ b d / Γ b + r − d / Γ g + l.o.t. , [ e , d (cid:63) / ]( ϑ · ϑ ) = Γ b e ( ϑ · ϑ ) + Γ b e ( ϑ · ϑ ) + r − d / ( ϑ · ϑ ) = r − d / (cid:0) Γ b · Γ g (cid:1) + l.o.t.We deduce, ignoring the lower order terms, e d (cid:63) / µ = − e ( d (cid:63) / d/ ζ ) − d (cid:63) / (cid:0) d/ β − κρ + Γ g · Γ b (cid:1) + r − Γ b d / Γ b (cid:1) + r − d / (cid:0) Γ b Γ g (cid:1) + r − d / Γ g = − e ( d (cid:63) / d/ ζ ) − d (cid:63) / d/ β + 32 κ d (cid:63) / ρ + r − d / Γ g + r − d / ≤ (Γ b · Γ b ) . Hence, e ( d (cid:63) / d/ ζ ) = − e ( d (cid:63) / µ ) − d (cid:63) / d/ β + r − d / Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t. (7.1.35)and thus, back to (7.1.34),2 d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:16) e ( d (cid:63) / µ ) + 2 d (cid:63) / d/ β (cid:17) − d (cid:63) / d/ e (cid:0) e θ ( κ ) (cid:1) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t. (7.1.36)Applying d (cid:63) / and commuting once more with e , i.e.,2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:16) e ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) − d (cid:63) / d (cid:63) / d/ e (cid:0) e θ ( κ ) (cid:1) + κ [ d (cid:63) / , e ] d (cid:63) / µ + r − d / Γ g · (cid:16) e ( d (cid:63) / µ ) + 2 d (cid:63) / d/ β (cid:17) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) . (7.1.37)Note that, in view of (7.1.36) we can write, e ( d (cid:63) / µ ) = 2 κ − d (cid:63) / d (cid:63) / d/ e (cid:0) e θ ( κ ) (cid:1) − d (cid:63) / d/ β + 2 κ − d (cid:63) / d/ d/ d (cid:63) / η = r − d / ≤ Γ b + l.o.t. (7.1.38)Hence, r − d / Γ g · (cid:16) e ( d (cid:63) / µ ) + 2 d (cid:63) / d/ β (cid:17) = r − d / Γ g · d / ≤ Γ b . Similarly,[ d (cid:63) / , e ] d (cid:63) / µ = Γ b · e d (cid:63) / µ + Γ b e d (cid:63) / µ + r − d / µ + l.o.t.= r − Γ b · d / ≤ Γ b + Γ b (cid:0) d (cid:63) / e µ + [ e , d (cid:63) / ] µ (cid:1) + r − d / Γ g + l.o.t. This is to avoid the presence of e , e derivatives in the error terms. CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Thus, making use of the equation for e µ and combining with the estimate above, κ [ d (cid:63) / , e ] d (cid:63) / µ + r − d / Γ g · (cid:16) e ( d (cid:63) / µ ) + 2 d (cid:63) / d/ β (cid:17) = r − Γ b · d / ≤ Γ b + r − d / ≤ Γ g . Back to (7.1.37) we deduce,2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:16) e ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) − d (cid:63) / d (cid:63) / d/ e (cid:0) e θ ( κ ) (cid:1) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b )as desired.To prove the second part we start with the formula for d/ d (cid:63) / ξ in Corollary 7.1.122 d/ d (cid:63) / ξ = κ (cid:0) e ( ζ ) − β (cid:1) − e ( e θ ( κ )) + r − d / ≤ Γ g + r − d / (Γ b · Γ b ) . Applying d (cid:63) / d/ and proceeding exactly as before in the derivation of (7.1.34) we derive,2 d (cid:63) / d/ d/ d (cid:63) / ξ = − e ( d (cid:63) / d/ e θ ( κ )) + κ (cid:0) e ( d (cid:63) / d/ ζ ) − d (cid:63) / d/ β (cid:1) + r − d / ≤ Γ g + r − d / (Γ b · d Γ b ) . (7.1.39)Making use of (7.1.35) we deduce, as in (7.1.36),2 d (cid:63) / d/ d/ d (cid:63) / ξ = − e ( d (cid:63) / d/ e θ ( κ )) + κ (cid:16) − e ( d (cid:63) / µ ) − d (cid:63) / d/ β (cid:17) + r − d / Γ g + r − d / ≤ (Γ b · d Γ b ) + l.o.t. (7.1.40)Applying d (cid:63) / and proceeding as in the derivation of (7.1.37), by making use of (7.1.39)and (7.1.38) we obtain2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / ξ = − e ( d (cid:63) / d (cid:63) / d/ e θ ( κ )) − κ (cid:16) e ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t.The identity d (cid:63) / d/ = d/ d (cid:63) / + 2 K yields, together with the bootstrap assumptions,2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / ξ = − e (( d (cid:63) / d/ + 2 K ) d (cid:63) / e θ ( κ )) − κ (cid:16) e ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t.= e (( d (cid:63) / d/ + 2 K ) d (cid:63) / d (cid:63) / ( κ )) − κ (cid:16) e ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t.as desired. .2. STRUCTURE OF THE PROOF OF THEOREM M4 We rephrase the statement of Theorem M4 as follows.
Theorem 7.2.1.
Let M = ( int ) M ∪ ( ext ) M be a GCM admissible spacetime . Under thebasic bootstrap assumptions and the results of Theorems M1-M4 (all encoded in Ref1 – Ref4 ) the following estimates hold true, for all k ≤ k small + 8 , everywhere on ( ext ) M , (cid:107) Γ g (cid:107) ∞ ,k (cid:46) (cid:15) min (cid:110) r − u − − δ dec , r − u − − δ dec (cid:111) , (cid:107) e Γ g (cid:107) ∞ ,k − (cid:46) (cid:15) r − u − − δ dec , (cid:107) Γ b (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (7.2.1) and, (cid:107) β (cid:107) ∞ ,k (cid:46) (cid:15) min (cid:110) r − ( u + 2 r ) − − δ dec , r − ( u + 2 r ) − − δ dec (cid:111) , (cid:107) e β (cid:107) ∞ ,k − (cid:46) (cid:15) r − ( u + 2 r ) − − δ dec , (cid:107) ˇ ρ (cid:107) ∞ ,k (cid:46) (cid:15) min (cid:110) r − u − − δ dec , r − u − − δ dec (cid:111) , (cid:107) e ˇ ρ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) ˇ µ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) β (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec . (7.2.2) Moreover, everywhere in ( ext ) M , (cid:107) α (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec . (7.2.3)Here is a short sketch of the proof of the theorem.1. Estimates on Σ ∗ . To start with we only have good estimates for q , α and α ,according to Ref2 . To proceed we make use in an essential way of all the GCMconditions (7.1.3)–(7.1.5) on the spacelike boundary Σ ∗ to estimate all the Ricci andcurvature coefficients along Σ ∗ . The main result is stated in Proposition 7.3.9. Theproof is divided in the following intermediary steps. In particular the conditions (7.1.1)–(7.1.5) hold on the spacelike boundary Σ ∗ . See Remark 7.1.4 for the definition of Γ g , Γ b used here. i.e estimates in terms of (cid:15) . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) (a) In Proposition 7.3.5 we derive flux type estimates along Σ ∗ for the quantities β, θ, η, ξ . These estimates take advantage in an essential way of the improvedflux estimates for q in Ref 2 , equation (7.1.15). This step also makes use ofProposition 7.1.11 and the identities of Proposition 7.3.4 for η, ξ .(b) We next estimate the (cid:96) = 1 modes of the Ricci and curvature coefficients inProposition 7.3.8. Besides the information provided by the estimates for q , α, α and the GCM conditions, an important ingredient in the proof is the vanishingof the (cid:96) = 1 mode of e θ ( K ), i.e. (cid:82) e θ ( K ) e Φ = 0. The flux estimates derived inProposition 7.3.5 play an essential role in deriving the desired estimate for the (cid:96) = 1 mode of β .(c) We make use of the previous steps to complete the proof all the desired esti-mates on Σ ∗ in Proposition 7.3.9. This step also uses, in addition to the GCMconditions, Proposition 7.1.10 relating q to d (cid:63) / d (cid:63) / ρ , the Codazzi equations andelliptic estimates on 2 surfaces.2. First Estimates in ( ext ) M . We make use of the propagation equations in e andthe estimates on Σ ∗ to derive some of the desired estimates of Theorem 7.2.1, moreprecisely the better estimates in powers of r for the Γ g quantities. Note that theseestimates decay only like u − / − δ dec in powers of u .(a) We first prove the desired estimates for ˇ κ, ˇ µ by simply integrating the cor-responding e equations. Note that these estimates are also well behaved interms of powers of u . This is done in subsection 7.4.3.(b) We derive spacetime estimates for all the (cid:96) = 1 modes in Lemma 7.4.7. Thisis done by propagating them from the last slice in the e direction, combinedwith Codazzi equations and the vanishing of the (cid:96) = 1 mode of e θ ( K ).(c) We provide all the optimal estimates in terms of powers of r for the quan-tities ϑ, ζ, η, ˇ κ, β, ˇ ρ . This is achieved in Proposition 7.4.5 with the help of theestimates on the last slice, the propagation equation for these quantities andthe estimates for the (cid:96) = 1 modes derived in the previous step.3. Optimal u -decay estimates in ( ext ) M . We derive all the remaining estimates ofTheorem 7.2.1 for all but the quantities ξ, ˇ ω, ˇΩ , ˇ ς . The main remaining difficultyis to get the top decay in powers of u for ϑ, ζ, η, ˇ κ, β, ˇ ρ, β . The result is stated inProposition 7.5.1. We proceed as follows.(a) One would like to start with ϑ by using the equation e ϑ + κϑ = − α . Thisunfortunately cannot work by integration starting from the last slice Σ ∗ . These estimates also provide weak decay in u , i.e. u − − δ dec decay. It would work however if instead we would integrate from the interior, but we don’t possess informa-tion about optimal u decay in the interior, for example on the timelike boundary T of ( ext ) M . .2. STRUCTURE OF THE PROOF OF THEOREM M4 ζ , β , ˇ ρ . On the other hand the quantities ˇ κ and ϑ could in principle be propagated using their corresponding e equations fromΣ ∗ , but unfortunately they are strongly coupled with the other quantities forwhich we don’t yet have information. For example we have, e ˇ κ + 12 κ ˇ κ + 12 ˇ κκ = − d/ ζ + 2 ˇ ρ + ˇΓ g · ˇΓ b , and therefore we cannot derive the estimate for ˇ κ , by integration, before es-timating d/ ζ and ˇ ρ . To circumvent this difficulty we proceed by an indirectmethod as follows.(b) We can derive optimal decay information on various mixed quantity. For ex-ample making use of the equation e α + (cid:18) κ − ω (cid:19) α = − d (cid:63) / β − ϑρ + 5 ζβ, we infer the desired decay in u for the quantity d (cid:63) / β − ϑρ . Other such informa-tions can be derived from the Codazzi equations for ϑ, ϑ , the Bianchi identityfor β and the identity (7.1.28) of Lemma 7.1.10.(c) We combine the control we have for α, ˇ κ, ˇ µ with the control for the mixedquantities mentioned above with a propagation equation for an intermediaryquantity, Ξ := r (cid:0) e θ ( κ ) + 4 r d (cid:63) / d/ ζ − r d (cid:63) / d/ β (cid:1) . We show in the crucial Lemma 7.5.2 that Ξ is a also a good mixed quantity,i.e. it has optimal decay in u . It is important to note that this estimate doesnot depend linearly on α for which we only have information on the last sliceand T .(d) We can combine the control of Ξ with all other available information mentionedabove, to derive good estimates, simultaneously, for d (cid:63) / d (cid:63) / κ, d (cid:63) / ζ and d (cid:63) / β . Thisis achieved in a sequence of crucial Lemma in subsection 7.5.2. Unfortunatelythis step is heavily dependent on the estimate of Ref 2 for α and therefore theestimates we derive are only useful on T .(e) We also show that we have good estimates for d (cid:63) / (cid:16) ζ, d (cid:63) / ˇ κ, β, β, d (cid:63) / ˇ ρ (cid:17) . To es-timate ˇ κ, ζ, β, β, ˇ ρ from d (cid:63) / (cid:16) ζ, d (cid:63) / ˇ κ, β, β, d (cid:63) / ˇ ρ (cid:17) we rely on the elliptic HodgeLemma 7.1.7 and the control we have for the (cid:96) = 1 modes from Lemma 7.4.7derived earlier. We obtain estimates for η, ϑ, ϑ as well. This establishes all theestimates of Proposition 7.5.1 on T .46 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) (f) The estimates mentioned above on T can now be propagated by integratingforward the e null structure and null Bianchi equations. This ends the proofof Proposition 7.5.1 in ( ext ) M .4. In Proposition 7.6.1 we derive improved decay estimates for e ( β, ϑ, ζ, ˇ κ, ˇ ρ ) andestimates for ξ, ˇ ω, ˇΩ , ˇ ς in terms of u − − δ dec decay. The estimates for ˇ ω and ξ arepropagated from the last slice using their e propagation equations. The estimatefor ˇ ω can be easily derived by integrating e (ˇ ω ) = ˇ ρ + Γ g · Γ b form the last sliceΣ ∗ . The estimate for ξ follows by integrating e ( ξ ) = − e ( ζ ) + β − κζ + Γ b · Γ b andmaking use of the previously derived estimates for e ζ, β, ζ . The estimates for ˇΩ , ˇ ς follow easily from the equations (2.2.19). Σ ∗ We shall make use of the following norms on Σ ∗ . (cid:107) ψ (cid:107) ∗∞ ,k ( u, r ) := (cid:88) j ≤ k (cid:107) d j ∗ ψ (cid:107) L ∞ ( S ( u,r )) , d j ∗ = (cid:88) j + j ≤ j d / j ( ν ∗ ) j . (7.3.1)Recall that ν ∗ = ν (cid:12)(cid:12)(cid:12) Σ ∗ = e + a ∗ e , is the tangent vector to Σ ∗ and (see (7.1.8) (7.1.9)),along Σ ∗ , a ∗ = − ς + Υ − r A = − ς − Ω . Based on on our assumptions
Ref 1-2 we deduce (cid:12)(cid:12)(cid:12) a ∗ + 1 + 2 mr (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)u − − δ dec . (7.3.2)As immediate consequence of the commutation Corollary 7.1.5 we derive the following, Lemma 7.3.1.
We have, schematically, [ d /, ν ∗ ] ψ = r Γ b ( ν ∗ ψ ) + d ≤ Γ b · d ψ. (7.3.3) .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ Proof.
Indeed, see Lemma 7.1.5,[ d /, e ] ψ = Γ g d (cid:37) ψ, [ d /, e ] ψ = r Γ b e ψ + Γ b d (cid:37) ψ + l.o.t. (7.3.4)Hence, since d /a ∗ ∈ r d / Γ b ,[ d /, ν ∗ ] ψ = [ d /, e + a ∗ e ] ψ = r Γ b e ψ + Γ b d (cid:37) ψ + a ∗ Γ g d (cid:37) ψ + d /a ∗ e ψ = r Γ b ( ν ∗ ψ − a ∗ e ψ ) + a ∗ Γ g d (cid:37) ψ + d /a ∗ e ψ = r Γ b ν ∗ ψ − a ∗ (Γ b d (cid:37) ψ + Γ g d (cid:37) ψ ) + d / Γ b · d ψ = r Γ b ν ∗ ψ + d ≤ Γ b · d ψ as desired.To estimate derivatives of the (cid:96) = 1 modes on Σ ∗ we make use of the following. Lemma 7.3.2.
For every scalar function h we have the formula ν ∗ (cid:18)(cid:90) S h (cid:19) = ( ς ) − (cid:90) S ς ( ν ∗ ( h ) + ( κ + a ∗ κ ) h ) . (7.3.5) In particular ν ∗ ( r ) = r ς ) − ς ( κ + a ∗ κ ) . (7.3.6) Proof.
We consider the coordinates u S , θ S along Σ with ν S ( θ S ) = 0. In these coordinateswe have, ν S = 2 ς S ∂ u S . The lemma follows easily by expressing the volume element of the surfaces S ⊂ Σ withrespect to the coordinates u S , θ S (see also the proof of Proposition 2.2.9). Lemma 7.3.3.
Given ψ ∈ s ( M ) we have the formula, ν ∗ (cid:18)(cid:90) S ψe Φ (cid:19) = (cid:82) S ( ν ∗ ψ ) e Φ + ( κ + a ∗ κ ) (cid:82) S ψe Φ + Err [ ψ, ν ∗ ] (7.3.7) with error termErr [ ψ, ν ∗ ] = − ς − ˇ ς (cid:90) S (cid:18) ν ∗ ψ − r ψ (cid:19) e Φ + ς − (cid:90) S ˇ ς (cid:18) ν ∗ ψ − r ψ (cid:19) e Φ + O ( (cid:15) u − − δ dec ) (cid:90) S | ψ | . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Proof.
We have ν ∗ (cid:18)(cid:90) S ψe Φ (cid:19) = ς − (cid:90) S ς (cid:0) ν ∗ ( ψe Φ ) + ( κ + a ∗ κ ) ψe Φ (cid:1) = ς − (cid:90) S ς (cid:0) ν ∗ ψe Φ + e − Φ ν ∗ ( e Φ ) + κ + a ∗ κ (cid:1) ψe Φ . Recalling that e (Φ) = ( κ − ϑ ), e (Φ) = ( κ − ϑ ) we deduce e − Φ ν ∗ ( e Φ ) + κ + a ∗ κ = 32 ( κ + a ∗ κ ) −
12 ( ϑ − a ∗ ϑ ) . Hence, writing also ς = ς + ˇ ς and then κ = κ + ˇ κ , κ = κ + ˇ κν ∗ (cid:18)(cid:90) S ψe Φ (cid:19) = ς − (cid:90) S ς (cid:18) ν ∗ ψ + 32 ( κ + a ∗ κ ) ψ (cid:19) e Φ − ς − (cid:90) S ς ( ϑ − a ∗ ϑ ) ψe Φ = ς − ς (cid:90) S (cid:18) ν ∗ ψ + 32 ( κ + a ∗ κ ) ψ (cid:19) e Φ + ς − (cid:90) S ˇ ς (cid:18) ν ∗ ψ + 32 ( κ + a ∗ κ ) (cid:19) e Φ − ς − (cid:90) S ς ( ϑ − a ∗ ϑ ) ψe Φ = (cid:90) S (cid:18) ν ∗ ψ + 32 ( κ + a ∗ κ ) ψ (cid:19) e Φ + ς − (cid:90) S ˇ ς (cid:18) ν ∗ ψ + 32 ( κ + a ∗ κ ) ψ (cid:19) e Φ + ( ς − ς − (cid:90) S (cid:18) ν ∗ ψ + 32 ( κ + a ∗ κ ) ψ (cid:19) e Φ − ς − (cid:90) S ς ( ϑ − a ∗ ϑ ) ψe Φ = (cid:90) S ( ν ∗ ψ ) e Φ + 32 ( κ + a ∗ κ ) (cid:90) S ψe Φ + Err[ ψ, ν ∗ ]where,Err[ ψ, ν ∗ ] = ς − (cid:90) S ˇ ς (cid:18) ν ∗ ψ + 32 ( κ + a ∗ κ ) ψ (cid:19) e Φ − ς − ˇ ς (cid:90) S (cid:18) ν ∗ ψ + 32 ( κ + a ∗ κ ) ψ (cid:19) e Φ + 32 (cid:90) S (ˇ κ + a ∗ ˇ κ ) ψe Φ − ς − (cid:90) S ς ( ϑ − a ∗ ϑ ) ψe Φ . (7.3.8)Using Ref1-Ref3 and (7.3.2) κ + a ∗ κ = − r + ( − − mr ) 2 r + O ( (cid:15)r − u − − δ dec ) = − r + O ( (cid:15)r − u − − δ dec ) , we deduce,Err[ ψ, ν ∗ ] = − ς − ˇ ς (cid:90) S (cid:18) ν ∗ ψ − r ψ (cid:19) e Φ + ς − (cid:90) S ˇ ς (cid:18) ν ∗ ψ − r ψ (cid:19) e Φ + O ( (cid:15)u − − δ dec ) (cid:90) S | ψ | as stated. .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ Σ ∗ Recall our our GCM conditions on Σ ∗ κ = 2 r , d (cid:63) / d (cid:63) / µ = 0 , d (cid:63) / d (cid:63) / κ = 0 , (cid:90) S ηe Φ = 0 , (cid:90) S ξe Φ = 0 . (7.3.9)Also, on S ∗ , the last cut of Σ ∗ , (cid:90) S ∗ βe Φ = 0 , (cid:90) S ∗ e θ ( κ ) e Φ = 0 . (7.3.10)The goal of the subsection is to derive identities involving the GCM conditions which willbe used later, see Lemma 7.3.7. Proposition 7.3.4.
The following identities hold true on Σ ∗ . d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:16) ν ∗ ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) − d (cid:63) / d (cid:63) / d/ ν ∗ ( e θ ( κ ))+ r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t. , (7.3.11)2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / ξ = ν ∗ (cid:16) ( d (cid:63) / d/ + 2 K ) d (cid:63) / d (cid:63) / κ ) (cid:17) − κ (cid:16) ν ∗ ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t. (7.3.12) The quadratic terms denoted l.o.t. are lower order both in terms of decay in r, u as wellin terms of number of derivatives.In particular, if the GCM conditions (7.3.9) are verified, we deduce, d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / η = κ d (cid:63) / d (cid:63) / d/ β + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t. ,d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / ξ = − κ d (cid:63) / d (cid:63) / d/ β + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t. (7.3.13) Proof.
The proof is a straightforward application of Proposition 7.1.14. Indeed accordingto (7.1.32) we have2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:16) e ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) − d (cid:63) / d (cid:63) / d/ e ( e θ ( κ ))+ r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t.On the other hand since ν ∗ = e + a ∗ e with a ∗ = ς ∗ − Υ + r A , see (7.1.8), e ( d (cid:63) / d (cid:63) / µ ) = ν ∗ ( d (cid:63) / d (cid:63) / µ ) − a ∗ e ( d (cid:63) / d (cid:63) / µ )= ν ∗ ( d (cid:63) / d (cid:63) / µ ) − a ∗ ( d (cid:63) / d (cid:63) / e µ + [ e , d (cid:63) / d (cid:63) / ]ˇ µ ) . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Also, in the same fashion , d (cid:63) / d (cid:63) / d/ e ( e θ ( κ )) = d (cid:63) / d (cid:63) / d/ [ ν ∗ ( e θ ( κ )) − a ∗ e e θ κ ]= d (cid:63) / d (cid:63) / d/ [( ν ∗ ( e θ ( κ ))] − a ∗ d (cid:63) / d (cid:63) / d/ ( e e θ κ ) + r − (cid:88) i + j =2 d / i a ∗ d / j ( e e θ κ )= d (cid:63) / d (cid:63) / d/ [ ν ∗ ( e θ ( κ )) − a ∗ e θ e κ − [ e , e θ ] κ ]= r − (cid:88) i + j =2 d / i Γ b d / j ( e θ ( e κ ) + [ e θ , e ] κ ) . Thus, after using the transport equations for e µ, e κ and the commutator lemma appliedto [ e , e θ ] we easily deduce,2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:16) ν ∗ ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) − d (cid:63) / d (cid:63) / d/ ν ∗ ( e θ ( κ ))+ r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t.which confirms the first identity of the proposition.The second part of the proposition can be derived in the same manner starting with theidentity (7.1.33)2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / ξ = e (cid:16) ( d (cid:63) / d/ + 2 K ) d (cid:63) / d (cid:63) / κ ) (cid:17) − κ (cid:16) e ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t.This concludes the proof of the proposition. Σ ∗ The goal of this subsection is to establish the following.
Proposition 7.3.5.
The following estimate holds true for all k ≤ k small + 15 (cid:90) Σ ∗ ( u,u ) r (cid:12)(cid:12) d k ∗ β (cid:12)(cid:12) + (cid:12)(cid:12) d k ∗ ( ϑ, η, ξ ) (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . (7.3.14) We also have the weaker estimates for k ≤ k small + 13 . (cid:107) β (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) ( ϑ, η, ξ ) (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − − δ dec . (7.3.15) Note that in view of (7.1.8) we have d /a ∗ ∈ r Γ b . .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ In the process we also prove the following estimates for the (cid:96) = 1 modes of ν ∗ - derivativesof ξ, η (see (7.3.21) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ η ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ ξ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r u − − δ dec , k ≤ k small + 15 . Proof.
We concentrate our attention on deriving (7.3.14). Following its proof below onecan easily also check the weaker estimates (7.3.15) which require in fact much less work.We divide the proof in the following steps.
Step 1.
We first prove the corresponding estimates for β away from its (cid:96) = 1 mode.More precisely we prove. Lemma 7.3.6.
The following estimates holds true for all k ≤ k small + 15 (cid:90) Σ ∗ ( u,u ) r (cid:12)(cid:12) d (cid:63) / ( d k ∗ β ) (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . (7.3.16)We postpone the proof of the lemma to Step 7 in this subsection. Step 2.
We make use of the result of Lemma 7.3.6 first prove the desired estimate for ϑ i.e., (cid:90) Σ ∗ ( u,u ) (cid:12)(cid:12) d k ∗ ϑ | (cid:46) (cid:15) u − − δ dec , k ≤ k small + 15 . (7.3.17) Proof.
One starts with the Codazzi equation d/ ϑ = − β − d (cid:63) / ( κ ) − ζκ + Γ g · Γ b . Differentiating w.r.t. d (cid:63) / and then taking tangential derivatives d k ∗ we derive, d k ∗ d (cid:63) / d/ ϑ = − d k ∗ d (cid:63) / β − d k ∗ d (cid:63) / d (cid:63) / ( κ ) − d k ∗ (cid:2) r − d / Γ g + r − d / (Γ g · Γ b ) (cid:3) . Making use of the GCM condition d (cid:63) / d (cid:63) / κ = 0 along Σ ∗ and our assumptions Ref1-Ref2 we deduce , for all k ≤ k small + 15, d k ∗ d (cid:63) / d/ ϑ = − d k ∗ d (cid:63) / β + r − d k +1 Γ g + r − d k +1 Γ g · Γ b = − d k ∗ d (cid:63) / β + (cid:15)r − + l.o.t. Note that we use (7.1.13) of
Ref1 to estimate the linear term r − d / Γ g . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) or, since r ≥ (cid:15)(cid:15) u δ dec , d k ∗ d (cid:63) / d/ ϑ = − d k ∗ d (cid:63) / β + (cid:15) r − u − − δ dec . Moreover, d (cid:63) / d/ d k ∗ ϑ = − d k ∗ d (cid:63) / β + (cid:15) r − u − / − δ dec + [ d k ∗ , d (cid:63) / d/ ] ϑ. Using the commutator estimates of Lemma 7.3.1 we derive, d (cid:63) / d/ d k ∗ ϑ = − d k ∗ d (cid:63) / β + (cid:15) r − u − − δ dec . Integrating and using the previously derived estimate for β we deduce, (cid:90) Σ ∗ ( u,u ∗ ) r (cid:12)(cid:12) d (cid:63) / d/ d k ∗ ϑ (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 15 . In view of the coercivity of d (cid:63) / d/ we infer that, (cid:90) Σ ∗ ( u,u ∗ ) (cid:12)(cid:12) d k ∗ ϑ (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 15as desired. Step 3.
We next derive a weak estimate for the (cid:96) = 1 mode of β with the help of theCodazzi equation for ϑ , 2 β = − d/ ϑ + e θ ( κ ) − κζ + Γ g · Γ b = − d/ ϑ + e θ (ˇ κ ) + 2Υ r ζ + Γ g · Γ b . Projecting on the (cid:96) = 1 mode and using the bootstrap assumptions
Ref1-Ref2 we deduce,2 (cid:90) S βe Φ = 2Υ r (cid:90) S ζe Φ + (cid:90) S e θ (ˇ κ ) e Φ + (cid:90) S Γ g · Γ b e Φ = O ( (cid:15) ) + O ( (cid:15) u − / − δ dec ) . The same estimate holds true for the tangential derivatives. More precisely taking tan-gential derivatives and projecting on the (cid:96) = 1 mode,2 (cid:90) S ( d k ∗ β ) e Φ = (cid:90) S d k ∗ (cid:104) e θ (ˇ κ ) + 2Υ r ζ + Γ g · Γ b (cid:105) e Φ − (cid:90) S (cid:2) d k ∗ , d/ (cid:3) ϑe Φ . i.e. of order O ( (cid:15) ). .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ Ref1-Ref2 and the commutator Lemma 7.3.1 we de-duce, (cid:90) S ( d k ∗ β ) e Φ = O ( (cid:15) ) . (7.3.18) Step 4.
We combine the result of Lemma 7.3.6 with 7.3.18 to deduce (cid:90) Σ ∗ ( u,u ) r (cid:12)(cid:12) d k ∗ β (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 15 . (7.3.19)Indeed, according the last elliptic estimate of Lemma 7.1.7, (cid:90) S r (cid:12)(cid:12) d k ∗ β (cid:12)(cid:12) (cid:46) r (cid:90) S (cid:12)(cid:12) d (cid:63) / ( d k ∗ β ) (cid:12)(cid:12) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( d k ∗ β ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r (cid:90) S (cid:12)(cid:12) d (cid:63) / ( d k ∗ β ) (cid:12)(cid:12) + (cid:15) r − . Since r ≥ ( (cid:15)(cid:15) − ) u − / − δ dec on Σ ∗ we deduce, (cid:90) S r (cid:12)(cid:12) d k ∗ β (cid:12)(cid:12) (cid:46) r (cid:90) S (cid:12)(cid:12) d (cid:63) / ( d k ∗ β ) (cid:12)(cid:12) + (cid:15) u − − δ dec . Thus, integrating and making use of estimate (7.3.16), (cid:90) Σ( u,u ∗ ) r (cid:12)(cid:12) d k ∗ β (cid:12)(cid:12) (cid:46) (cid:90) Σ( u,u ∗ ) r (cid:12)(cid:12) d (cid:63) / ( d k ∗ β ) (cid:12)(cid:12) + (cid:15) u − − δ dec (cid:46) (cid:15) u − − δ dec as stated. This establishes the estimate for β in Proposition 7.3.5. Step 5.
To finish the proof of Proposition 7.3.5 it remains to establish the estimates for η and ξ . As for β we prove separately estimates for d (cid:63) / ( η, ξ ) and estimates for the (cid:96) = 1modes of the tangential derivatives with respect to ν ∗ of η, ξ (recall that the (cid:96) = 1 modesof η, ξ were set to zero). We then combine them, as in the case of β , to provide the desiredestimates. Lemma 7.3.7.
We have (cid:90) Σ ∗ ( u,u ∗ ) r (cid:16) | d (cid:63) / ( d k ∗ η ) | + | d (cid:63) / ( d k ∗ ξ ) | (cid:17) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 15 . We postpone the proof of the lemma to Step 8 of this section.54
CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Step 6.
We provide below estimates for the (cid:96) = 1 modes of ξ and η and use them, incombination with Lemma 7.3.7, to close the estimates of Proposition 7.3.5, i.e. we showthat, (cid:90) Σ ∗ ( u,u ∗ ) (cid:16) | d k ∗ η | + | d k ∗ ξ | (cid:17) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 15 . (7.3.20)In the process we also prove the following estimates for the (cid:96) = 1 modes of ν ∗ - derivativesof ξ, η , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ η ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ ξ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r u − − δ dec , k ≤ k small + 15 . (7.3.21) Proof.
We prove separately the estimates for ξ and η . Step 6a.
We prove first the estimate for ξ . Since the (cid:96) = 1 mode of ξ vanishes we have,in view of the elliptic estimates Lemma 7.1.7 and the estimates of Lemma 7.3.7, (cid:90) Σ( u,u ∗ ) (cid:12)(cid:12) d /ξ (cid:12)(cid:12) + (cid:12)(cid:12) ξ (cid:12)(cid:12) (cid:46) (cid:90) Σ( u,u ∗ ) r (cid:12)(cid:12) d (cid:63) / ξ (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . (7.3.22)According to our GCM conditions we have (cid:82) S ξe Φ = 0. In view of Lemma 7.3.3 we deduce, (cid:90) S ( ν ∗ ξ ) e Φ = Err[ ξ, ν ∗ ]where,Err[ ξ, ν ∗ ] = − ς − ˇ ς (cid:90) S (cid:18) ν ∗ ξ − r ξ (cid:19) e Φ + ς − (cid:90) S ˇ ς (cid:18) ν ∗ ξ − r ξ (cid:19) e Φ + O ( (cid:15) u − − δ dec ) (cid:90) S | ξ | . We deduce, (cid:90) S ( ν ∗ ξ ) e Φ = ς − (cid:20) − ˇ ς (cid:90) S ( ν ∗ ξ ) e Φ + (cid:90) S ˇ ς ( ν ∗ ξ ) e Φ (cid:21) + O ( (cid:15) u − − δ dec ) (cid:90) S | ξ | . Hence, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν ∗ ξ ) e Φ − ς − (cid:90) S ˇ ς ( ν ∗ ξ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) O ( (cid:15) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν ∗ ξ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + O ( (cid:15) ) ru − − δ dec i.e., setting B := (cid:82) S ( ν ∗ ξ ) e Φ − and I = ς − (cid:82) S ˇ ς ( ν ∗ ξ ) e Φ , (cid:12)(cid:12) B − I (cid:12)(cid:12) (cid:46) (cid:15)B + (cid:15) ru − − δ dec . (7.3.23) .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ I we decompose ν ∗ ξ into its (cid:96) = 1 mode B = (cid:82) S ( ν ∗ ξ ) e Φ and its orthogonalcomplement ν ∗ ξ = B (cid:82) S e e Φ + ( ν ∗ ξ ) ⊥ . Thus, in view of our elliptic estimates, (cid:90) S (cid:12)(cid:12)(cid:12) ( ν ∗ ξ ) ⊥ (cid:12)(cid:12)(cid:12) (cid:46) r (cid:90) S (cid:12)(cid:12)(cid:12) d (cid:63) / ( ν ∗ ξ ) ⊥ (cid:12)(cid:12)(cid:12) = r (cid:90) S (cid:12)(cid:12)(cid:12) d (cid:63) / ( ν ∗ ξ ) (cid:12)(cid:12)(cid:12) . We deduce, I = (cid:90) S ˇ ς ( ν ∗ ξ ) e Φ = B (cid:82) S e (cid:90) ˇ ςe + (cid:90) S ˇ ς ( ν ∗ ξ ) ⊥ e Φ (cid:46) (cid:15) | B | + (cid:15)r (cid:18)(cid:90) S (cid:12)(cid:12)(cid:12) ( ν ∗ ξ ) ⊥ (cid:12)(cid:12)(cid:12) (cid:19) (cid:46) (cid:15) | B | + (cid:15)r (cid:18)(cid:90) S (cid:12)(cid:12)(cid:12) d (cid:63) / ( ν ∗ ξ ) (cid:12)(cid:12)(cid:12) (cid:19) . Hence, | B | (cid:46) (cid:15)r (cid:18)(cid:90) S (cid:12)(cid:12)(cid:12) d (cid:63) / ( ν ∗ ξ ) (cid:12)(cid:12)(cid:12) (cid:19) / + (cid:15) ru − − δ dec . (7.3.24)Finally, (cid:90) S (cid:12)(cid:12) ν ∗ ξ (cid:12)(cid:12) (cid:46) r (cid:90) S (cid:12)(cid:12) d (cid:63) / ( ν ∗ ξ ) (cid:12)(cid:12) + r − B (cid:46) r (cid:90) S (cid:12)(cid:12) d (cid:63) / ( ν ∗ ξ ) (cid:12)(cid:12) + (cid:15) r − u − − δ dec or, since r ≥ u / , (cid:90) S (cid:12)(cid:12) ν ∗ ξ (cid:12)(cid:12) (cid:46) r (cid:90) S (cid:12)(cid:12) d (cid:63) / ( ν ∗ ξ ) (cid:12)(cid:12) + (cid:15) u − − δ dec . Integrating we derive, (cid:90) Σ( u,u ∗ ) (cid:12)(cid:12) ν ∗ ξ (cid:12)(cid:12) (cid:46) (cid:90) Σ( u,u ∗ ) r (cid:12)(cid:12) d (cid:63) / ( ν ∗ ξ ) (cid:12)(cid:12) + (cid:15) u − − δ dec . Thus, making use of (7.3.7), (cid:90) Σ( u,u ∗ ) (cid:12)(cid:12) ν ∗ ξ (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) which, together with (7.3.22), yields (cid:90) Σ( u,u ∗ ) (cid:12)(cid:12) d ∗ ξ (cid:12)(cid:12) + (cid:12)(cid:12) ξ (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . Moreover we as desired. Returning to (7.3.24) we also infer that, | B | (cid:46) (cid:15)r (cid:18)(cid:90) S (cid:12)(cid:12)(cid:12) d (cid:63) / ( ν ∗ ξ ) (cid:12)(cid:12)(cid:12) (cid:19) / + (cid:15) ru − − δ dec (cid:46) (cid:15) ru − − δ dec . This proves the estimates (7.3.20), (7.3.21) for ξ and k ≤
1. The proof for the highertangential derivatives can be derived in the same manner.
Step 6b.
To prove the desired estimates for η we make use of the GCM condition (cid:82) S ηe Φ = 0. We deduce, by elliptic estimates, (cid:90) S | η | + | d /η | (cid:46) (cid:90) S r | d (cid:63) / η | + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ηe Φ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) S r | d (cid:63) / η | . Hence, integrating and using the estimates of Lemma 7.3.7, (cid:90) Σ( u,u ∗ ) | η | + | d /η | (cid:46) (cid:15) u − − δ dec (7.3.25)which is the exact analogue of (7.3.22) in Step 6a.To estimate (cid:82) S | ν ∗ η | we proceed exactly as in step 6a by making use of the GCM condition (cid:82) S ξe Φ = 0 and Lemma 7.3.3 to deduce (cid:90) S ( ν ∗ η ) e Φ = Err[ η, ν ∗ ]= − ς − ˇ ς (cid:90) S (cid:18) ν ∗ η − r η (cid:19) e Φ + ς − (cid:90) S ˇ ς (cid:18) ν ∗ η − r η (cid:19) e Φ + O ( (cid:15) u − − δ dec ) (cid:90) S | η | . The term Err[ η, ν ∗ ] can be dealt with precisely as Err[ ξ, ν ∗ ] in Step 6a. Proceeding exactlyas before in Step 6a we deduce, (cid:90) Σ( u,u ∗ ) | ν ∗ η | (cid:46) (cid:15) u − − δ dec . Thus, combined with (7.3.25), we infer that, (cid:90) Σ( u,u ∗ ) | η | + | d ∗ η | (cid:46) (cid:15) u − − δ dec .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν ∗ η ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ru − − δ dec which establishes the estimates (7.3.20), (7.3.21) for η and k ≤
1. The estimates for thehigher tangential derivatives can be proved in the same manner. This completes the proofof both inequalities (7.3.20) and (7.3.21).At this point we have reduced all estimates of Proposition 7.3.5 to the proofs of Lemmas7.3.6 and 7.3.7.
Step 7.
We prove Lemma 7.3.6 i.e. we show that, (cid:90) Σ ∗ ( u,u ) r (cid:12)(cid:12) d (cid:63) / ( d k ∗ β ) (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 15 . We make use of Proposition 7.1.11 according to which e ( r q ) = r (cid:40) d (cid:63) / d (cid:63) / d/ β − κρα − ρ d (cid:63) / d (cid:63) / κ − κρ d (cid:63) / ζ + 34 (2 ρ − κκρ ) ϑ (cid:41) + Err[ e ( r q )]where, Err[ e ( r q )] = r d ≤ (cid:0) Γ b · Γ g (cid:1) . It is easy to check, with the help of the estimates
Ref1 and
Ref2 and the product Lemma7.1.6, that for all k ≤ k small + 16, (cid:107) Err[ e ( r q )] (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:15) (cid:0) r − / u − − δ dec + u − / − δ dec (cid:1) . (7.3.26)We can also check, making use of the estimates (7.1.13) of Ref1 for large r , (cid:107) ρ d (cid:63) / d (cid:63) / κ, κρ d (cid:63) / ζ, κκρϑ, ρ ϑ (cid:107) ∞ ,k (cid:46) (cid:15) (cid:16) r − + r − u − − δ dec (cid:17) . In view of our assumption for r on Σ ∗ we have r ≥ (cid:15)(cid:15) u δ dec , we thus deduce for all k ≤ k small + 17 (cid:13)(cid:13)(cid:13) e ( r q ) − r (cid:18) d (cid:63) / d (cid:63) / d/ β − κρα (cid:19) (cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15)r − + (cid:15) (cid:0) r − / u − − δ dec + u − / − δ dec (cid:1) (cid:46) (cid:15) u − / − δ dec . which hold for all k ≤ k large − CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
We infer that, (cid:107) r d (cid:63) / d (cid:63) / d/ β (cid:107) ∞ ,k (cid:46) (cid:107) r − e ( r q ) (cid:107) ∞ ,k + (cid:107) α (cid:107) ∞ ,k + (cid:15) r − u − / − δ dec . Thus integrating on the last slice Σ ∗ and making use of the assumptions (7.1.15) and(7.1.18), i.e. (cid:90) Σ ∗ ( u,u ∗ ) | e d k q | + r − | q | + | d k α | (cid:46) (cid:15) (1 + u ) − − δ dec , k ≤ k small + 14 , we deduce, (cid:90) Σ ∗ ( u,u ∗ ) r (cid:12)(cid:12) d k ( d (cid:63) / d (cid:63) / d/ β ) (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . In particular, (cid:90) Σ ∗ ( u,u ∗ ) r (cid:12)(cid:12) d k ∗ ( d (cid:63) / d (cid:63) / d/ β ) (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec where d k ∗ = ν k ∗ d / k denote the tangential derivatives to Σ ∗ . Taking into account thecommutator Lemma 7.3.1 we deduce, for k ≤ k small + 14, (cid:90) Σ ∗ ( u,u ∗ ) r (cid:12)(cid:12) d (cid:63) / d (cid:63) / d/ ( d k ∗ β ) (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . (7.3.27)Since d (cid:63) / d/ = d/ d (cid:63) / + 2 K, we infer that (cid:90) Σ ∗ ( u,u ∗ ) r (cid:12)(cid:12) ( d (cid:63) / d/ + 2 K ) d (cid:63) / ( d k ∗ β ) (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . In view of the coercivity of d (cid:63) / d/ + 2 K we deduce, (cid:90) Σ ∗ ( u,u ∗ ) r (cid:12)(cid:12) d (cid:63) / ( d k ∗ β ) (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 16 . This concludes the proof of Lemma 7.3.6. In view of the results derived in the first threesteps this also establishes, unconditionally, the estimate, (cid:90) Σ ∗ ( u,u ) r (cid:12)(cid:12) d k ∗ β (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 15 (7.3.28) .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ Step 8.
We prove Lemma 7.3.7 based on the identities of Proposition 7.3.4. To derivethe desired flux estimate for η we make use of the first part of Proposition 7.3.4 accordingto which we have,2 d (cid:63) / d (cid:63) / d/ d/ d (cid:63) / η = κ (cid:16) ν ∗ ( d (cid:63) / d (cid:63) / µ ) + 2 d (cid:63) / d (cid:63) / d/ β (cid:17) − d (cid:63) / d (cid:63) / d/ ν ∗ ( e θ ( κ ))+ r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t.Since, d (cid:63) / d/ = d/ d (cid:63) / + 2 K , we deduce, d (cid:63) / ( d/ d (cid:63) / + 2 K ) d/ d (cid:63) / η = 12 (cid:104) κν ∗ ( d (cid:63) / d (cid:63) / µ ) − d (cid:63) / d (cid:63) / d/ ν ∗ ( e θ ( κ )) (cid:105) + κ d (cid:63) / ( d/ d (cid:63) / + 2 K ) β + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t.or, ( d/ d (cid:63) / + 2 K ) (cid:2) d (cid:63) / d/ d (cid:63) / η − κ d (cid:63) / β (cid:3) = 12 (cid:104) κν ∗ ( d (cid:63) / d (cid:63) / µ ) − d (cid:63) / d (cid:63) / d/ ν ∗ ( e θ ( κ )) (cid:105) + r − d / ≤ Γ g + r − d / ≤ (Γ b · Γ b ) + l.o.t.Taking higher tangential derivatives and using our GCM assumptions on Σ ∗ d k ∗ ( d/ d (cid:63) / + 2 K ) (cid:104) d (cid:63) / d/ d (cid:63) / η − κ d (cid:63) / β (cid:105) = d k ∗ (cid:104) r − d / ≤ Γ g + r − d / ≤ (Γ b · d Γ b ) (cid:105) + l.o.t.Making use of the commutation Lemma 7.3.1 we can rewrite, r ( d/ d (cid:63) / + 2 K ) (cid:104) d (cid:63) / d/ d (cid:63) / ( d k ∗ η ) − κ d (cid:63) / ( d k ∗ β ) (cid:105) = (cid:88) j ≤ k d / ≤ (cid:104) r − d / ≤ d j ∗ Γ g + r − d / ≤ d j ∗ (Γ b · d Γ b ) (cid:105) . Using the ellipticity of the operator ( d/ d (cid:63) / +2 K ), assumptions Ref 2- Ref 4 , interpolationLemma 7.1.6 and condition r ≥ ( (cid:15)(cid:15) − ) u / δ dec on Σ ∗ we derive, d (cid:63) / d/ d (cid:63) / ( d k ∗ η ) = κ d (cid:63) / ( d k ∗ β ) + O (cid:16) (cid:15)r − + (cid:15) r − u − − δ dec (cid:17) = κ d (cid:63) / ( d k ∗ β ) + O (cid:16) (cid:15) r − u − / − δ dec (cid:17) . Using also the ellipticity of d (cid:63) / d/ and the assumption r ≥ (cid:15) ( (cid:15) − ) u / δ dec on Σ ∗ , (cid:107) d (cid:63) / ( d k ∗ η ) (cid:107) L ( S ) (cid:46) r (cid:107) ( d k ∗ β ) (cid:107) L ( S ) + (cid:15) r − u − / − δ dec . Finally, squaring, integrating on Σ ∗ and taking into account the flux estimate for β in(7.3.28) we deduce (cid:90) Σ ∗ ( u,u ∗ ) r (cid:12)(cid:12) d (cid:63) / ( d k ∗ η ) (cid:12)(cid:12) (cid:46) (cid:90) Σ ∗ ( u,u ∗ ) r (cid:12)(cid:12) ( d k ∗ β ) (cid:12)(cid:12) + (cid:15) u − − δ dec (cid:46) (cid:15) u − − δ dec . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Hence, (cid:90) Σ ∗ ( u,u ∗ ) r (cid:12)(cid:12) d (cid:63) / ( d k ∗ η ) (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec (cid:46) (cid:15) u − − δ dec as stated. This completes the proof of Lemma 7.3.7 for η . The proof for ξ is very similarand will thus be omitted. This therefore also completes the proof of Proposition 7.3.5. (cid:96) = 1 modes on Σ ∗ In the previous subsection we have already derived estimates for the (cid:96) = 1 modes of β, η, ξ .In what follows we derive estimates for the remaining (cid:96) = 1 modes. We summarize theresults in the following proposition.
Proposition 7.3.8.
The following estimates hold true for all k ≤ k small + 16 , r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ η ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ ξ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ β ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ ζ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ ( e θ κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν k ∗ ( e θ ρ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν k ∗ ( e θ µ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ β ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r − u − − δ dec , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν k ∗ ( e θ ω ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ru − − δ dec . (7.3.29) Proof.
Note that the estimates in the first line of (7.3.29) have already been derived inthe previous subsection. The proof of the remaining ones is done in a series of stepsstarting with the estimate for the (cid:96) = 1 mode of e θ ( κ ) and ending with that for the (cid:96) = 1mode of β . To derive the correct estimate for the latter we need to derive in fact strongerintermediary estimates than those stated in the proposition. Step 1.
We prove the desired estimate for (cid:82) S ζe Φ . Indeed, in view of the Codazziequations and the GCM condition on κ , d/ ϑ = − β + ( e θ ( κ ) + ζκ ) + Γ g · Γ g = − β + 2 r ζ + Γ g · Γ g . Integrating, (cid:90) S ζe Φ = − r (cid:90) S βe Φ + r Γ g · Γ g . .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ Ref1-3 for β and Γ g , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ζe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r / − δ B / + (cid:15) u − − δ dec . The higher derivative estimates can be proved in the same manner. Hence, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ ζ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r / − δ B / + (cid:15) u − − δ dec , k ≤ k small + 16 . (7.3.30)In particular, for r ≥ (cid:16) (cid:15)(cid:15) (cid:17) u δ dec , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ ζ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r(cid:15) u − − δ dec , k ≤ k small + 16as stated in the proposition. Step 2.
We prove here the following stronger estimate for the (cid:96) = 1 mode of e θ κ . (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν k ∗ e θ κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 16 . (7.3.31)To control the (cid:96) = 1 mode of e θ ( κ ) on Σ ∗ we need the precise identity of Proposition2.2.19 e ( e θ ( κ )) = − d/ d (cid:63) / ξ + κ (cid:0) e ( ζ ) − β (cid:1) + κ ζ − κe θ κ + 6 ρξ − ωe θ ( κ ) + Err[ d/ d (cid:63) / ξ ] , Err[ d/ d (cid:63) / ξ ] = (cid:18) d/ ξ + 12 κ ϑ + 2 ηξ − ϑ (cid:19) η + 2 e θ ( ηξ ) − e θ ( ϑ )+ κ (cid:18) ϑζ − ϑξ (cid:19) − ϑe θ κ − ϑϑξ − ζ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) + ξ (cid:16) − ϑϑ − d/ ζ + 2 ζ (cid:17) − ηζξ − e θ ( ζξ ) . The error term can be written schematically as,Err[ d/ d (cid:63) / ξ ] = (cid:18) d/ ξ + 12 κ ϑ + 2 ηξ − ϑ (cid:19) η + 2 e θ ( ηξ ) − e θ ( ϑ )+ r − d / ≤ (Γ g · Γ b ) + Γ g · Γ b · Γ b . Note also that we can write, schematically, κ ζ − κe θ κ + 6 ρξ − ωe θ ( κ ) = r − ζ + r − e θ ( κ ) + r − ξ + r − d / ≤ (Γ g · Γ b ) + Γ g · Γ b · Γ b . Note that the schematic form of Corollary 7.1.12 is not suitable here. CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Also, using the transport equation for e ( κ ), e ( e θ ( κ )) = e θ e κ + [ e , e θ ] κ = e θ (cid:20) − κ κ − d/ ζ + 2 ρ + Γ g · Γ b (cid:21) + 12 κe θ κ + l.o.t.= r − e θ κ + d (cid:63) / d/ ζ + e θ ( ρ ) + r − d / (Γ g · Γ b )= r − e θ κ + ( d/ d (cid:63) / + 2 K ) ζ + e θ ( ρ ) + r − d / (Γ g · Γ b ) . We can also write, since ν ∗ = e + a ∗ e e ( ζ ) = ν ∗ ( ζ ) − a ∗ e ( ζ ) = ν ∗ ( ζ ) − a ∗ ( − κζ − β + Γ g · Γ g ) . Therefore, writing also κ = − r + Γ g , ν ∗ ( e θ ( κ )) = − d/ d (cid:63) / ξ − r (cid:0) ν ∗ ( ζ ) − β (cid:1) + E + E + E ,E = r − ζ + r − e θ ( κ ) + r − ξ + d/ d (cid:63) / ζ + r − β + e θ ( ρ ) ,E = (cid:18) d/ ξ + 12 κ ϑ + 2 ηξ − ϑ (cid:19) η + 2 e θ ( ηξ ) , − e θ ( ϑ ) E = r − d / ≤ (Γ g · Γ b ) + Γ g · Γ b · Γ b . (7.3.32)Projecting on the (cid:96) = 1 mode we derive, (cid:90) S ν ∗ (cid:16) e θ ( κ ) − κζ (cid:17) e Φ = − r (cid:90) S βe Φ + (cid:90) S ( E + E + E ) e Φ . (7.3.33)On the other hand, according to Lemma 7.3.3 ν ∗ (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) = (cid:82) S ( ν ∗ e θ ( κ )) e Φ + ( κ + a ∗ κ ) (cid:82) S e θ ( κ ) e Φ + Err[ e θ ( κ ) , ν ∗ ]with error termErr[ e θ ( κ ) , ν ∗ ] = − ς − ˇ ς (cid:90) S (cid:18) ν ∗ e θ ( κ ) − r e θ ( κ ) (cid:19) e Φ + ς − (cid:90) S ˇ ς (cid:18) ν ∗ e θ ( κ ) − r e θ ( κ ) (cid:19) e Φ + O ( (cid:15) u − − δ dec ) (cid:90) S | e θ ( κ ) | . We easily deduce Err[ e θ ( κ ) , ν ∗ ] = r d / ≤ Γ b · Γ b + l.o.t. (7.3.34) .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ ν ∗ (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19) = (cid:90) S ( ν ∗ e θ ( κ )) e Φ + O ( r − ) (cid:90) S e θ ( κ ) e Φ + r d / ≤ Γ b · Γ b + l.o.t.= (cid:90) S ( ν ∗ e θ ( κ )) e Φ + r d / Γ g + r d / ≤ Γ b · Γ b + l.o.t.Similarly, ν ∗ (cid:18)(cid:90) S ( κζ ) e Φ (cid:19) = (cid:82) S ( ν ∗ κζ ) e Φ + r d / Γ g + r d / ≤ Γ b · Γ b + l.o.t.Combining with (7.3.33) we deduce, ν ∗ (cid:90) S (cid:16) e θ ( κ ) − κζ (cid:17) e Φ = − r (cid:90) S βe Φ + (cid:90) S ( E + E + E ) e Φ + r d / Γ g + r d / ≤ Γ b · Γ b + l.o.t.Now schematically, in view of Ref 1-2 , (cid:90) S E e Φ = (cid:90) S (cid:16) r − ζ + r − e θ ( κ ) + r − β + e θ ( ρ ) (cid:17) e Φ = r d / ≤ Γ g (cid:90) S E e Φ = r d / ≤ (Γ g · Γ b ) + r Γ g · Γ b · Γ b . Hence, ν ∗ (cid:90) S (cid:16) e θ ( κ ) − κζ (cid:17) e Φ = − r (cid:90) S βe Φ + r d / ≤ Γ g + (cid:90) S E e Φ + r d / ≤ Γ b · Γ b + r Γ g · Γ b · Γ b + l.o.t. (7.3.35)In particular, in view of the estimate for the (cid:96) = 1 mode of β derived in the previoussubsection (see (7.3.18)) and our assumptions on Γ g , Γ b , ν ∗ (cid:18)(cid:90) S (cid:16) e θ ( κ ) − κζ (cid:17) e Φ (cid:19) = (cid:90) S E e Φ + O ( (cid:15)r − )or, in view of the condition r ≥ ( (cid:15)(cid:15) − ) u δ dec , ν ∗ (cid:18)(cid:90) S (cid:16) e θ ( κ ) − κζ (cid:17) e Φ (cid:19) = (cid:90) S E e Φ + O (cid:16) (cid:15) u − − δ dec (cid:17) . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Integrating along Σ ∗ from S ∗ and making use of the vanishing of the (cid:96) = 1 mode of e θ ( κ )on S ∗ and the estimate for the (cid:96) = 1 mode of ζ derived earlier in step 0, see (7.3.30), wededuce, (cid:90) S ( u ) (cid:16) e θ ( κ ) − κζ (cid:17) e Φ = (cid:90) S ∗ (cid:16) e θ ( κ ) − κζ (cid:17) e Φ + O (cid:16) (cid:15) u − − δ dec (cid:17) + (cid:90) Σ( u,u ∗ ) E e Φ = 2Υ r (cid:90) S ∗ ζe Φ + O (cid:16) (cid:15) u − − δ dec (cid:17) + (cid:90) Σ( u,u ∗ ) E e Φ = O ( (cid:15)r − / − δ B ) + O (cid:16) (cid:15) u − − δ dec (cid:17) + (cid:90) Σ( u,u ∗ ) E e Φ . Since min Σ ∗ r ≥ ( (cid:15)(cid:15) − ) u and δ B ≥ δ dec we infer that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( u ) (cid:16) e θ ( κ ) − κζ (cid:17) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec + (cid:82) Σ( u,u ∗ ) r | E | . It remains to estimate the term (cid:82) Σ( u,u ∗ ) r | E | where, recall, E = (cid:0) d/ ξ + κ ϑ + 2 ηξ − ϑ (cid:1) η + 2 e θ ( ηξ ) . − e θ ( ϑ ) . Hence, appealing to Proposition 7.3.5 (cid:90) Σ( u,u ∗ ) r (cid:12)(cid:12) E (cid:12)(cid:12) (cid:46) (cid:90) Σ( u,u ∗ ) (cid:16)(cid:12)(cid:12) d / ≤ ϑ (cid:12)(cid:12) + (cid:12)(cid:12) d / ≤ η (cid:12)(cid:12) + (cid:12)(cid:12) d / ≤ ξ (cid:12)(cid:12) (cid:17) (cid:46) (cid:15) u − − δ dec . Thus, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( u ) | e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( u ) κζe Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15) u − − δ dec (cid:46) r − / (cid:15) + (cid:15) u − − δ dec (cid:46) (cid:15) u − − δ dec which establishes the desired estimate in this case.To derive a similar estimate for the first tangential derivative we go back to (7.3.33) (cid:90) S ν ∗ (cid:16) e θ ( κ ) − κζ (cid:17) e Φ = − r (cid:90) S βe Φ + (cid:90) S ( E + E + E ) e Φ from which we easily derive, as before, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν ∗ (cid:16) e θ ( κ ) − κζ (cid:17) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − + (cid:15) u − − δ dec (cid:46) (cid:15) u − − δ dec .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν ∗ e θ ( κ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − / + (cid:15) u − − δ dec (cid:46) (cid:15) u − − δ dec . To estimate the second tangential derivative we make use once more of Lemma 7.3.3.Setting ψ := ν ∗ (cid:16) e θ ( κ ) − κζ (cid:17) we deduce, (cid:90) S ( ν ∗ ψ ) e Φ = ν ∗ (cid:18)(cid:90) S ψe Φ (cid:19) − Err[ ψ, ν ∗ ]= ν ∗ (cid:18) − r (cid:90) S βe Φ + (cid:90) S Ee Φ (cid:19) − Err[ ψ, ν ∗ ]with E = E + E + E and error termErr[ ψ, ν ∗ ] = − ς − ˇ ς (cid:90) S ( e ψ + 32 κψ ) e Φ + ς − (cid:90) S ˇ ς (cid:0) e ψ + 32 κψ (cid:1) e Φ ) + O ( (cid:15) u − − δ dec ) (cid:90) S | ψ | which can be estimated as in 7.3.34 and shown to be lower order. Note that, using againLemma 7.3.3, ν ∗ (cid:18) − r (cid:90) S βe Φ + (cid:90) S Ee Φ (cid:19) = − r (cid:90) S ( ν ∗ β ) e Φ + (cid:90) S ( ν ∗ E ) e Φ + l.o.t.= O ( (cid:15)r − ) + O ( (cid:15) u − − δ dec ) = O ( (cid:15) u − − δ dec )and we then proceed as before. All higher tangential derivatives can be treated in thesame manner. Step 3.
We establish the estimate for the (cid:96) = 1 mode of β . Recall that we had, see(7.3.18), 2 (cid:90) S βe Φ = 2Υ r (cid:90) S ζe Φ + (cid:90) S e θ (ˇ κ ) e Φ + (cid:90) S Γ g · Γ b e Φ . Thus, using the estimates already derived in Steps 0 and 1, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . The higher derivative estimates are derived in the same manner.
Step 4.
We establish the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν k ∗ e θ ( ρ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r − u − − δ dec + (cid:90) S (cid:12)(cid:12) ν k ∗ d / ( ϑϑ ) (cid:12)(cid:12) . (7.3.36)66 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
We start by differentiating the Gauss equation K = − ρ − κ κ + ϑϑ . we derive Usingthe GCM condition for κ we derive, e θ ( ρ ) = − e θ ( K ) − r e θ ( κ ) + 14 e θ ( ϑϑ ) . We make use of the vanishing of the (cid:96) = 1 mode of e θ ( K ) (see Lemma 2.1.29) to derive (cid:90) S e θ ( ρ ) e Φ = − r (cid:90) S e θ ( κ ) e Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ . (7.3.37)Using the previously deduced estimate for the (cid:96) = 1 mode of e θ ( κ ) we deduce (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( ρ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r − u − − δ dec + (cid:90) S | ϑ || d /ϑ | + | ϑ || d /ϑ | . (7.3.38)Note that in particular we deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( ρ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r − u − / − δ dec but we shall need the precise form (7.3.38) in Step 7.To estimate (cid:82) S ( ν ∗ e θ ( ρ )) e Φ we apply ν ∗ to (7.3.38) and make use of Lemma 7.3.3 as inStep 2. (cid:90) S ( ν ∗ e θ ( ρ )) e Φ = ν ∗ (cid:18)(cid:90) S e θ ( ρ ) e Φ (cid:19) −
32 ( κ + a ∗ κ ) (cid:90) S e θ ( ρ ) e Φ − Err[ e θ ( ρ ) , ν ∗ ]= ν ∗ (cid:20) − r (cid:90) S e θ ( κ ) e Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ (cid:21) + O ( r − ) (cid:90) S e θ ( ρ ) e Φ − Err[ e θ ( ρ ) , ν ∗ ]= − r ν ∗ (cid:16) (cid:90) S e θ ( κ ) e Φ (cid:17) + 14 (cid:90) S ν ∗ (cid:2) e θ ( ϑϑ ) (cid:3) e Φ + l.o.t.Hence, in view of the results derived in the first step, we deduce (7.3.36) for k = 1. Thecase of higher derivatives can be treated in the same manner. Step 5.
To estimate the (cid:96) = 1 mode of µ we differentiate the relation µ = − div ζ − ρ +Γ g · Γ b , e θ ( µ ) = d (cid:63) / d/ ζ − e θ ( ρ ) + r − d / (Γ g · Γ b )= ( d/ d (cid:63) / + 2 K ) ζ − e θ ( ρ ) + r − d / (Γ g · Γ b )= d/ d (cid:63) / ζ + 2 r ζ − e θ ( ρ ) + r − d / (Γ g · Γ b ) + l.o.t. .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ (cid:90) S e θ ( µ ) e Φ = 2 r − (cid:90) S ζe Φ − (cid:90) S e θ ( ρ ) e Φ + r d / (Γ g · Γ b ) . (7.3.39)Using the previous estimates for the (cid:96) = 1 modes of ζ and e θ ρ we deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( µ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r − u − − δ dec . The higher ν ∗ derivatives can be derived exactly as in the previous steps by differentiating(7.3.39) and applying Lemma 7.3.3. Step 6.
We estimate the (cid:96) = 1 mode of ω making use of the first identity in Corollary7.1.12 2 d (cid:63) / ω = (cid:18) κ + 2 ω (cid:19) η + e ( ζ ) − β + 12 κξ + r − Γ g + Γ b = ( − Υ r + 2 mr ) η + e ( ζ ) − β + 1 r ξ + r − Γ g + Γ b + l.o.t.= ( − Υ r + 2 mr ) η + ν ∗ ( ζ ) − a ∗ e ζ − β + 1 r ξ + r − Γ g + Γ b + l.o.t.= − η + ν ∗ ( ζ ) − β + 1 r ξ + r − Γ g + Γ b + l.o.t.Hence,2 (cid:90) S d (cid:63) / ( ω ) e Φ = − (cid:90) S ηe Φ + (cid:90) S ν ∗ ζe Φ − (cid:90) S βe Φ + r − (cid:90) S ξe Φ + r Γ g + r Γ b · Γ b . Taking into account the prescribed GCM conditions for the (cid:96) = 1 modes of η and ξ andthe previous results for the (cid:96) = 1 modes of ν ∗ ( ζ ) , β we deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( ω ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) ru − / − δ dec . The higher ν ∗ derivatives can be treated as in the previous steps. Step 7.
It remains to estimate the (cid:96) = 1 mode of β and show (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν k ∗ ( β ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r − u − − δ dec . (7.3.40)We start with the e β equation which we write in the form e ( β ) = e θ ( ρ ) + 3 ρ η + J + O (cid:0) r − ( α, β ) (cid:1) , J := 3 η ˇ ρ − ϑβ. CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Also, taking into account the e equation for β , ν ∗ ( β ) = e ( β ) + a ∗ e β = e ( β ) + a ∗ ( − κβ + d/ α + ζα ))= e θ ( ρ ) + 3 ρ η + J + O (cid:0) r − ( β, d / ≤ α ) (cid:1) + l.o.t.Projecting on the (cid:96) = 1 mode, (cid:90) S ν ∗ ( β ) e Φ = (cid:90) S e θ ( ρ ) e Φ + (cid:90) S J e Φ + O (cid:0) r ( β, d / ≤ α ) (cid:1) . (7.3.41)On the other hand, making use of Lemma 7.3.3, ν ∗ (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S ( ν ∗ β ) e Φ + 32 ( κ + a ∗ κ ) (cid:90) S βe Φ + Err[ β, ν ∗ ]= (cid:90) S ( ν ∗ β ) e Φ + O (cid:0) r β (cid:1) + Err[ β, ν ∗ ]with error termErr[ β, ν ∗ ] = − ς − ˇ ς (cid:90) S ( e β + 32 κβ ) e Φ + ς − (cid:90) S ˇ ς (cid:0) e β + 32 κβ (cid:1) e Φ ) + (cid:15) u − − δ dec (cid:90) S | β | . Note that e β = e θ ρ + l.o.t. = r − d / ≤ Γ g . Hence,Err[ β, ν ∗ ] = r ( r − Γ b · d / ≤ Γ g ) + l.o.t.Hence, ν ∗ (cid:18)(cid:90) S βe Φ (cid:19) = (cid:82) S ( ν ∗ β ) e Φ + O (cid:0) r β (cid:1) + r Γ b · d / ≤ Γ g + l.o.t. (7.3.42)Thus combining to (7.3.41), ν ∗ (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S e θ ( ρ ) e Φ + 3 ρ (cid:90) S ηe Φ + (cid:90) S J e Φ + O (cid:0) r ( β, d / ≤ α ) (cid:1) + r Γ b · d / ≤ Γ g and, since ν ∗ ( r ) = O (1) ν ∗ (cid:18) r (cid:90) S βe Φ (cid:19) = r (cid:90) S e θ ( ρ ) e Φ + 3 rρ (cid:90) S ηe Φ + r (cid:90) S J e Φ + O (cid:0) r ( β, d / ≤ α ) (cid:17) + r Γ b · d / ≤ Γ g . Note that, based on
Ref1-Ref2 and condition min Σ ∗ r ≥ ( (cid:15)(cid:15) − ) r and δ B ≥ δ dec , r (cid:12)(cid:12) ( β, d / ≤ α ) (cid:12)(cid:12) (cid:46) (cid:15)r − / − / δ B (cid:46) (cid:15) u − − δ dec ,r (cid:12)(cid:12) Γ b · d / ≤ Γ g (cid:12)(cid:12) (cid:46) (cid:15)r − (cid:46) (cid:15) u − − δ dec . .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ (cid:96) = 1 mode of e θ ( ρ ) has been estimated in (7.3.38) (cid:12)(cid:12)(cid:12) r (cid:90) S e θ ( ρ ) e Φ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec + r (cid:90) S | ϑ || d /ϑ | + | ϑ || d /ϑ | . Therefore, using also that the (cid:96) = 1 mode of η vanishes on Σ ∗ due to our GCM conditions, (cid:12)(cid:12)(cid:12) ν ∗ (cid:18) r (cid:90) S βe Φ (cid:19) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec + r (cid:90) S | ϑ || d /ϑ | + | ϑ || d /ϑ | + r | J | . (7.3.43)Hence, integrating on Σ ∗ starting from S ∗ and using the GCM condition (cid:82) S ∗ βe Φ = 0, wededuce (cid:12)(cid:12)(cid:12) r (cid:90) S ( u ) βe Φ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec + (cid:90) Σ( u,u ∗ ) r (cid:16) | ϑ || d /ϑ | + | ϑ || d /ϑ | + r | J | (cid:17) . Using the flux estimates of Proposition 7.3.5 we infer that (cid:90) Σ( u,u ∗ ) r ( | ϑ || d /ϑ | + | ϑ || d /ϑ | ) (cid:46) (cid:18)(cid:90) Σ( u,u ∗ ) | ϑ | + | d /ϑ | (cid:19) / (cid:18)(cid:90) Σ( u,u ∗ ) r (cid:0) | ϑ | + | d /ϑ | (cid:1)(cid:19) / (cid:46) (cid:15) u − − δ dec (cid:18)(cid:90) Σ( u,u ∗ ) r (cid:0) | ϑ | + | d /ϑ | (cid:1)(cid:19) . On the other hand, according to
Ref 1 , (cid:90) Σ( u,u ∗ ) r (cid:0) | ϑ | + | d /ϑ | (cid:1) (cid:46) (cid:15) (cid:90) Σ( u,u ∗ ) r − u − − δ dec (cid:46) (cid:15) . Hence, (cid:90) Σ( u,u ∗ ) r ( | ϑ || d /ϑ | + | ϑ || d /ϑ | ) (cid:46) (cid:15)(cid:15) u − − δ dec . Similarly, recalling the definition of J = 3 η ˇ ρ − ϑβ , (cid:90) Σ( u,u ∗ ) r | J | (cid:46) (cid:90) Σ( u,u ∗ ) r (cid:0)(cid:12)(cid:12) η (cid:12)(cid:12)(cid:12)(cid:12) ˇ ρ (cid:12)(cid:12) + (cid:12)(cid:12) ϑ (cid:12)(cid:12)(cid:12)(cid:12) β (cid:12)(cid:12)(cid:1) (cid:46) (cid:16) (cid:90) Σ( u,u ∗ ) (cid:0) | η | + r | β | (cid:1) (cid:17) / · (cid:16) (cid:90) Σ( u,u ∗ ) r (cid:0) | ϑ | + r | ˇ ρ | (cid:1) (cid:17) / (cid:46) (cid:15) u − − δ dec · (cid:16) (cid:90) Σ( u,u ∗ ) r (cid:0) | ϑ | + r | ˇ ρ | (cid:1) (cid:17) / (cid:46) (cid:15) (cid:15)u − − δ dec . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r − u − − δ dec as desired. To derive (7.3.40) for k = 1 we go back to (7.3.42) which we write in the form (cid:90) S ( ν ∗ β ) e Φ = r − ν ∗ (cid:18) r (cid:90) S βe Φ (cid:19) + O (cid:0) r β (cid:1) + r Γ b · d / ≤ Γ g + l.o.t.= r − ν ∗ (cid:18) r (cid:90) S βe Φ (cid:19) + O ( (cid:15)r − / ) . Hence, making use of (7.3.43), the assumptions
Ref1-2 and the condition on r , r ≥ ( (cid:15)(cid:15) − ) u δ dec , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( ν ∗ β ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec + r (cid:90) S (cid:0) | ϑ || d /ϑ | + | ϑ || d /ϑ | + r | J | (cid:1) + (cid:15)r − / (cid:46) (cid:15) r − u − − δ dec . The higher derivative estimates are derived in the same manner. This completes the proofof Proposition 7.3.8. Σ ∗ The goal here is to prove the following proposition.
Proposition 7.3.9.
The following estimates hold true along Σ ∗ for all k ≤ k small + 19 (cid:107) ϑ, η, ξ, ˇ ω, rβ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) ˇΩ , Ω + Υ , ˇ ς, ς − (cid:107) ∗∞ ,k (cid:46) (cid:15) u − − δ dec , (cid:107) ˇ κ, ˇ κ, r ˇ µ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) ϑ, ζ, r ˇ ρ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − / − δ dec , (cid:107) β (cid:107) ∗∞ ,k (cid:46) (cid:15) r − (2 r + u ) − / − δ dec . (7.3.44) Proof.
Recall that we have already derived the desired estimates for β, ϑ, η, ξ , see (7.3.15),in Proposition 7.3.5. To prove the remaining estimates we proceed in steps as follows.
Step 1.
In this step we prove the estimate, (cid:107) ˇ κ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − − δ dec . (7.3.45) .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ d (cid:63) / d (cid:63) / ˇ κ = 0, the control of the (cid:96) = 1 mode of d (cid:63) / ˇ κ provided by Lemma 7.3.8, the bootstrap assumptions and our dominance condition on r along Σ ∗ . We also make use of the commutation properties of the tangential derivativeson Σ ∗ with the GCM condition.According to part 4 of the elliptic Hodge Lemma 7.1.7 we derive, (cid:107) d (cid:63) / ˇ κ (cid:107) h k +1 ( S ) (cid:46) r (cid:107) d (cid:63) / d (cid:63) / ˇ κ (cid:107) h k ( S ) + r − (cid:12)(cid:12)(cid:12) (cid:90) S e Φ d (cid:63) / ˇ κ (cid:12)(cid:12)(cid:12) = r − (cid:12)(cid:12)(cid:12) (cid:90) S e Φ d (cid:63) / ˇ κ (cid:12)(cid:12)(cid:12) . According to Proposition 7.3.8 we have, for all 0 ≤ k ≤ k small + 20 (cid:12)(cid:12)(cid:12) (cid:90) S e Φ d (cid:63) / ˇ κ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . Therefore, (cid:107) d (cid:63) / ˇ κ (cid:107) h k +1 ( S ) (cid:46) (cid:15) r − u − − δ dec . Thus, (cid:107) ˇ κ (cid:107) h k +2 ( S ) (cid:46) (cid:15) r − u − − δ dec and therefore, for k ≤ k small + 20 (cid:107) d / ≤ k ˇ κ (cid:107) L ∞ ( S ) (cid:46) (cid:15) r − u − − δ dec . We next commute with ν ∗ . Since ν ∗ is tangent to Σ ∗ and r d (cid:63) / d (cid:63) / ˇ κ = 0, making also use ofthe commutator Lemma 7.3.1, d (cid:63) / ( ν ∗ d (cid:63) / κ ) = r − [ ν ∗ , r d (cid:63) / ] d (cid:63) / ˇ κ = Γ b ν ∗ ( d (cid:63) / κ )) + l.o.t.Proceeding as before, (cid:90) S (cid:12)(cid:12) ν ∗ ( d (cid:63) / ˇ κ ) (cid:12)(cid:12) (cid:46) r (cid:90) S | d (cid:63) / ( ν ∗ d (cid:63) / ˇ κ ) | + r − (cid:12)(cid:12)(cid:12) (cid:90) S e Φ ν ∗ ( d (cid:63) / ˇ κ ) (cid:12)(cid:12)(cid:12) (cid:46) r (cid:90) S (cid:12)(cid:12)(cid:12) Γ b ν ∗ ( d (cid:63) / κ ) (cid:12)(cid:12)(cid:12) + (cid:15) r − u − − δ dec (cid:46) (cid:15) (cid:90) S (cid:12)(cid:12) ν ∗ ( d (cid:63) / ˇ κ ) (cid:12)(cid:12) + (cid:15) r − u − − δ dec . We can then proceed as before making use of the estimate for the (cid:96) = 1 mode of ν ∗ e θ κ inProposition 7.3.8, i.e. for all 0 ≤ k ≤ k small + 20. The higher ν ∗ estimates are proved inthe same manner.72 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Step 2.
In exactly the same manner, making use of the GCM condition d (cid:63) / d (cid:63) / µ = 0 andthe estimate for the (cid:96) = 1 mode of e θ µ in Lemma 7.3.8, we deduce (cid:107) ˇ µ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − − δ dec . (7.3.46) Step 3.
At this point we have established the estimates of Proposition 7.3.9 for β, ϑ, ξ, η and ˇ κ, ˇ µ . Next we derive estimates for ˇ ρ by making use of the quantity q . We make useof Proposition 7.1.10 according to which we have, relative to the background frame of ( ext ) M , r (cid:18) d (cid:63) / d (cid:63) / ρ + 34 κρϑ + 34 κρϑ (cid:19) = q + r d / ≤ (Γ b · Γ g ) + l.o.t.Using the estimates Ref1-2 for the Ricci and curvature coefficients as well as the productLemma 7.1.6 we derive, (cid:107) q (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − / − δ dec , (cid:107) ϑ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) r d / ≤ (Γ b · Γ g ) (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − − δ dec . Also, using our condition on r along Σ ∗ (cid:107) ϑ (cid:107) ∗∞ ,k (cid:46) (cid:15)r − (cid:46) (cid:15) r − u − − δ dec . We deduce, (cid:107) d (cid:63) / d (cid:63) / ˇ ρ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − / − δ dec . Also, after commuting ν ∗ with r d (cid:63) / , (cid:107) d (cid:63) / ( ν i ∗ d (cid:63) / ρ ) (cid:107) ∗∞ ,k − i (cid:46) (cid:15) r − u − / − δ dec . Since, according to Lemma 7.3.8, we also control the (cid:96) = 1 modes of ν i ∗ e jθ ( ρ ) we canproceed as before to deduce the desired estimate (cid:107) ˇ ρ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − / − δ dec , ∀ k ≤ k small + 20 . (7.3.47) Step 4.
From the definition of µ = − d/ ζ − ρ + ϑϑ , we have, d/ ζ = − ˇ µ − ρ + Γ g · Γ b . i.e. passing to L , using part 4 of Lemma 7.1.7, and then going back to L ∞ norms as before .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ k ≤ k small + 19, (cid:107) d (cid:63) / d/ ζ (cid:107) ∗∞ ,k (cid:46) (cid:107) d (cid:63) / ˇ µ (cid:107) ∗ k + (cid:107) d (cid:63) / ˇ ρ (cid:107) ∗∞ ,k + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − u − / − δ dec . Since according to Proposition 7.3.8 we control the (cid:96) = 1 modes of ζ and their ν ∗ deriva-tives we deduce, passing as before through L norms, (cid:107) ζ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − / − δ dec , ∀ k ≤ k small + 19 . (7.3.48) Step 5.
We next use the Codazzi equations, the bootstrap assumption for β and theabove result for ζ to deduce, d/ ϑ = − β + ( e θ ( κ ) + ζκ ) + Γ g · Γ g , = − β + 2 r ζ + Γ g · Γ g . Making use of the bootstrap assumption for β and the previous estimate for ζ , (cid:13)(cid:13) d/ ϑ (cid:13)(cid:13) ∗∞ ,k (cid:46) (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k + r − (cid:13)(cid:13) ζ (cid:13)(cid:13) ∗∞ ,k + (cid:15) r − u − − δ dec (cid:46) (cid:15)r − / − δ dec + (cid:15) r − u − / − δ dec from which we derive, using also the condition min Σ ∗ r ≥ ( (cid:15)(cid:15) − ) u δ dec , (cid:13)(cid:13) ϑ (cid:13)(cid:13) ∗∞ ,k (cid:46) (cid:15) r − u − / − δ dec , ∀ k ≤ k small + 19 . (7.3.49) Step 6.
We now estimate β from the equation, e α + (cid:18) κ − ω (cid:19) α = − d (cid:63) / β − ϑρ + ( ζ + 4 η ) β. Hence, (cid:13)(cid:13) d (cid:63) / β (cid:107) ∗∞ ,k − (cid:46) (cid:13)(cid:13) e α (cid:13)(cid:13) ∗∞ ,k − + r − (cid:13)(cid:13) α (cid:13)(cid:13) ∗∞ ,k − + r − (cid:13)(cid:13) ϑ (cid:107) ∗∞ ,k − + (cid:15) r − (2 r + u ) − / − δ dec (cid:46) (cid:15) r − (2 r + u ) − / − δ dec . We deduce, in particular, using our previous estimate for the (cid:96) = 1 mode of β (cid:107) β (cid:107) h k +1 ( S ) (cid:46) r (cid:13)(cid:13) d (cid:63) / β (cid:107) h k ( S ) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r − (2 r + u ) − / − δ dec + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − (2 r + u ) − / − δ dec . We deduce, arguing as before, (cid:13)(cid:13) β (cid:107) ∗∞ ,k (cid:46) (cid:15) r − (2 r + u ) − / − δ dec , ∀ k ≤ k small + 19 . (7.3.50)74 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Remark 7.3.10.
We can in fact prove a stronger estimate by making use of the fullstrength of the estimate for α in Ref 2 , | α | (cid:46) log(1 + u ) r − (2 r + u ) − / − δ extra , | e α | (cid:46) r − (2 r + u ) − / − δ extra . Hence, (cid:13)(cid:13) d (cid:63) / β (cid:107) ∗∞ ,k (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − / − δ extra + (cid:15) r − u − / − δ dec and thus (cid:13)(cid:13) β (cid:107) ∗∞ ,k (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − / − δ extra . (7.3.51) Step 7.
Next, we estimate ˇ ω by making use of the first identity in Proposition 2.2.19, d (cid:63) / ω = − κ ξ + 12 e ζ − β + 12 κζ + 12 ϑζ − ϑ ξ. We deduce, using (7.3.48) and the dominance condition for r on Σ ∗ , (cid:13)(cid:13) d (cid:63) / ˇ ω (cid:13)(cid:13) ,k − (cid:46) r − (cid:13)(cid:13) ξ (cid:13)(cid:13) ,k − + (cid:13)(cid:13) e ζ (cid:13)(cid:13) ,k − + r − (cid:13)(cid:13) ζ (cid:13)(cid:13) ,k − + (cid:13)(cid:13) β (cid:13)(cid:13) ,k − + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − u − − δ dec . Thus, using the Poincare and Sobolev inequalities, (cid:13)(cid:13) ˇ ω (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec . (7.3.52) Step 8.
We estimate ˇ ς, ς − a ∗ = 2 ς − Υ + r A. Also, recall that, see (7.1.9),Ω = e ( r ) = − Υ + r A, on Σ ∗ . Step 9.
We determine ˇ ς, ˇ ω using the equations, see (2.2.19), e θ log ς = 12 ( η − ζ ) ,e θ ( ˇΩ) = − ξ − ( η − ζ )Ω . This is a precise version of the identity in Proposition 7.1.12. .3. DECAY ESTIMATES ON THE LAST SLICE Σ ∗ η, ζ, ξ and the bootstrap assumptions already derived, (cid:107) ˇ ς (cid:107) h k +1 (cid:46) (cid:15) u − − δ dec , (cid:107) ˇΩ (cid:107) h k +1 (cid:46) (cid:15) u − − δ dec . Step 10.
We estimate ς − ∗ using the equations (7.1.8), (7.1.9), a ∗ = − ς + Υ − r A, Ω = − Υ + r A, and the GCM condition for a ∗ , see (7.1.1), a ∗ (cid:12)(cid:12)(cid:12) SP = − − m S r S . We deduce, 2 ς (cid:12)(cid:12) SP = 2 − r A (cid:12)(cid:12)(cid:12) SP = 2 − (Ω + Υ) (cid:12)(cid:12) SP i.e., 2 (cid:32) − ς (cid:12)(cid:12) SP (cid:33) = Ω (cid:12)(cid:12) SP + Υ . Since ς = ς (cid:12)(cid:12) SP + ˇ ς (cid:12)(cid:12) SP , Ω = Ω (cid:12)(cid:12) SP + ˇΩ (cid:12)(cid:12) SP we deduce, in view of the above estimates for ˇ ς, ˇΩ, (cid:12)(cid:12) ς − (cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12) Ω + Υ (cid:12)(cid:12)(cid:12) + (cid:15) u − − δ dec . It thus remains to estimate Ω + Υ = r A. This can be done with the help of the following identity, A S = 1 ς S (cid:16) ˇ ς S ˇ κ S − ˇ ς S ˇ A S (cid:17) (7.3.53)76 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) from which we infer that, (cid:12)(cid:12)(cid:12)
Ω + Υ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec and therefore also (cid:12)(cid:12) ς − (cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec . To prove (7.3.53) we apply Lemma 7.3.2 to deduce, e ( r ) + a ∗ = ν ∗ ( r ) = r ς − ς ( κ + a ∗ κ ) = r ς − (cid:0) ςκ + ˇ ς ˇ κ (cid:1) + ς − ςa ∗ = − Υ ς − ς + r ς − ˇ ς ˇ κ + ς − ςa ∗ . Since e ( r ) = − Υ + r A we infer that − Υ + r A + a ∗ = − Υ ς − ς + r ς − ˇ ς ˇ κ + ς − ςa ∗ and therefore A = 2 r (cid:16) Υ − a ∗ − Υ ς − ς + r ς − ˇ ς ˇ κ + ς − ςa ∗ (cid:17) . In particular, multiplying by ς and taking the average, we infer that ςA = ˇ ς ˇ κ, and hence A = 1 ς (cid:16) ˇ ς ˇ κ − ˇ ς ˇ A (cid:17) as stated. The estimates for all other derivatives of ˇ ς, ˇΩ , ˇ ς − , Ω + Υ can be derived asbefore. This concludes the proof of Proposition 7.3.9. ( ext ) M , Part I Commutation Lemmas
Here and below we write schematically d / = r d/, d (cid:37) = { re , d / } , d = ( e , re , r d/ ) , T = 12 ( e + Υ e ) . .4. CONTROL IN ( EXT ) M , PART I Lemma 7.4.1.
We have, schematically, [ T , e ] = r − Γ b d (cid:37) , [ d /, e ] = ˇΓ g d (cid:37) + Γ g . (7.4.1) Also, T ( r ) = 12 r A = r − Γ b . Proof.
The identity for [ d /, e ] has already been discussed in Corollary 7.1.5. Accordingto lemma 2.2.14 we have,[ T , e ] = (cid:18)(cid:16) ω − mr (cid:17) − m r (cid:18) κ − r (cid:19) + e ( m ) r (cid:19) e + ( η + ζ ) e θ , In view of
Ref 4 and bootstrap assumptions
Ref 2 the factors of e and e θ , on the righthand side behave at worst like Γ b . Thus schematically [ T , e ] = r − Γ b d (cid:37) . Transport Lemmas
The following lemma will be used repeatedly in what follows.
Lemma 7.4.2. If f verifies the transport equation e ( f ) + p κf = F, we have for fixed u and any r ≤ r ≤ r ∗ , r p (cid:107) f (cid:107) ∞ ( u, r ) (cid:46) r p (cid:107) f (cid:107) ∞ ( u, r H ) + (cid:90) rr λ p (cid:107) F (cid:107) ∞ ( u, λ ) dλ,r p (cid:107) f (cid:107) ∞ ( u, r ) (cid:46) r p ∗ (cid:107) f (cid:107) ∞ ( u, r ∗ ) + (cid:90) r ∗ r λ p (cid:107) F (cid:107) ∞ ( u, λ ) dλ, (7.4.2) where r is the area radius at fixed u .Proof. According to Corollary 2.2.12 we have e ( r p f ) = r p F The desired estimates followeasily by integration with respect to the affine parameter s , recall that e ( s ) = 1. Proposition 7.4.3.
The following inequalities hold true for all k ≤ k large − , r ≤ r ≤ r ∗ r p (cid:107) f (cid:107) ∞ ,k ( u, r ) (cid:46) r p ∗ (cid:107) f (cid:107) ∞ ,k ( u, r ∗ ) + (cid:90) r ∗ r λ p (cid:107) F (cid:107) ∞ ,k ( u, λ ) dλ,r p (cid:107) f (cid:107) ∞ ,k ( u, r ) (cid:46) r p (cid:107) f (cid:107) ∞ ,k ( u, r ) + (cid:90) rr λ p (cid:107) F (cid:107) ∞ ,k ( u, λ ) dλ. (7.4.3)78 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Proof.
Commuting the equation e ( r p f ) = r p F with d / , applying the commutation Lemma7.4.1and our bootstrap assumptions on Γ g we derive, e ( r p | d /f | ) (cid:46) r p | d /F | + r p | d /F | + (cid:15)r − r p (cid:0) | d /f | + re ( r p f ) (cid:1) (cid:46) r p ( | d /F | + | F | ) + (cid:15)r − r p ( | d /f | + | f | ) . Similarly, commuting with T , e (cid:0) T ( r p f ) (cid:1) = T ( r p F ) − [ T , e ]( r p f ) = r p F − pr p − T ( r ) F − r − Γ b d (cid:37) r p f. Hence, e (cid:0) r p T f (cid:1) = r p T ( F ) − pr p − T ( r ) F − r − Γ b d (cid:37) r p f − pe (cid:0) r p − T ( r ) f (cid:1) i.e., e (cid:0) r p | T f | (cid:1) (cid:46) r p (cid:0) | T F | + | F | (cid:1) + O ( (cid:15)r − ) (cid:0) r p | F | + | d /f | + | f | (cid:1) . Similarly, commuting the equation with re we derive, e (cid:0) r p | rf | (cid:1) (cid:46) r p (cid:0) | re F | + | F | (cid:1) + O ( (cid:15)r − ) (cid:0) r p | F | + | d /f | + | f | (cid:1) . Integrating the inequalities, e ( r p | d /f | ) (cid:46) r p ( | d /F | + | F | ) + (cid:15)r − r p ( | d /f | + | f | ) e (cid:0) r p | T f | (cid:1) (cid:46) r p (cid:0) | T F | + | F | (cid:1) + (cid:15)r − r p (cid:0) | d /f | + | f | (cid:1) e (cid:0) r p | rf | (cid:1) (cid:46) r p (cid:0) | re F | + | F | (cid:1) + (cid:15)r − (cid:0) r p | F | + | d /f | + | f | (cid:1) and applying Gronwall we derive the desired estimates in (7.4.3) for k = 1.Repeating the procedure for (cid:101) d k , any combination of derivatives of the form (cid:101) d k = T k d / k with k + k = k , estimating the corresponding commutators using our assumptions Ref1 , we deduce for all 0 ≤ k ≤ k large − e ( r p | (cid:101) d ≤ k f | ) (cid:46) r p | (cid:101) d ≤ k F | + (cid:15)r − r p | (cid:101) d ≤ k f | and the desired estimates follow by integration. Transport equations for (cid:96) = 1 modes
To estimate (cid:96) = 1 modes we make use of the following. .4. CONTROL IN ( EXT ) M , PART I Lemma 7.4.4.
The following equation holds true for reduced scalars ψ ∈ s ( ( ext ) M ) . e (cid:18)(cid:90) S ψe Φ (cid:19) = (cid:90) S e ( ψ ) e Φ + 32 κ (cid:90) S ψe Φ + (cid:90) S
12 (3ˇ κ − ϑ ) ψ ) e Φ . (7.4.4) Proof.
This is an immediate consequence of Proposition (2.2.9). Indeed according to itand e Φ = ( κ − ϑ ), e (cid:18)(cid:90) S ψe Φ (cid:19) = (cid:90) S ( e ( ψe Φ ) + κψe Φ ) = (cid:90) S (cid:18) e ( ψ ) + 12 (3 κ − ϑ ) ψ (cid:19) e Φ = (cid:90) S (cid:18) e ( ψ ) + 32 κψ (cid:19) e Φ + (cid:90) S
12 (3ˇ κ − ϑ ) ψ ) e Φ = (cid:90) S e ( ψ ) e Φ + 32 κ (cid:90) S ψe Φ + (cid:90) S
12 (3ˇ κ − ϑ ) ψ ) e Φ as desired. In what follows we prove the stronger estimates in terms of powers of r for the quantitiesˇ κ, ˇ µ, ϑ, ζ, ˇ κ, β, ˇ ρ . More precisely we establish the following. Proposition 7.4.5.
The following estimates hold true in ( ext ) M for all k ≤ k small + 20 (cid:13)(cid:13) ˇ κ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:13)(cid:13) ˇ µ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec . (7.4.5) Also, for all k ≤ k small + 18 (cid:13)(cid:13) ϑ, ζ, ˇ κ, r ˇ ρ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − (2 r + u ) − / − δ dec , (cid:13)(cid:13) e β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , (cid:13)(cid:13) e θ K (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec . (7.4.6) Remark 7.4.6.
Note that in fact β admits the stronger estimate, for all k ≤ k small + 18 , (cid:13)(cid:13) β (cid:107) ∞ ,k (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − / − δ extra . (7.4.7)80 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) ˇ κ, ˇ µ in ( ext ) M Step 1.
We prove the following estimates for ˇ κ in ( ext ) M . (cid:13)(cid:13) ˇ κ (cid:107) ∞ ,k (cid:46) (cid:15) r − / u − − δ dec , k ≤ k small + 20 . (7.4.8)We make use of the equation e (ˇ κ ) + κ ˇ κ = F := −
12 ˇ κ −
12 ˇ κ − (cid:16) ϑ − ϑ (cid:17) . In view of our assumptions
Ref1-2 and Lemma 7.1.6 (cid:13)(cid:13)(cid:13) F (cid:13)(cid:13) ∞ ,k ( u, λ ) (cid:46) (cid:15) λ − / u − − δ dec . Applying Proposition 7.4.3 we deduce, r (cid:107) ˇ κ (cid:107) ∞ ,k ( u, r ) (cid:46) r ∗ (cid:107) ˇ κ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) u − − δ dec (cid:90) r ∗ r λ λ − / dλ (cid:46) r ∗ (cid:107) ˇ κ (cid:107) ∞ ( u, r ∗ ) + (cid:15) r − / u − − δ dec . In view of the control on the last slice we infer that, everywhere in ( ext ) M , (cid:107) ˇ κ (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:15) r − / u − − δ dec . Step 2.
We prove the estimate, (cid:13)(cid:13) ˇ µ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 20 . (7.4.9)Recall that we have e (ˇ µ ) + 32 κ ˇ µ = − µ ˇ κ + F,F : = −
32 ˇ µ ˇ κ + 12 ˇ µ ˇ κ + Err[ e ˇ µ ] − Err[ e ˇ µ ] , Err[ e ˇ µ ] = − κϑ − ϑ d (cid:63) / ζ − ϑζ + (cid:18) e θ ( κ ) − β + 32 κζ (cid:19) ζ. In view of Lemma 7.1.6 we check, (cid:107) F (cid:107) ∞ ,k ( u, λ ) (cid:46) (cid:15) λ − / u − − δ dec . Applying Proposition 7.4.3 and the estimates on the last slice for ˇ µ we deduce r (cid:107) (cid:101) d k ˇ µ (cid:107) ∞ ,k ( u, λ ) (cid:46) r ∗ (cid:107) (cid:101) d k ˇ µ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) u − − δ dec (cid:90) r ∗ r λ λ − / (cid:46) r ∗ (cid:107) (cid:101) d k ˇ µ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) u − − δ dec r − / (cid:46) (cid:15) u − − δ dec from which the desired estimate (7.4.9) follows. .4. CONTROL IN ( EXT ) M , PART I (cid:96) = 1 modes in ( ext ) M We extend the validity of Lemma 7.3.8 to the entire region ( ext ) M . Lemma 7.4.7.
The following estimates hold true on ( ext ) M or all k ≤ k small + 19 , (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S βe Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k ( u, r ) (cid:46) (cid:15) r − u − − δ dec , (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S ζe Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k ( u, r ) (cid:46) (cid:15) ru − − δ dec , (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S e θ ( ρ ) e Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k ( u, r ) (cid:46) (cid:15) r − u − − δ dec , (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S e θ ( κ ) e Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k ( u, r ) (cid:46) (cid:15) u − − δ dec , (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S βe Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k ( u, r ) (cid:46) (cid:15) u − − δ dec . (7.4.10) Proof.
We first note that the estimate for the (cid:96) = 1 mode of ˇ µ is an immediate consequenceof the estimate (7.4.9). To prove the remaining estimates we proceed in steps as follows. Step 1.
Observe that the estimates of Lemma 7.3.8 remain valid when we replaces thenorms (cid:107) (cid:107) ∗∞ ,k by (cid:107) (cid:107) ∞ ,k . To show this it suffices to prove estimates for re of all (cid:96) = 1modes. This can easily be achieved with the help of Lemma 7.4.4 and our e transportequations for ζ, ˇ ρ, ˇ µ, ˇ κ, β . Step 2.
We establish the estimate, (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S βe Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 20 . (7.4.11)In view of (7.4.4) and the Bianchi identity for e ( β ) e (cid:18)(cid:90) S βe Φ (cid:19) = (cid:90) S e βe Φ + 32 κ (cid:90) S βe Φ + (cid:90) S
12 (3ˇ κ − ϑ ) β ) e Φ = (cid:90) S ( − κβ + d (cid:63) / α + ζα ) e Φ + 32 κ (cid:90) S βe Φ + (cid:90) S
12 (3ˇ κ − ϑ ) β ) e Φ = − κ (cid:90) S βe Φ + (cid:90) S (cid:18) ζα + 12 ( − ˇ κ + ϑ ) β (cid:19) e Φ , CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) and hence e (cid:18)(cid:90) S βe Φ (cid:19) + κ (cid:90) S βe Φ = (cid:90) S (cid:18) ζα + 12 ( − ˇ κ + ϑ ) β (cid:19) e Φ . (7.4.12)Recall that (cid:12)(cid:12) ( α, β ) (cid:12)(cid:12) (cid:46) (cid:15)r − (2 r + u ) − / − δ dec . We deduce, (cid:12)(cid:12)(cid:12) e (cid:18) r (cid:90) S βe Φ (cid:19) (cid:12)(cid:12)(cid:12) (cid:46) r(cid:15) r − u − / − δ dec r − (2 r + u ) − / − δ dec (cid:90) S | e Φ | (cid:46) (cid:15) r − u − / − δ dec (2 r + u ) − / − δ dec (cid:46) (cid:15) r − − δ u − − δ dec i.e., in view of the estimate on Σ ∗ , everywhere on ( ext ) M , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec . (7.4.13)Commuting with T , d / and re we also easily deduce, (cid:13)(cid:13)(cid:13)(cid:101) d k ( re ) k (cid:90) S βe Φ (cid:13)(cid:13)(cid:13) ∞ (cid:46) (cid:15)r − u − − δ dec , ∀ k + k ≤ k small + 20from which (7.4.11) follows. Step 3.
We prove the estimate, (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S ζe Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r / u − − δ dec , k ≤ k small + 19 (7.4.14)which is better than the desired estimate in Lemma 7.4.7. This follows, as for the corre-sponding estimate on Σ ∗ , by projecting the Codazzi equation for ϑ on the (cid:96) = 1 mode (cid:90) S ζe Φ = r (cid:18) (cid:90) S βe Φ − (cid:90) S e θ ( κ ) e Φ − (cid:90) S ϑζe Φ − (cid:90) S (cid:18) κ − r (cid:19) ζe Φ (cid:19) . Note that in view of the estimates for ˇ κ in (7.4.8) already established we have, (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S e θ ( κ ) e Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − / u − − δ dec , k ≤ k small + 19 . Note that the estimate for ˇ κ is stronger in powers of r than the corresponding bootstrap assumption. .4. CONTROL IN ( EXT ) M , PART I (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S βe Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 19 . Thus, since Υ is bounded away from zero in ( ext ) M , we easily deduce (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S ζe Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r / u − − δ dec , k ≤ k small + 19 . Step 4.
We prove the estimate, (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S e θ ( ρ ) e Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 19 . (7.4.15)We proceed as in Step 4 of the proof of Lemma 7.3.8. In view of the definition of µ andthe identity d (cid:63) / d/ = d/ d (cid:63) / + 2 K we write, (cid:90) S e θ ( ρ ) e Φ = − (cid:90) S e θ ( µ ) e Φ + (cid:90) S d (cid:63) / d/ ζe Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ = − (cid:90) S e θ ( µ ) e Φ + (cid:90) S ( d/ d (cid:63) / + 2 K ) ζe Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ = − (cid:90) S e θ ( µ ) e Φ + 2 r (cid:90) S ζe Φ + 2 (cid:90) S (cid:18) K − r (cid:19) ζe Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ . Together with the above estimate for the (cid:96) = 1 mode of ζ , the estimate (7.4.9) for ˇ µ andthe bootstrap assumptions, we infer that (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S e θ ( ρ ) e Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec . Step 5.
We prove the estimate, (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S e θ ( κ ) e Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) u − − δ dec , k ≤ k small + 19 . (7.4.16)As in the corresponding estimate on the last slice we make use of the remarkable identityfor the (cid:96) = 1 mode of e θ ( K ), i.e. (cid:90) S e θ ( ρ ) e Φ + 14 (cid:90) S e θ ( κκ ) e Φ − (cid:90) S e θ ( ϑϑ ) e Φ = 0 . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
We infer (cid:90) S e θ ( κ ) e Φ = − r (cid:90) S e θ ( ρ ) e Φ − r (cid:90) S κe θ ( κ ) e Φ + r (cid:90) S e θ ( ϑϑ ) e Φ − r (cid:90) S (cid:18) κ − r (cid:19) e θ ( κ ) e Φ . The estimate (7.4.16) follows easily from with the above estimate for the (cid:96) = 1 mode of e θ ( ρ ), the estimate for ˇ κ in (7.4.8) and the bootstrap assumptions. Step 6.
We prove the estimate, (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S βe Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) u − − δ dec , k ≤ k small + 19 . (7.4.17)Projecting the Codazzi for ϑ on the (cid:96) = 1 mode, we have − (cid:90) S βe Φ + (cid:90) S e θ ( κ ) e Φ − (cid:90) S κζe Φ + (cid:90) S ϑζe Φ = 0and hence (cid:90) S βe Φ = 12 (cid:90) S e θ ( κ ) e Φ + Υ r (cid:90) S ζe Φ − (cid:90) S (cid:18) κ + 2Υ r (cid:19) ζe Φ + 12 (cid:90) S ϑζe Φ . The desired estimate follows easily in view of the above estimates for the (cid:96) = 1 mode of e θ ( κ ), the (cid:96) = 1 mode of ζ and the bootstrap assumptions. We prove the second part of Proposition 7.4.5, i.e. we prove for all k ≤ k small + 18 (cid:13)(cid:13) ϑ, ζ, ˇ κ, r ˇ ρ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − (2 r + u ) − / − δ dec , (cid:13)(cid:13) e β (cid:13)(cid:13) k, ∞ (cid:46) (cid:15) r − u − / − δ dec , (cid:13)(cid:13) e θ K (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec . (7.4.18)We also prove the stronger estimate for β (see Remark 7.4.6) (cid:13)(cid:13) β (cid:107) ∞ ,k (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − / − δ extra . (7.4.19) .4. CONTROL IN ( EXT ) M , PART I Proof.
We proceed in steps as follows.
Step 1.
We derive the estimate, (cid:13)(cid:13) ϑ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , ∀ k ≤ k small + 19 , (7.4.20)with the help of the equation e ϑ + κϑ = F := − α − ˇ κϑ and the corresponding estimateon the last slice.Note that, (cid:13)(cid:13) α (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − − δ (2 r + u ) − / − δ dec where δ > δ < δ extra − δ dec . Thus, using also the product estimatesof Lemma 7.1.6, we easily check that, (cid:107) F (cid:107) ∞ ,k (cid:46) (cid:15) r − − δ u − / − δ dec + (cid:15) r − / u − − δ dec , k ≤ k small + 20 . Making use of Proposition 7.4.3 we deduce, for all k ≤ k small + 19, r (cid:107) d k ϑ (cid:107) ∞ ,k ( u, r ) (cid:46) r ∗ (cid:107) d k ϑ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) u − / − δ dec (cid:90) r ∗ r λ − − δ dλ. Thus, in view of the results on the last slice Σ ∗ , we deduce, (cid:107) d k ϑ (cid:107) ∞ ( u, r ) (cid:46) r − u − / − δ dec . Step 2.
We derive the estimate, (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − (2 r + u ) − / − δ dec , ∀ k ≤ k small + 19 . (7.4.21)We proceed exactly as in the estimates for β on the last slice Σ ∗ by making use of theBianchi identity e α + (cid:0) κ − ω (cid:1) α = − d (cid:63) / β − ϑρ + 5 ζβ , from which we deduce, (cid:107) d (cid:63) / β (cid:107) ∞ ,k − (cid:46) (cid:13)(cid:13) e α (cid:13)(cid:13) ∞ ,k − + r − (cid:13)(cid:13) α (cid:13)(cid:13) ∞ ,k − + r − (cid:13)(cid:13) ϑ (cid:107) ∞ ,k − + (cid:15) r − u − − δ dec . Thus, in view of the above estimate for ϑ and Ref 2 for α , (cid:107) d (cid:63) / β (cid:107) ∞ ,k − (cid:46) (cid:15) r − (2 r + u ) − / − δ dec + (cid:15) r − u − / − δ dec . On the other hand we have, according to (7.4.11), (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) S βe Φ (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 20 . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Estimate (7.4.21) follows then easily, according to the part 4 of the elliptic Hodge Lemma7.1.7.As mentioned in Remark 7.4.6 we can prove a stronger estimate for β . Indeed we have,in view of Ref 2. | α | (cid:46) log(1 + u ) r − (2 r + u ) − / − δ extra , | e α | (cid:46) r − (2 r + u ) − / − δ extra . Hence, using the equation e α + (cid:0) κ − ω (cid:1) α = − d (cid:63) / β − ϑρ + 5 ζβ , (cid:13)(cid:13) d (cid:63) / β (cid:107) ∞ ,k (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − / − δ extra + (cid:15) r − u − / − δ dec . According to Lemma 7.1.7 (cid:107) β (cid:107) h k +1 ( S ) (cid:46) r (cid:107) d (cid:63) / β (cid:107) h k ( S ) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e Φ β (cid:12)(cid:12)(cid:12)(cid:12) and thus, in view of the estimate (7.4.11) for the (cid:96) = 1 mode of β , (cid:107) β (cid:107) h k +1 ( S ) (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − / − δ extra + (cid:15) r − u − − δ dec (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − / − δ extra . The estimates for the T and e derivatives are derived in the same manner. and hence, (cid:13)(cid:13) β (cid:107) ∞ ,k (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − / − δ extra , ∀ k ≤ k small + 19 . (7.4.22)This improvement is needed in the next step. Step 3.
We derive the estimate (cid:13)(cid:13) ζ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , ∀ k ≤ k small + 19 . (7.4.23)For this we make use of the transport equation for ζ , e ζ + κζ = F := − β + Γ g · Γ g and the improved estimate for β in the previous step. Thus, making use of the productLemma 7.1.6, (cid:13)(cid:13) F (cid:13)(cid:13) ∞ ,k (cid:46) (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k + (cid:15) r − / u − − δ dec (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − / − δ extra + (cid:15) r − / u − − δ dec (cid:46) (cid:15) r − − δ u − / − δ dec + (cid:15) r − / u − − δ dec . .4. CONTROL IN ( EXT ) M , PART I r (cid:107) d k ζ (cid:107) ∞ ,k ( u, r ) (cid:46) r ∗ (cid:107) d k ζ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) u − / − δ dec (cid:90) r ∗ r λ − − δ dλ. Thus, in view of the estimates on the last slice, r (cid:107) d k ζ (cid:107) ∞ ( u, r ) (cid:46) (cid:15) u − / − δ dec , k ≤ k small + 19as desired. Step 4.
We derive the estimate (cid:13)(cid:13) ˇ ρ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , ∀ k ≤ k small + 18 . (7.4.24)We make use of the definition of µ from which we infer that,ˇ µ = − d/ ζ − ˇ ρ + Γ g · Γ b . Hence, in view of the product Lemma and the estimates already derived, for all k ≤ k small + 18, (cid:13)(cid:13) ˇ ρ (cid:13)(cid:13) ∞ ,k (cid:46) r − (cid:13)(cid:13) ζ (cid:13)(cid:13) ∞ ,k +1 + (cid:13)(cid:13) ˇ µ (cid:13)(cid:13) ∞ ,k + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − u − / − δ dec as desired. Step 5.
We derive the estimate (cid:13)(cid:13) ˇ κ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , ∀ k ≤ k small + 18 . (7.4.25)We make use of the equation e ˇ κ + 12 κ ˇ κ = F := − d/ ζ −
12 ˇ κκ + 2 ˇ ρ + Γ g · Γ b . In view of the previously derived estimates, (cid:13)(cid:13) F (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , k ≤ k small + 18 . Making use of Proposition 7.4.3 we deduce, for all k ≤ k small + 18, r (cid:107) d k ˇ κ (cid:107) ∞ ,k ( u, r ) (cid:46) r ∗ (cid:107) d k ˇ κ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) u − / − δ dec (cid:90) r ∗ r λ − dλ (cid:46) r ∗ (cid:107) d k ˇ κ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) r − u − / − δ dec . Thus, in view of the estimates on the last slice, r (cid:107) d k ˇ κ (cid:107) ∞ ( u, r ) (cid:46) (cid:15) ( r ∗ ) − u − / − δ dec + (cid:15) r − u − / − δ dec from which the desired estimate easily follows.88 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Step 6.
We derive the estimate (cid:13)(cid:13) e β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , ∀ k ≤ k small + 18 . (7.4.26)making use of the equation e β + ( κ − ω ) β = − d (cid:63) / ρ + 3 ζρ + Γ g β + Γ b α and the estimatesderived above for β , d (cid:63) / ρ , ζ . Hence, (cid:13)(cid:13) e β (cid:13)(cid:13) ∞ ,k (cid:46) r − (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k + (cid:13)(cid:13) d (cid:63) / ρ (cid:13)(cid:13) ∞ ,k + r − (cid:13)(cid:13) ζ (cid:13)(cid:13) ∞ ,k + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − u − / − δ dec . Step 7.
As a corollary of the above estimates (see also
Ref 4 ) we also derive, in ( ext ) M , (cid:13)(cid:13) K − K (cid:13)(cid:13) ∞ ,k − (cid:46) (cid:15) r − u − / − δ dec , k ≤ k small + 19 , (cid:13)(cid:13) K − r (cid:13)(cid:13) ∞ ,k − (cid:46) (cid:15) r − u − / − δ dec , k ≤ k small + 19 . (7.4.27)In view of the definition of K we have, e θ ( K ) = − e θ (cid:18) ˇ ρ − κ ˇ κ − κ ˇ κ + 14 ϑϑ (cid:19) . Thus, in view of the above estimates, (cid:13)(cid:13) e θ K (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec from which the desired estimate easily follows. ( ext ) M , Part II We derive the crucial decay estimates which imply, in particular, decay of order u − − δ dec for all quantities in Γ and ˇ R (except ξ, ˇ ω, ˇΩ which will be treated separately) in theinterior. More precisely we prove the following, Proposition 7.5.1.
The following estimates hold true in ( ext ) M , for all k ≤ k small + 8 . (cid:13)(cid:13) ϑ, ζ, η, ˇ κ, ϑ, rβ, r ˇ ρ, rβ, α (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec . (7.5.1)To prove the proposition we make use of the fact that we already have good decay esti-mates in terms of powers of u for ˇ κ, ˇ µ . We also derive below decay estimates for variousrenormalized quantities. .5. CONTROL IN ( EXT ) M , PART II η We start with the following simple estimate for η in terms of ζ . (cid:107) η (cid:107) ∞ ,k (cid:46) (cid:107) ζ (cid:107) ∞ ,k + (cid:15) r − u − − δ dec , k ≤ k small + 17 . (7.5.2)This can be derived by propagation from the last slice with the help of the equation, e ( η − ζ ) + 12 κ ( η − ζ ) = − ϑ ( η − ζ ) = Γ g · Γ b . Note that (cid:107) Γ g · Γ b (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec . Thus making use of Proposition 7.4.3 we deduce r (cid:107) η − ζ (cid:107) ∞ ,k ( u, r ) (cid:46) r ∗ (cid:107) η − ζ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:90) r ∗ r λ (cid:107) Γ g · Γ b (cid:107) ∞ ,k ( u, λ ) (cid:46) r ∗ (cid:107) η − ζ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) u − − δ dec with r ∗ the value of r on C ( u ) ∩ Σ ∗ . On the last slice we have derived the estimates,recorded in Proposition 7.3.5 and Proposition 7.4.5 (cid:107) η (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) ζ (cid:107) ∗∞ ,k (cid:46) (cid:15) r − u − / − δ dec . In view of the dominance condition on r on Σ ∗ we deduce, (cid:107) η − ζ (cid:107) ∗∞ ,k ( u, r ) (cid:46) (cid:15) r − u − − δ dec and therefore also, r ∗ (cid:107) η − ζ (cid:107) ∞ ,k ( u, r ∗ ) (cid:46) (cid:15) u − − δ dec . Therefore, r (cid:107) η (cid:107) ∞ ,k ( u, r ) (cid:46) r (cid:107) ζ (cid:107) ∞ ,k ( u, r ) + (cid:15) u − − δ dec as desired.90 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
We start with the following lemma.
Lemma 7.5.2.
The s ( M ) reduced tensor Ξ : = r (cid:0) e θ ( κ ) + 4 r d (cid:63) / d/ ζ − r d (cid:63) / d/ β (cid:1) (7.5.3) verifies in ( ext ) M the estimate, (cid:13)(cid:13) Ξ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) u − − δ dec , ∀ k ≤ k small + 13 . (7.5.4) Proof.
To calculate e Ξ we make use of the equations, e ( κ ) + 12 κκ = − d/ ζ + 2 ρ − ϑϑ + 2 ζ ,e ζ + κζ = − β − ϑζ,e β + 2 κβ = d/ α + ζα. Since we already have an estimate for ˇ µ we re-express ρ = − µ − d/ ζ + ϑϑ and derive, e ( κ ) + 12 κκ = − µ − d/ ζ + 2 ζ . Commuting with d (cid:63) / and making use of [ d (cid:63) / , e ] = ( κ + ϑ ) d (cid:63) / we derive, e ( d (cid:63) / κ ) + κ d (cid:63) / κ + 12 κ d (cid:63) / κ = − d (cid:63) / ˇ µ − d (cid:63) / d/ ζ + 2 d (cid:63) / ( ζ ) + ϑ d (cid:63) / κ. (7.5.5)Hence, since e ( r ) = r κ , e ( r d (cid:63) / κ ) = r ( κ − κ ) d (cid:63) / κ − r κ d (cid:63) / ˇ κ − r d (cid:63) / d/ ζ − r d (cid:63) / ˇ µ + r ( d (cid:63) / ( ζ ) + ϑ d (cid:63) / κ )= − r κ d (cid:63) / ˇ κ − r d (cid:63) / d/ ζ + Err where, Err : = − r κ d (cid:63) / ˇ κ − r d (cid:63) / ˇ µ + r (cid:0) ˇ κ d (cid:63) / κ + d (cid:63) / ( ζ ) + ϑ d (cid:63) / κ (cid:1) . In view of the estimate already established for , ˇ κ, ˇ µ and the product Lemma 7.1.6 wecheck, (cid:13)(cid:13) Err (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 19 . To simplify notation we introduce the following. .5. CONTROL IN ( EXT ) M , PART II Definition 7.5.3.
We say that a quantity ψ ∈ s k ( M ) is r − p Good a provided that it verifiesthe estimate, everywhere in ( ext ) M , (cid:13)(cid:13) ψ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − p u − − δ dec , ∀ k ≤ k small + a. (7.5.6)Using this notation we write, e ( r d (cid:63) / κ ) = − r d (cid:63) / d/ ζ + r − Good . (7.5.7)Using the same notation the transport equation for ζ can be written in the form, e ζ + κζ = − β − ϑζ − ˇ κζ = − β + r − / Good . Commuting with ( r d (cid:63) / )( r d/ ) (making us of Lemma 7.4.1) we derive e ( r d (cid:63) / d/ ζ ) + κ ( r d (cid:63) / d/ ζ ) = − r d (cid:63) / d/ β + r − / Good . Since e ( r ) = rκ we deduce, e ( r d (cid:63) / d/ ζ ) = − κr d (cid:63) / d/ ζ − r d (cid:63) / d/ β + r − / Good . (7.5.8)Similarly the transport equation for β takes the form e β + 2 κβ = d/ α + ζα − κβ = d/ α + r − / Good and, e ( r d (cid:63) / d/ β ) + 2 κr d (cid:63) / d/ β = r d (cid:63) / d/ d/ α + r − / Good . As before, since e ( r ) = rκ , we deduce, e ( r d (cid:63) / d/ β ) = − κr d (cid:63) / d/ β + r d (cid:63) / d/ d/ α + r − / Good . (7.5.9)Combining (7.5.7)–(7.5.9) we deduce, e Ξ = e (cid:104) r (cid:0) − d (cid:63) / κ + 4 r d (cid:63) / d/ ζ − r d (cid:63) / d/ β (cid:1) (cid:105) = 4 r d (cid:63) / d/ ζ + 4 (cid:18) − κr d (cid:63) / d/ ζ − r d (cid:63) / d/ β (cid:19) − (cid:0) − κr d (cid:63) / d/ β + r d (cid:63) / d/ d/ α (cid:1) + r − Good = − (cid:18) κ − r (cid:19) r d (cid:63) / d/ ζ + 2 r (cid:18) κ − r (cid:19) d (cid:63) / d/ β − r d (cid:63) / d/ d/ α + r − Good . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Making use of
Ref 4 estimates for κ − r and the estimates for α in Ref 2 , i.e., r | d (cid:63) / d/ d/ α | (cid:46) (cid:15) r − (2 r + u ) − − δ extra (cid:46) (cid:15) r − − δ u − − δ extra + δ , < δ < δ extra i.e., r d (cid:63) / d/ d/ α = r − − δ Good we thus deduce, e Ξ = r − − δ Good . We deduce, (cid:107) Ξ (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:107) Ξ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) u − − δ dec (cid:90) r ∗ r λ − − δ dλ, ∀ k ≤ k small + 13 . In view of the estimates on the last slice it is easy to check that (cid:107) Ξ (cid:107) ∞ ,k ( u, r ∗ ) (cid:46) (cid:15) u − − δ dec , ∀ k ≤ k small + 13 . Indeed, on the last slice, (cid:107) d (cid:63) / κ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , (cid:107) d (cid:63) / d/ ζ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , (cid:107) d (cid:63) / d/ β (cid:107) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec . Hence, since r (cid:29) u on Σ ∗ , (cid:107) Ξ (cid:107) ∞ ,k ( u, r ∗ ) (cid:46) (cid:15) r − u − / − δ dec (cid:46) (cid:15) u − − δ dec . Thus everywhere on ( ext ) M , (cid:107) Ξ (cid:107) ∞ ,k (cid:46) (cid:15) u − − δ dec , ∀ k ≤ k small + 13 (7.5.10)as desired.In the following lemma, we make use of the control we have already established for q , α, α, ˇ κ, ˇ µ in ( ext ) M to derive two nontrivial relations between angular derivatives of ζ, ˇ κ and β . Remark 7.5.4.
According to Theorem M3 we only have good estimates for α along T andon the last slice Σ ∗ . To keep track of this fact we denote by r − p Good a ( α ) those r − p Good a terms which depend linearly on α and their derivatives. .5. CONTROL IN ( EXT ) M , PART II Lemma 7.5.5.
Let
A, B be the operators A := d (cid:63) / d/ − ρ , B = d (cid:63) / d/ + 2 K . The followingidentities hold true, AB d (cid:63) / ζ − κ ρ d (cid:63) / e θ ( κ ) ∈ r − Good ( α ) A B d (cid:63) / β + 98 (cid:0) κ ρ (cid:1) d (cid:63) / e θ ( κ ) ∈ r − Good ( α ) (7.5.11) Proof.
In view of the improved control for α in Theorem M2, α in Theorem M3, and q in Theorem M1, the bootstrap assumptions and product lemma, and the control we havealready derived for ˇ κ and ˇ µ in ( ext ) M , we obtain d/ ϑ + 2 β − κζ ∈ r − Good , Codazzi and control of ˇ κ,d/ ϑ + 2 β − e θ ( κ ) + κζ ∈ r − Good , Codazzi ,d (cid:63) / β + 32 ρϑ ∈ r − Good , Bianchi and control of α,d (cid:63) / β + 32 ρϑ ∈ r − Good ( α ) , Bianchi and control of α,d (cid:63) / d (cid:63) / ρ + 34 κ ρϑ + 34 κ ρϑ ∈ r − Good , (7.1.28) and control of q , (7.5.12)where we used Codazzi for the two first inequalities, Bianchi for the third and fourthinequalities, the definition of µ for the fifth one, and the identity relating q and d (cid:63) / d (cid:63) / ρ forthe last one.Combining the first statement with the third and the second with the fourth we inferthat, ( d (cid:63) / d/ − ρ ) ϑ − κ d (cid:63) / ζ ∈ r − Good , ( d (cid:63) / d/ − ρ ) ϑ − d (cid:63) / e θ ( κ ) + κ d (cid:63) / ζ ∈ r − Good ( α ) , or, setting A := d (cid:63) / d/ − ρ,Aϑ − κ d (cid:63) / ζ ∈ r − Good ,Aϑ − d (cid:63) / e θ ( κ ) + κ d (cid:63) / ζ ∈ r − Good ( α ) . (7.5.13)From the fifth equations we deduce, A (cid:18) d (cid:63) / d (cid:63) / ρ + 34 κ ρϑ + 34 κ ρϑ (cid:19) ∈ r − Good CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) i.e.,
A d (cid:63) / d (cid:63) / ρ + 34 κ ρ Aϑ + 34 κ ρ Aϑ ∈ r − Good . Making use of (7.5.13) we deduce,
A d (cid:63) / d (cid:63) / ρ + 34 κ ρ (cid:0) κ d (cid:63) / ζ (cid:1) + 34 κ ρ (cid:0) d (cid:63) / e θ ( κ ) − κ d (cid:63) / ζ (cid:1) ∈ r − Good ( α ) . Hence, simplifying,
A d (cid:63) / d (cid:63) / ρ + 34 κ ρ d (cid:63) / e θ ( κ ) ∈ r − Good ( α ) . (7.5.14)Next, in view of the identity d (cid:63) / d/ = d/ d (cid:63) / + 2 K ,( d (cid:63) / d/ + 2 K ) d (cid:63) / ζ = d (cid:63) / d/ d (cid:63) / ζ + 2 K d (cid:63) / ζ = d (cid:63) / ( d/ d (cid:63) / ζ + 2 Kζ ) − d (cid:63) / Kζ = d (cid:63) / d (cid:63) / d/ ζ + r − / Good . Recalling the definition of µ = − d/ ζ − ρ + ϑϑ and the product Lemma we write d (cid:63) / d/ ζ = − d (cid:63) / µ − d (cid:63) / ρ + 14 d (cid:63) / ( ϑϑ ) = − d (cid:63) / µ − d (cid:63) / ρ + r − Good . In view of the estimates for ˇ µ we have already established we deduce, d (cid:63) / d/ ζ = − d (cid:63) / ρ + r − Good . Thus, ( d (cid:63) / d/ + 2 K ) d (cid:63) / ζ = − d (cid:63) / d (cid:63) / ρ + r − Good . (7.5.15)Therefore, making use of (7.5.14) A ( d (cid:63) / d/ + 2 K ) d (cid:63) / ζ = − A d (cid:63) / d (cid:63) / ρ + r − Good ( α )= 34 κ ρ d (cid:63) / e θ ( κ ) + r − Good ( α )i.e., A ( d (cid:63) / d/ + 2 K ) d (cid:63) / ζ − κ ρ d (cid:63) / e θ ( κ ) = r − Good ( α ) (7.5.16)as desired. .5. CONTROL IN ( EXT ) M , PART II d (cid:63) / d/ + 2 K ) Aϑ = κ ( d (cid:63) / d/ + 2 K ) d (cid:63) / ζ + r − Good ( α ) . Hence applying A and making use of (7.5.16), A ( d (cid:63) / d/ + 2 K ) Aϑ = κA ( d (cid:63) / d/ + 2 K ) d (cid:63) / ζ + r − Good ( α )= 34 κ ρ d (cid:63) / e θ ( κ ) + r − Good ( α ) . Finally, making use of the relation d (cid:63) / β + ρ ϑ ∈ r − Good , we have A ( d (cid:63) / d/ + 2 K ) d (cid:63) / β = A ( d (cid:63) / d/ + 2 K ) A d (cid:63) / β + r − Good ( α )= − ρA ( d (cid:63) / d/ + 2 K ) Aϑ + r − Good ( α )= − ρ (cid:18) κ ρ d (cid:63) / e θ ( κ ) + r − Good ( α ) (cid:19) + r − Good = − κ ρ d (cid:63) / e θ ( κ ) + r − Good ( α )as desired. This concludes the proof of the lemma. Corollary 7.5.6.
The s ( M ) tensor e θ ( κ ) = − d (cid:63) / ˇ κ verifies the following fifth order ellipticequation in ( ext ) M A d (cid:63) / ( e θ κ ) − mr A d (cid:63) / e θ ( κ ) + 36 m r d (cid:63) / e θ ( κ ) ∈ r − Good ( α ) . (7.5.17) Proof.
According to Lemma 7.5.5
AB d (cid:63) / ζ − κ ρ d (cid:63) / e θ ( κ ) ∈ r − Good ( α ) ,A B d (cid:63) / β + 98 (cid:0) κ ρ (cid:1) d (cid:63) / e θ ( κ ) ∈ r − Good ( α ) , we have A B d (cid:63) / ζ = 34 κ ρ A d (cid:63) / e θ ( κ ) + r − Good ( α ) ,A B d (cid:63) / β = − (cid:0) κ ρ (cid:1) d (cid:63) / e θ ( κ ) + r − Good ( α ) . (7.5.18)In view of Lemma 7.5.2 we have on ( ext ) M , e θ ( κ ) + 4 r d (cid:63) / d/ ζ − r d (cid:63) / d/ β ∈ r − Good . (7.5.19)96 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Thus, A d (cid:63) / (cid:16) e θ ( κ ) + 4 r d (cid:63) / d/ ζ − r d (cid:63) / d/ β (cid:17) ∈ r − Good . Making use of d (cid:63) / d (cid:63) / d/ = d (cid:63) / (cid:16) d/ d (cid:63) / + 2 K (cid:17) = ( d (cid:63) / d/ + 2 K ) d (cid:63) / − e θ ( K ) , we deduce, A d (cid:63) / ( e θ κ ) = − rA d (cid:63) / d (cid:63) / d/ ζ + 2 r A d (cid:63) / d (cid:63) / d/ β + r − Good = − rA ( d (cid:63) / d/ + 2 K ) d (cid:63) / ζ + 2 r A ( d (cid:63) / d/ + 2 K ) d (cid:63) / β + r − Good = − rA B d (cid:63) / ζ + 2 r A B d (cid:63) / β + r − Good . Thus, in view of the lemma, A d (cid:63) / ( e θ κ ) = − r (cid:0) κ ρA d (cid:63) / e θ ( κ ) + r − Good (cid:1) − r (cid:16)(cid:0) κ ρ (cid:1) d (cid:63) / e θ ( κ ) + r − Good ( α ) (cid:17) + r − Good . We deduce, A d (cid:63) / e θ κ + 3 r ( κ ρ ) A d (cid:63) / e θ ( κ ) + 94 r ( κ ρ ) d (cid:63) / e θ ( κ ) ∈ r − Good ( α ) . Finally, A d (cid:63) / e θ κ − mr A d (cid:63) / e θ ( κ ) + 36 m r d (cid:63) / e θ ( κ ) ∈ r − Good ( α )as desired. Lemma 7.5.7.
We have the following Poincar´e inequality on ( ext ) M for f ∈ s ( M ) with A = ( d (cid:63) / d/ − ρ ) (cid:90) S f (cid:18) A − mr A + 36 m r (cid:19) f ≥ r (cid:90) S ( d/ f ) + 9 r (cid:90) S f . Proof.
Recall that we have the following Poincar´e inequality for d/ (cid:90) S ( d/ f ) ≥ (cid:90) S Kf . .5. CONTROL IN ( EXT ) M , PART II (cid:12)(cid:12) K − r − (cid:12)(cid:12) (cid:46) (cid:15)r − , (cid:90) S f Af = (cid:90) S f ( d (cid:63) / d/ − ρ ) f ≥ (cid:90) S (4 K − ρ ) f = (cid:18) r + 6 mr + O ( r − (cid:15) ) (cid:19) (cid:90) S f . Since A is positive self-adjoint, (cid:90) S f A f = (cid:90) S ( A / f ) A ( A / f ) = (cid:18) r + 6 mr + O ( r − (cid:15) ) (cid:19) (cid:90) S | A / f | = (cid:18) r + 6 mr + O ( r − (cid:15) ) (cid:19) (cid:90) S f Af This yields (cid:90) S f (cid:18) A f − mr Af (cid:19) = (cid:18) r + 6 mr − mr + O ( r − (cid:15) ) (cid:19) (cid:90) S f Af = (cid:18) r − mr + O ( r − (cid:15) ) (cid:19) (cid:90) S f Af, and therefore, (cid:90) S f (cid:18) A − mr A + 36 m r (cid:19) ≥ (cid:18) r − mr + O ( r − (cid:15) ) (cid:19) (cid:90) S f Af + 36 m r (cid:90) S f = (cid:18) r (cid:18) − m r (cid:19) + O ( r − (cid:15) ) (cid:19) (cid:90) S f Af + 36 m r (cid:90) S f . Note that for r > m we have, 1 − m r > . We deduce, for sufficiently small (cid:15) , everywhere in ( ext ) M , (cid:90) S f (cid:18) A − mr A + 36 m r (cid:19) > r (cid:18)(cid:90) S f Af + 36 m r (cid:90) S f (cid:19) . Now, since (cid:12)(cid:12)(cid:12) ρ + mr (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r − (cid:90) S f Af = (cid:90) S f ( d (cid:63) / d/ − ρ ) f = (cid:90) S (cid:18) | d/ f | + (cid:18) mr + O ( r − (cid:15) ) (cid:19) | f | (cid:19) . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Hence, (cid:90) S f Af + 36 m r (cid:90) S f > (cid:90) S (cid:16) | d/ f | + (cid:0) mr + 36 m r (cid:1) | f | (cid:17) > (cid:90) S | d/ f | or, since (cid:82) S ( d/ f ) ≥ (cid:82) S r (cid:82) S | f | + O ( (cid:15) r − ) (cid:82) S | f | . We deduce, (cid:90) S f (cid:18) A − mr A + 36 m r (cid:19) ≥ r (cid:90) S ( d/ f ) + 9 r (cid:90) S f as desired. This concludes the proof of the lemma.Applying the lemma to f = d (cid:63) / e θ κ in (7.5.17), i.e. A d (cid:63) / ( e θ κ ) − mr A d (cid:63) / e θ ( κ ) + 36 m r d (cid:63) / e θ ( κ ) ∈ r − Good ( α )or, in any region where (cid:107) α (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 16 , we have (cid:13)(cid:13)(cid:13) A d (cid:63) / ( e θ κ ) − mr A d (cid:63) / e θ ( κ ) + 36 m r d (cid:63) / e θ ( κ ) (cid:13)(cid:13)(cid:13) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 8 . We deduce, by L -elliptic estimates, (cid:107) d (cid:63) / e θ ˇ κ (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 12 . (7.5.20)Since we control the (cid:96) = 1 mode of e θ ˇ κ we infer that, (cid:107) e θ ˇ κ (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 13i.e., (cid:107) ˇ κ (cid:107) ,k (cid:46) (cid:15) u − − δ dec , k ≤ k small + 14 . Therefore, using the Sobolev embedding, (cid:107) ˇ κ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec k ≤ k small + 12 . This proves the following,
Proposition 7.5.8.
In any region of ( ext ) M where, (cid:107) α (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 16 , we also have, (cid:107) ˇ κ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec k ≤ k small + 12 . (7.5.21) .5. CONTROL IN ( EXT ) M , PART II We first prove Proposition 7.5.1 in the region where the estimate (cid:107) α (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 16 , (7.5.22)holds true. Step 1.
We prove the estimates, (cid:107) ζ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 15 , (cid:107) β (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 12 . (7.5.23)According to (7.5.18) A B d (cid:63) / ζ = 34 κ ρ A d (cid:63) / e θ ( κ ) + r − Good ( α ) ,A B d (cid:63) / β = − (cid:0) κ ρ (cid:1) d (cid:63) / e θ ( κ ) + r − Good ( α ) . In view of (7.5.20) we deduce, in L norms, (cid:107) A B d (cid:63) / ζ (cid:107) ,k (cid:46) r − (cid:107) A d (cid:63) / e θ κ (cid:107) ,k + (cid:15) r − u − − δ dec , k ≤ k small + 12 , (cid:107) A B d (cid:63) / β (cid:107) ,k (cid:46) r − (cid:107) d (cid:63) / e θ κ (cid:107) ,k + (cid:15) r − u − − δ dec , k ≤ k small + 9 . Thus, in view of the estimates for ˇ κ derived above, (cid:107) A B d (cid:63) / ζ (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 12 , (cid:107) A B d (cid:63) / β (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 9 . Thus, by elliptic estimates, (cid:107) d (cid:63) / ζ (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 16 , (cid:107) d (cid:63) / β (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 13 . In view of the estimates for the (cid:96) = 1 modes of ζ, β we deduce, (cid:107) ζ (cid:107) ,k (cid:46) (cid:15) u − − δ dec , k ≤ k small + 17 , (cid:107) β (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 14 . Passing to L ∞ norms we derive (cid:107) ζ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 15 , (cid:107) β (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 13 . (7.5.24)00 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Step 2.
We prove the estimate (cid:107) η (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 15 . This follows immediately from the estimate from ζ and the previously derived estimate(7.5.2). Indeed, (cid:107) η (cid:107) ∞ ,k (cid:46) (cid:107) ζ (cid:107) ∞ ,k + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − u − − δ dec . Step 3.
We derive the estimate, (cid:107) ϑ (cid:107) ∞ ,k (cid:46) r − u − − δ dec , k ≤ k small + 11 . (7.5.25)This follows easily in view of the equation (see (7.5.12)) d/ ϑ + 2 β − κζ ∈ r − Good from which, in view of Step 1, (cid:107) d/ ϑ (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 12 . The desired estimate follows by elliptic estimates and Sobolev.
Step 4.
We derive the intermediate estimate for ϑ , (cid:107) ϑ (cid:107) ∞ ,k (cid:46) (cid:15) u − − δ dec , k ≤ k small + 12 . (7.5.26)To show this we combine the equations (see (7.5.12)) d/ ϑ + 2 β − e θ ( κ ) + κζ ∈ r − Good ,d (cid:63) / β + 32 ρϑ ∈ r − Good , to deduce, d (cid:63) / d/ ϑ − ρϑ = d (cid:63) / e θ κ + κ d (cid:63) / ζ + r − Good , and hence, (cid:107) Aϑ (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 12 . Thus, (cid:107) ϑ (cid:107) ,k (cid:46) (cid:15) ru − − δ dec , k ≤ k small + 14 .5. CONTROL IN ( EXT ) M , PART II (cid:107) ϑ (cid:107) ∞ ,k (cid:46) (cid:15) u − − δ dec , k ≤ k small + 12as desired. Step 5.
We derive the estimate, (cid:107) ˇ ρ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 14 . (7.5.27)From, d (cid:63) / d (cid:63) / ρ + 34 κ ρϑ + 34 κ ρϑ ∈ r − Good , we deduce, (cid:107) d (cid:63) / d (cid:63) / ρ (cid:107) ,k (cid:46) r − ( (cid:107) θ (cid:107) ,k + (cid:107) θ (cid:107) ,k ) + (cid:15) r − u − − δ dec , k ≤ k small + 14 (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 14 . Since we control the (cid:96) = 1 mode of d (cid:63) / ρ (see Lemma 7.4.7) we infer that, (cid:107) ˇ ρ (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 16i.e., (cid:107) ˇ ρ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 14as desired. Step 6.
We derive the estimate, (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , ∀ k ≤ k small + 9 (7.5.28)with the help of the identity e ( r q ) = r (cid:40) d (cid:63) / d (cid:63) / d/ β − κρα − ρ d (cid:63) / d (cid:63) / κ − κρ d (cid:63) / ζ + 34 (2 ρ − κκρ ) ϑ (cid:41) + Err[ e ( r q )] , Err[ e ( r q )] = r ( e Γ b ) · d / ≤ β + r Γ b · q + r d / (Γ g · Γ b ) , of Proposition 7.1.11. In view of (7.3.26) we have, (cid:107) Err[ e ( r q )] (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:15) u − − δ dec , k ≤ k small + 16 . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
We can now make use of the estimates for ˇ κ, , ζ, ϑ, ϑ already derived and the
Ref 2 estimate for e ( q ) and α to deduce, for all k ≤ k small + 10, (cid:107) ρ d (cid:63) / d (cid:63) / κ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) κρ d (cid:63) / ζ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) ρ ϑ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) κκϑ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) κρα (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:107) e ( r q ) (cid:107) ∞ ,k (cid:46) (cid:15) u − − δ dec . Therefore, (cid:107) d (cid:63) / d (cid:63) / d/ β (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 10 , i.e., (cid:107) d (cid:63) / d (cid:63) / d/ β (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 10 . Making use of the identity, d (cid:63) / d/ = d/ d (cid:63) / + 2 K, we deduce (cid:13)(cid:13) ( d (cid:63) / d/ + K ) d (cid:63) / β (cid:13)(cid:13) ,k (cid:46) (cid:15) r − u − − δ dec . Since d (cid:63) / d/ + K is coercive we deduce, (cid:13)(cid:13) d (cid:63) / β (cid:13)(cid:13) ,k (cid:46) (cid:15) r − u − − δ dec , ∀ k ≤ k small + 10 . Since we control the (cid:96) = 1 mode of β (see Lemma 7.4.7 ) according to Lemma 7.3.8, (cid:13)(cid:13) β (cid:13)(cid:13) ,k (cid:46) (cid:15) r − u − − δ dec , ∀ k ≤ k small + 11 . Hence, (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , ∀ k ≤ k small + 9 . (7.5.29) Step 7.
Using the above estimate for β we can improve the estimate for ϑ derived inStep 4. We show, in the region where the estimate (7.5.22) for α holds, (cid:107) ϑ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 9 . (7.5.30) .5. CONTROL IN ( EXT ) M , PART II d/ ϑ + 2 β − e θ ( κ ) + κζ ∈ r − Good , we infer that, for all k ≤ k small + 11, (cid:107) d/ ϑ (cid:107) ,k (cid:46) (cid:107) β (cid:107) ,k + r − (cid:107) ˇ κ (cid:107) ,k +1 + r − (cid:107) ζ (cid:107) ,k + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − u − − δ dec . Thus, for all k ≤ k small + 12 (cid:107) ϑ (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec and hence, (cid:107) ϑ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 10 . (7.5.31)This ends the proof of Proposition 7.5.1 in the region for which the desired estimate(7.5.22) for α holds true.Since (7.5.22) for α holds true on T in view of Theorem M3, this ends the proof ofProposition 7.5.1 on T . We extend the validity of Proposition 7.5.1 to all of ( ext ) M propagating the estimatesderived in the first part on T . We also recall that we have good decay estimates for ˇ κ and ˇ µ everywhere on ( ext ) M . Step 1.
We first derive estimates for ϑ in M ext making use of the transport equation e ( ϑ ) + κϑ = − α − ( κ − κ ) ϑ = − α + Γ g · Γ g . Making use of Proposition 7.4.3 we derive, for all r ≥ r = r T , r (cid:107) ϑ (cid:107) ∞ ,k ( u, r ) (cid:46) r (cid:107) ϑ (cid:107) ∞ ,k ( u, r ) + (cid:90) rr λ (cid:107) α (cid:107) ∞ ,k ( u, λ ) dλ + (cid:15) u − − δ dec . We now make use of the estimate, (cid:107) α (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 20 Recall that r is bounded on T and that T ⊂ ( int ) M so that (7.5.22) holds true for ( int ) α on T inview of Theorem M3. Then, since we have ( ext ) α = ( ( ext ) Υ) int ) α on T , (7.5.22) holds indeed true for ( ext ) α on T . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) and, (cid:107) ϑ (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:15) u − − δ dec derived above in (7.5.25), to derive r (cid:107) ϑ (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:15) u − − δ dec + (cid:15) ru − − δ dec . Therefore, everywhere on ( ext ) M , (cid:107) ϑ (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:15) r − u − − δ dec . (7.5.32) Step 2.
Next, we estimate β from the equation, e β + 2 κβ = d/ α − ( κ − κ ) β + ˇΓ g · α = d/ α + Γ g · ( α, β )to deduce in the same manner r (cid:107) β (cid:107) ∞ ,k ( u, r ) (cid:46) r (cid:107) β (cid:107) ∞ ,k ( u, r ) + (cid:90) rr λ (cid:107) d/ α (cid:107) ∞ ,k ( u, λ ) dλ + (cid:15) ru − − δ dec . Thus, in view of the estimates for α in (7.5.24) and the estimates for α in Ref2 , i.e., for0 ≤ k ≤ k small + 20, (cid:107) α (cid:107) ∞ ,k (cid:46) (cid:15) min { r − log(1 + u )( u + 2 r ) − − δ extra , r − ( u + 2 r ) − − δ extra } . Thus we have with I ( u, r ) := (cid:82) rr λ (cid:107) d/ α (cid:107) ∞ ,k ( u, λ ) dλI ( u, r ) (cid:46) (cid:15) min (cid:110) log(1 + u ) (cid:90) rr λ ( u + 2 λ ) − − δ extra dλ, (cid:90) rr ( u + 2 λ ) − / − δ extra dλ (cid:111) . If r ≤ u we have, (cid:90) rr λ ( u + 2 λ ) − − δ extra dλ (cid:46) r u − − δ extra (cid:46) r ( u + 2 r ) − − δ extra and r − I ( u, r ) (cid:46) (cid:15) r − log(1 + u )( u + 2 r ) − − δ extra . If r ≥ u we have, (cid:90) rr ( u + 2 λ ) − / − δ extra (cid:46) ( u + 2 r ) / δ extra .5. CONTROL IN ( EXT ) M , PART II r − I ( u, r ) (cid:46) r − ( u + 2 r ) / δ extra (cid:46) r − ( u + 2 r ) − − δ extra . We deduce, (cid:107) β (cid:107) ∞ ,k (cid:46) r − (cid:107) β (cid:107) ∞ ,k ( u, r ) + (cid:15) r − log(1 + u )( u + 2 r ) − − δ extra . Thus in view of (7.5.24), (cid:107) β (cid:107) ∞ ,k (cid:46) (cid:15) r − log(1 + u )( u + 2 r ) − − δ extra . (7.5.33) Step 3.
We now estimate ζ using the equation e ( ζ ) + κζ = − β + ˇΓ g · ˇΓ g . This can be done exactly as in Step 1 making use of the estimates already derived for β and the estimate (7.5.24) for ζ along T . We thus derive, (cid:107) ζ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small +15 . Step 4.
We estimate ˇ ρ using equationˇ ρ = − d/ ζ − ˇ µ + ˇΓ g · ˇΓ b , the previous estimate for ζ and ˇ µ in ( ext ) M . We deduce, (cid:107) ˇ ρ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 14 . (7.5.34) Step 5.
We estimate ˇ κ using the equation, e ˇ κ + 12 κ ˇ κ + 12 ˇ κκ = − d/ ζ + 2 ˇ ρ + ˇΓ g · ˇΓ b . Making use of the estimates in ( ext ) M for ˇ κ , ζ and ˇ ρ as well as the estimates for ˇ κ on T in Proposition 7.5.8 we derive, everywhere on ( ext ) M , (cid:107) ˇ κ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 12 . (7.5.35)Alternatively we can make use of the estimate for Ξ = r ( e θ ( κ ) + 4 r d (cid:63) / d/ ζ − r d (cid:63) / d/ β )in Lemma 7.5.2, which holds everywhere on ( ext ) M , and the above estimates for ζ, β .06 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Step 6.
We estimate β everywhere on ( ext ) M with the help of the equation e β + κβ = − d/ ρ − ζρ − ϑβ − ( κ − κ ) β together with the estimate (7.5.29) for β on T and the above derived estimates for ˇ ρ, ζ in ( ext ) M to infer that, (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , ∀ k ≤ k small + 9 . (7.5.36) Step 7.
We extend the for ϑ everywhere on ( ext ) M by making use of the Codazzi equationfor ϑ in (7.5.12), d/ ϑ + 2 β − e θ ( κ ) + κζ ∈ r − Good . Using the estimates already derived above, we infer that, for all k ≤ k small + 11, (cid:107) d/ ϑ (cid:107) ,k (cid:46) (cid:107) β (cid:107) ,k + r − (cid:107) ˇ κ (cid:107) ,k +1 + r − (cid:107) ζ (cid:107) ,k + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − u − − δ dec . Hence, everywhere in ( ext ) M , (cid:107) ϑ (cid:107) ,k (cid:46) (cid:15) r − u − − δ dec , for all k ≤ k small + 12 , and therefore, (cid:107) ϑ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , for all k ≤ k small + 10 . Step 7.
We estimate α everywhere on ( ext ) M by making use of the equation e α + 12 κα = − d (cid:63) / β − ϑρ − ζβ −
12 ( κ − κ ) α as well as the estimate (7.5.22) for α on T and the above estimates in all ( ext ) M for β and ϑ . Proceeding as before we derive, (cid:107) α (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec f orallk ≤ k small + 8 . (7.5.37)This concludes the proof of Proposition 7.5.1. .6. CONCLUSION OF THE PROOF OF THEOREM M4 So far we have established the following estimates, for all k ≤ k small + 8 (cid:13)(cid:13) ˇ κ, r ˇ µ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:13)(cid:13) ϑ, ζ, ˇ κ, r ˇ ρ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − / − δ dec , (cid:13)(cid:13) ϑ, ζ, η, ˇ κ, ϑ, rβ, r ˇ ρ, rβ, α (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:13)(cid:13) β, re β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − (2 r + u ) − / − δ dec . (7.6.1)It only remains to derive improved decay estimates for e ( β, ϑ, ζ, ˇ κ, ˇ ρ ) and the estimatesfor ξ, ˇ ω, ˇ ς, ˇΩ as well as ς + 1 and Ω + Υ in terms of u − − δ dec decay. More precisely itremains to prove the following. Proposition 7.6.1.
The following estimates hold true on ( ext ) M for all k ≤ k small + 7 . (cid:13)(cid:13) e ( ϑ, ζ , ˇ κ ) , re β, re ˇ ρ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:13)(cid:13) ξ, ˇ ω (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , (cid:13)(cid:13) ˇ ς, ˇΩ , ς + 1 , Ω + Υ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) u − − δ dec . Proof.
We proceed in steps as follows.
Step 1.
We make use of the equation e ϑ = − κ ϑ + 2 ωϑ − d (cid:63) / η − κ ϑ + 2 η and thepreviously derived estimates to derive, (cid:13)(cid:13) e ϑ (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 9 . (7.6.2) Step 2.
We make use of the equation e β + ( κ − ω ) β = − d (cid:63) / ρ + 3 ηρ + Γ g β + Γ b α andthe previously derived estimates for β, ˇ ρ, β to derive, (cid:13)(cid:13) e β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 9 . Step 3.
To estimate e ζ in the next step we actually need a stronger estimate for e β than the one derived above. At the same time we derive an improved estimate for β . Weshow in fact, for some 0 < δ , (cid:13)(cid:13) β (cid:13)(cid:13) ∞ ,k (cid:46) (cid:15) r − − δ u − − δ dec , k ≤ k small + 10 , (cid:13)(cid:13) e β (cid:13)(cid:13) ∞ ,k − (cid:46) (cid:15) r − − δ u − − δ dec , k ≤ k small + 10 . (7.6.3)08 CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
This makes use of the equation e β + 2 κβ = d/ α + Γ g · α = F := d/ α + Γ g · α − κβ and the estimates for α in Ref 2.
Thus, for some 0 < δ < δ extra − δ dec , (cid:107) F (cid:107) ∞ ,k (cid:46) log(1 + u ) r − (2 r + u ) − − δ extra + (cid:15) r − u − − δ dec (cid:46) (cid:15) u − − δ dec r − − δ . Integrating from T , where r = r T = r (cid:46)
1, we deduce with the help of Proposition 7.4.3 r (cid:107) β (cid:107) ∞ ,k ( u, r ) (cid:46) r (cid:107) β (cid:107) ∞ ,k ( u, r ) + (cid:90) rr λ (cid:107) F (cid:107) ∞ ,k ( u, λ ) dλ (cid:46) (cid:107) β (cid:107) ∞ ,k ( u, r ) + (cid:15) (cid:90) rr λ − δ dλ. Based on the previously derived estimate for β we have (cid:107) β (cid:107) ∞ ,k ( u, r H ) (cid:46) (cid:15) u − − δ dec . Hence, (cid:107) β (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:15) r − u − − δ dec + (cid:15) r − r − δ u − − δ dec (cid:46) (cid:15) r − − δ u − − δ dec as desired.To prove the second estimate in (7.6.3) we commute the transport equation for β with T and make use of the corresponding estimate for T α (which follows from Ref 2 . (cid:107) T α (cid:107) ∞ ,k (cid:46) (cid:15) log(1 + u ) r − (2 r + u ) − − δ extra (cid:46) (cid:15) u − − δ dec r − − δ as well as the fact that we control T β on T , i.e. (cid:107) T β (cid:107) ∞ ,k − ( u, r ) (cid:46) (cid:15) u − − δ dec . Step 4.
We make use of the equation e ζ + κζ = − β + Γ g · Γ g to derive, (cid:107) e ζ (cid:107) k, ∞ (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 9 . (7.6.4)Indeed commuting the equation with T we derive, e T ζ + κ T ζ = F := − T β + [ T , e ] ζ + ζ T κ + T (Γ g · Γ g ) . It is easy to check, in view of the commutation Lemma 7.4.1, (cid:107) F (cid:107) ∞ ,k − (cid:46) (cid:107) T β (cid:107) ∞ ,k − + (cid:15) r − u − − δ dec . Thus, in view of the estimate for e ζ derived in Step 3 and the estimate for e ζ we inferthat, (cid:107) F (cid:107) ∞ ,k − (cid:46) (cid:15) r − − δ u − − δ dec . .6. CONCLUSION OF THE PROOF OF THEOREM M4 T and using the previously derived estimate (cid:107) ζ (cid:107) k, ∞ (cid:46) (cid:15) r − u − − δ dec r (cid:107) T ζ (cid:107) ∞ ,k − (cid:46) r (cid:107) T ζ (cid:107) ∞ ,k − ( u, r ) + (cid:15) u − − δ dec (cid:90) rr λ − − δ dλ (cid:46) (cid:107) T ζ (cid:107) ∞ ,k − ( u, r ) + (cid:15) r − δ u − − δ dec (cid:46) (cid:15) u − − δ dec . Hence (cid:107) T ζ (cid:107) ∞ ,k − (cid:46) (cid:15) r − u − − δ dec from which the desired estimate easily follows. Step 5.
We make use of the equation e (ˇ ω ) = ˇ ρ + Γ g · Γ b and the previously derivedestimates for ˇ ρ as well as the estimates of ˇ ω on the last slice (see Proposition 7.3.9) toderive the estimate (cid:107) ˇ ω (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 9 . (7.6.5)Indeed, (cid:107) e ˇ ω (cid:107) ∞ ,k (cid:46) (cid:107) ˇ ρ (cid:107) ∞ ,k + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − u − − δ dec . Thus, applying Proposition 7.4.3, integrating from Σ ∗ and using the previously derivedestimate for ˇ ω on Σ ∗ , (cid:107) ˇ ω (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:107) ˇ ω (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) u − − δ dec (cid:90) r ∗ r λ − dλ (cid:46) (cid:15) r − u − − δ dec as desired. Step 6.
We derive the estimate, (cid:107) ξ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 9 (7.6.6)by making use of the transport equation e ( ξ ) = F := − e ( ζ ) + β − κ ( ζ + η ) + Γ b · Γ b .In view of the previously derived estimates for e ζ, β, ζ, η we derive, (cid:107) F (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec . Integrating from Σ ∗ and making use of the estimate for ξ on Σ ∗ (see Proposition 7.3.9)we derive, (cid:107) ξ (cid:107) ∞ ,k ( u, r ) (cid:46) (cid:107) ξ (cid:107) ∞ ,k ( u, r ∗ ) + (cid:15) r − u − − δ dec (cid:46) (cid:15) r − u − − δ dec . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Step 7.
We derive the estimate (cid:107) ˇΩ (cid:107) ∞ ,k (cid:46) (cid:15) u − − δ dec , k ≤ k small + 8 . (7.6.7)This follows immediately from the the equation e θ (Ω) = − ξ − ( η − ζ )Ω, see (2.2.19), andthe previous estimate for ξ . Note that Ω has been estimated in Lemma 3.4.1. Step 8.
We derive the estimate (cid:107) ς − (cid:107) ∞ ,k (cid:46) (cid:15) u − − δ dec , k ≤ k small + 8 . (7.6.8)The estimate follows from the propagation equation e ( ς ) = 0 and the estimate for ς − ∗ . Step 9.
We derive the estimate, (cid:107) e ˇ ρ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 8 (7.6.9)with the help of the equation (see Proposition 7.1.8) e ˇ ρ = r − d / ≤ Γ b + r − Γ b · Γ b and the previously derived estimates for β, ˇ κ, ˇ ρ, ˇΩ , ˇ ς . Step 10.
We derive the estimate, (cid:107) e ˇ κ (cid:107) ∞ ,k (cid:46) (cid:15) r − u − − δ dec , k ≤ k small + 8 (7.6.10)using the equation (see Proposition 7.1.8) e ˇ κ = r − d / ≤ Γ b + Γ b · Γ b and the previously derived estimates for ˇ κ, ξ, ˇ ω, ˇΩ , ˇ ς . This ends the proof of Proposition7.6.1 and Theorem M4. Recall from Theorem M3 that we have obtained the following estimate for ( int ) α in ( int ) M max ≤ k ≤ k small +16 sup ( int ) M u δ dec | d k α | (cid:46) (cid:15) . (7.7.1) Step 1.
We consider the control of the other curvature components, as well as the Riccicomponents on T . Recall that the ( u, ( int ) s ) foliation is initialized on T as follows .7. PROOF OF THEOREM M5 • u and ( int ) s are defined on T by u = u and ( int ) s = ( ext ) s on T . In particular, the 2-spheres S ( u, ( int ) s ) coincide on T with the 2-sphere S ( u, ( ext ) s ). • In view of the above initialization, and the fact that T = { r = r T } , we infer that ( int ) r = ( ext ) r = r T , ( int ) m = ( ext ) m. • The null frame ( ( int ) e , ( int ) e , ( int ) e θ ) is defined on T by ( int ) e = ( ext ) λ ( ext ) e , ( int ) e = ( ( ext ) λ ) − ext ) e , ( int ) e θ = ( ext ) e θ on T where ( ext ) λ = 1 − ( ext ) m ( ext ) r . In particular, we deduce the following identities for the curvature components and Riccicoefficients on T . Lemma 7.7.1.
We have on T ( int ) ς = − ( ext ) κ + ( ext ) A ( ext ) κ λ − ext ) ς, ( int ) Ω = λ − λ ext ) κ ( ext ) κ + ( ext ) A − λ ( ext ) κ ( ext ) κ + ( ext ) A ( ext ) Ω . where λ = ( ext ) λ = 1 − ( ext ) m ( ext ) r . Moreover, we have on T ( int ) α = λ ext ) α, ( int ) β = λ ( ext ) β, ( int ) ρ = ( ext ) ρ, ( int ) β = λ − ext ) β, ( int ) α = λ − ext ) α, ( int ) ξ = 0 , ( int ) ω = 0 , ( int ) ζ = ( ext ) ζ, ( int ) η = − ( ext ) ζ, ( int ) κ = λ ( ext ) κ, ( int ) ϑ = λ ( ext ) ϑ, ( int ) κ = λ − ext ) κ, ( int ) ϑ = λ − ext ) ϑ, CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5) and ( int ) ξ = λ ext ) κ ( ext ) κ + ( ext ) A ( ( ext ) ζ − ( ext ) η ) , ( int ) ω = λ ( ext ) κ ( ext ) κ + ( ext ) A ( ext ) ω, ( int ) η = ( ext ) ζ − ( ext ) κ ( ext ) κ + ( ext ) A ( ext ) ξ. Proof.
The following vectorfield is tangent to T ν T := ( ext ) e − ( ext ) κ + ( ext ) A ( ext ) κ ( ext ) e , which can also be written as ν T = λ ( int ) e − ( ext ) κ + ( ext ) A ( ext ) κ λ − int ) e . Since ν T is tangent to T , and in view of the definition of u and ( int ) s , we immediatelyinfer ν T ( u ) = ν T ( u ) and ν T ( ( int ) s ) = ν T ( ( ext ) s ) on T and hence, using the identities ( ext ) e ( u ) = ( int ) e ( u ) = 0 , ( ext ) e ( ( ext ) s ) = 1 , ( int ) e ( ( int ) s ) = − , we deduce on T− ( ext ) κ + ( ext ) A ( ext ) κ λ − int ) e ( u ) = ( ext ) e ( u ) , − λ − ( ext ) κ + ( ext ) A ( ext ) κ λ − int ) e ( ( int ) s ) = ( ext ) e ( ( ext ) s ) − ( ext ) κ + ( ext ) A ( ext ) κ . In view of the definition of ( ext ) ς , ( int ) ς , ( ext ) Ω and ( int ) Ω, this yields ( int ) ς = − ( ext ) κ + ( ext ) A ( ext ) κ λ − ext ) ς, ( int ) Ω = λ − λ ext ) κ ( ext ) κ + ( ext ) A − λ ( ext ) κ ( ext ) κ + ( ext ) A ( ext ) Ω . .7. PROOF OF THEOREM M5 ( int ) M on T . From ( int ) e = λ ( ext ) e , ( int ) e = λ − ext ) e , ( int ) e θ = ( ext ) e θ on T , the fact that λ is constant on T , and the fact that ( ext ) e θ is tangent to T , we infer on T ( int ) α = λ ext ) α, ( int ) β = λ ( ext ) β, ( int ) ρ = ( ext ) ρ, ( int ) β = λ − ext ) β, ( int ) α = λ − ext ) α, and ( int ) ζ = ( ext ) ζ, ( int ) κ = λ ( ext ) κ, ( int ) ϑ = λ ( ext ) ϑ, ( int ) κ = λ − ext ) κ, ( int ) ϑ = λ − ext ) ϑ. Also, since the foliation of ( int ) M is ingoing geodesic, we have ( int ) ξ = 0 , ( int ) ω = 0 , ( int ) η = − ( int ) ζ. It remains to find identities for ( int ) ξ , ( int ) ω and ( int ) η . Since λ is constant on T and ν T tangent to T , we have on T D ν T ( int ) e = λD ν T ( ext ) e , D ν T ( int ) e = λ − D ν T ( ext ) e and hence g ( D ν T ( int ) e , ( int ) e θ ) = λg ( D ν T ( ext ) e , ( ext ) e θ ) ,g ( D ν T ( int ) e , ( int ) e ) = g ( D ν T ( ext ) e , ( ext ) e ) ,g ( D ν T ( int ) e , ( int ) e θ ) = λ − g ( D ν T ( ext ) e , ( ext ) e θ ) . We deduce2 λ ( int ) η − ( ext ) κ + ( ext ) A ( ext ) κ λ − int ) ξ = λ (cid:32) ( ext ) η − ( ext ) κ + ( ext ) A ( ext ) κ ( ext ) ξ (cid:33) , − λ ( int ) ω − ( ext ) κ + ( ext ) A ( ext ) κ λ − int ) ω = − ( ext ) ω − ( ext ) κ + ( ext ) A ( ext ) κ ( ext ) ω, λ ( int ) ξ − ( ext ) κ + ( ext ) A ( ext ) κ λ − int ) η = λ − (cid:32) ( ext ) ξ − ( ext ) κ + ( ext ) A ( ext ) κ ( ext ) η (cid:33) , and thus ( int ) ξ = λ ext ) κ ( ext ) κ + ( ext ) A ( ( ext ) ζ − ( ext ) η ) , ( int ) ω = λ ( ext ) κ ( ext ) κ + ( ext ) A ( ext ) ω, ( int ) η = ( ext ) ζ − ( ext ) κ ( ext ) κ + ( ext ) A ( ext ) ξ. This concludes the proof of the lemma.14
CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
Remark 7.7.2.
Since the 2-spheres S ( u, ( int ) s ) coincide on T with the 2-sphere S ( u, ( ext ) s ) ,the above lemma immediately yields ( int ) ˇ ρ = ( ext ) ˇ ρ, ( int ) ˇ κ = λ ( ext ) ˇ κ, ( int ) ˇ κ = λ − ext ) ˇ κ ( int ) ˇ µ = − ( ext ) ˇ µ − ( ext ) ˇ ρ + 12 ( ext ) ϑ ( ext ) ϑ − ( ext ) ϑ ( ext ) ϑ, ( int ) ˇ ω = λ ( ext ) κ (cid:32) ( ext ) ω ( ext ) κ + ( ext ) A − ( ext ) ω ( ext ) κ + ( ext ) A (cid:33) , and ( int ) ˇ ς = − λ ( ext ) κ (cid:16) ( ( ext ) κ + ( ext ) A ) ( ext ) ς − ( ( ext ) κ + ( ext ) A ) ( ext ) ς (cid:17) , ( int ) ˇΩ = − λ ext ) κ (cid:18) ( ext ) κ + ( ext ) A − ( ext ) κ + ( ext ) A (cid:19) − λ ( ext ) κ (cid:32) ( ext ) Ω ( ext ) κ + ( ext ) A − ( ext ) Ω ( ext ) κ + ( ext ) A (cid:33) . Together with the estimates on T for the outgoing geodesic foliation of ( ext ) M derived inTheorem M4, we infer the control of tangential derivatives to T , i.e. ( e θ , T T ) derivatives.Recovering the traversal derivative thanks to the transport equations in the direction e ,we infer for the ingoing geodesic foliation of ( int ) M on T max ≤ k ≤ k small +8 sup T u δ dec (cid:13)(cid:13)(cid:13) d k (cid:16) ( int ) α, ( int ) β, ( int ) ˇ ρ, ( int ) β, ( int ) ˇ µ, ( int ) ˇ κ, ( int ) ϑ, ( int ) ζ, ( int ) η, ( int ) ˇ κ, ( int ) ϑ, ( int ) ξ, ( int ) ˇ ω, ( int ) ˇ ς, ( int ) ˇΩ (cid:17)(cid:13)(cid:13)(cid:13) L ( S ) (cid:46) (cid:15) . Step 2.
Relying on the estimates of the ingoing geodesic foliation of ( int ) M on T derivedin Step 1, we propagate these estimates to ( int ) M thanks to transport equations in the e direction given by the null structure equations and Bianchi identities. Recalling that α has already been estimated in Theorem M3, see (7.7.1), quantities are recovered in thefollowing order1. We recover ˇ κ , with a control of k small + 8 derivatives, from e ˇ κ + κ ˇ κ = Err[ e ˇ κ ] .
2. We recover ϑ , with a control of k small + 8 derivatives, from e ( ϑ ) + κ ϑ = − α. .7. PROOF OF THEOREM M5 β , with a control of k small + 8 derivatives, from e β + 2 κ β = d/ α − ζα.
4. We recover ζ , with a control of k small + 8 derivatives, from e ( ζ ) + κζ = β − ϑζ.
5. We recover η , with a control of k small + 8 derivatives, from e ( η + ζ ) + 12 κ ( η + ζ ) = − ϑ ( η + ζ ) .
6. We recover ˇ µ , with a control of k small + 8 derivatives, from e ˇ µ + 32 κ ˇ µ + 32 µ ˇ κ = Err[ e ˇ µ ] .
7. We recover ˇ ρ , with a control of k small + 7 derivatives, from e ˇ ρ + 32 κ ˇ ρ + 32 ρ ˇ κ = d/ β + Err[ e ˇ ρ ] .
8. We recover ˇ κ , with a control of k small + 7 derivatives, from e ˇ κ + 12 κ ˇ κ + 12 ˇ κκ = 2 d/ ζ + 2 ˇ ρ + Err[ e ˇ κ ] .
9. We recover ϑ , with a control of k small + 7 derivatives, from e ϑ + 12 κ ϑ = − d (cid:63) / ζ − κ ϑ + 2 ζ .
10. We recover β , with a control of k small + 6 derivatives, from e β + κβ = e θ ( ρ ) + 3 ζρ − ϑβ.
11. We recover α , with a control of k small + 5 derivatives, from e α + 12 κα = − d (cid:63) / β − ϑρ + 5 ζβ.
12. We recover ˇ ω , with a control of k small + 7 derivatives, from e ˇ ω = ˇ ρ + Err[ e ˇ ω ] . CHAPTER 7. DECAY ESTIMATES (THEOREMS M4, M5)
13. We recover ˇΩ, with a control of k small + 7 derivatives, from e ( ˇΩ) = − ω + ˇ κ ˇΩ .
14. We recover ξ , with a control of k small + 6 derivatives, from e ( ξ ) = e ( ζ ) + β + 12 κ ( ζ − η ) + 12 ϑ ( ζ − η ) .
15. We recover ς , with a control of k small + 8 derivatives, from e ( ς −
1) = 0 . As the estimates are significantly simpler to derive and in the same spirit than thecorresponding ones in Theorem M4, we leave the details to the reader. This concludesthe proof of Theorem M5. Note that r is bounded on ( int ) M and that all quantities behave the same in ( int ) M . hapter 8INITIALIZATION ANDEXTENSION (Theorems M6, M7,M8) In this chapter, we prove M6 concerning initialization, Theorem M7 concerning extension,and Theorem M8 concerning the improvement of hight order weighted energies.
Step 1.
Let r such that r := 2 (cid:15) − , (8.1.1)and let δ > ◦ S of the initial data layer on C (1+ δ , L ) with area radius r . Then, denoting S ( u L , ( ext ) s L ) the spheres of the outgoingportion of the initial data layer, we have ◦ S = S ( ◦ u, ◦ s ) , ◦ u = 1 + δ , | ◦ s − r | (cid:46) (cid:15) . Relying on the control of the initial data layer given by (3.3.5), i.e. I k large +5 ≤ (cid:15) , we then invoke Theorem GCMS-II of section 3.7.4 with the choices ◦ δ = ◦ (cid:15) = (cid:15) , s max = k large + 5 , CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) to produce a unique GCM sphere S ∗ , which is a deformation of ◦ S , satisfying κ S ∗ = 2 r S ∗ , d (cid:63) / S ∗ d (cid:63) / S ∗ κ S ∗ = d (cid:63) / S ∗ d (cid:63) / S ∗ µ S ∗ = 0 , (cid:90) S ∗ β S ∗ e Φ = 0 , (cid:90) S ∗ e S ∗ θ ( κ S ∗ ) e Φ = 0 on S ∗ . Step 2.
Starting from S ∗ constructed in Step 1, and relying on the control of the initialdata layer, we then invoke Theorem GCMH of section 3.7.4 to produce a smooth spacelikehypersurface Σ ∗ included in the initial data layer, passing through the sphere S ∗ , and ascalar function u defined on Σ ∗ such that • The following GCM conditions holds κ = 2 r , d (cid:63) / d (cid:63) / κ = d (cid:63) / d (cid:63) / µ = 0 , (cid:90) S ηe Φ = (cid:90) S ξe Φ = 0 on Σ ∗ • We have, for some constant c Σ ∗ , u + r = c Σ ∗ , along Σ ∗ . • The following normalization condition holds true at the South Pole SP of everysphere S , a (cid:12)(cid:12)(cid:12) SP = − − mr where a is such that we have ν = e + ae , with ν the unique vectorfield tangent to the hypersurface Σ ∗ , normal to S , andnormalized by g ( ν, e ) = − max k ≤ k large +4 sup Σ ∗ r (cid:16) | d k f | + | d k f | + | d k log( λ ) | (cid:17) (cid:46) (cid:15) , (8.1.2) We have in fact max k ≤ k large +6 sup Σ ∗ (cid:16) (cid:107) d k f (cid:107) L ( S ) + (cid:107) d k f (cid:107) L ( S ) + (cid:107) d k log( λ ) (cid:107) L ( S ) (cid:17) (cid:46) (cid:15) , and then use the Sobolev embedding on the 2-spheres S foliating Σ ∗ to deduce (8.1.2). .1. PROOF OF THEOREM M6 Σ ∗ (cid:16) | m − m | + | r − r | (cid:17) (cid:46) (cid:15) , (8.1.3)where ( f, f , λ ) are the transition function from the frame of the initial data layer to theframe of Σ ∗ . Step 3.
Provided δ > ∗ of Step 2 intersects the curve of the south poles of the spheres foliating the outgoingcone C (1 , L ) of the initial data layer. We then call S the unique sphere of Σ ∗ such thatits south pole coincides with the south pole of a sphere of C (1 , L ) , and we calibrate u suchthat u = 1 on S . We then can compare ◦ u = 1 + δ to u ( S ∗ ) and obtain | u ( S ∗ ) − − δ | (cid:46) (cid:15) δ , so that 1 ≤ u ≤ u ( S ∗ ) on Σ ∗ where 1 < u ( S ∗ ) < δ . Together with the estimate (8.1.3), and in view of the choice (8.1.1) for r , we haveinf Σ ∗ r ≥ (cid:15) − ≥ (cid:15) − ( u ( S ∗ )) so that the dominant condition (3.3.4) for r is satisfied since 1 ≤ u ≤ δ on Σ ∗ . Step 4.
In view of Step 1 to Step 3, Σ ∗ satisfies all the required properties for the futurespacelike boundary of a GCM admissible spacetime, see item 3 of definition 3.1.2. Wenow control the outgoing geodesic foliation initialized on Σ ∗ and covering the region wedenote by ( ext ) M , which is included in the initial data layer. Let ( f, f , λ ) the transitionfunctions from the frame of the outgoing part of the initial data layer to the frame of ( ext ) M . Since both frames are outgoing geodesic, we may apply Corollary 2.3.7 whichyields for ( f , f, log( λ )) the following transport equations λ − e (cid:48) ( rf ) = E (cid:48) ( f, Γ) ,λ − e (cid:48) (log( λ )) = E (cid:48) ( f, Γ) ,λ − e (cid:48) (cid:16) rf − r e (cid:48) θ (log( λ )) + rf Ω (cid:17) = E (cid:48) ( f, f , λ, Γ) , where E (cid:48) ( f, Γ) = − r κf − r ϑf + l.o.t. ,E (cid:48) ( f, Γ) = f ζ − f ω − ηf − f κ + l.o.t. ,E (cid:48) ( f, f , λ, Γ) = − r κf + r (cid:18) ˇ κ − (cid:18) κ − r (cid:19)(cid:19) e (cid:48) θ (log( λ )) + r (cid:16) d/ (cid:48) ( f ) + λ − ϑ (cid:48) (cid:17) e (cid:48) θ (log( λ )) − r κ Ω f + rE ( f, f , Γ) − r e (cid:48) θ ( E ( f, Γ)) + r Ω E ( f, Γ) , CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and where E , E and E are given in Lemma 2.3.6. Integrating these transport equationsfrom Σ ∗ , using the control (8.1.2) of ( f, f , λ ) on Σ ∗ , and together with the assumption(3.3.5) for the Ricci coefficients of the foliation of the initial data layer, we obtainsup ( ext ) M ( r ≥ m (1+ δ H )) r (cid:16) | d ≤ k large +4 ( f, log( λ )) | + | d ≤ k large +3 f | (cid:17) (cid:46) (cid:15) . (8.1.4)Then, let T = { r = 2 m (1 + δ H ) } , i.e. we choose r T = 2 m (1 + δ H ). We initialize theingoing geodesic foliation of ( int ) M on T using the outgoing geodesic foliation of ( ext ) M as in item 4 of definition 3.1.2. Using the control of ( f, f , λ ) induced on T by (8.1.4), andusing the analog of Corollary 2.3.7 in the e direction for ingoing foliations, we obtainsimilarly, sup ( int ) M (cid:16) | d ≤ k large +3 ( f , log( λ )) | + | d ≤ k large +2 f | (cid:17) (cid:46) (cid:15) . (8.1.5)Then, in view of (8.1.4) (8.1.5), and the assumption (3.3.5) for the Ricci coefficients andcurvature components of the foliation of the initial data layer, and using the transforma-tion formulas of Proposition 2.3.4, we deducemax k ≤ k large (cid:40) sup ( ext ) M (cid:16) r + δ B ( | d k α | + | d k β | ) + r | d k ˇ ρ | + r | d k β | + r | d k α | (cid:17) + sup ( ext ) M r ( | d k ˇ κ | + | d k ϑ | + | d k ζ | + | d k ˇ κ | )+ sup ( ext ) M r ( | d k η | + | d k ϑ | + | d k ˇ ω | + | d k ξ | ) (cid:17)(cid:41) (cid:46) (cid:15) , and max k ≤ k large sup ( int ) M (cid:16) | d k ˇ R | + | d k ˇΓ | (cid:17) (cid:46) (cid:15) . In particular, we infer that N ( En ) k large + N ( Dec ) k small (cid:46) (cid:15) which concludes the proof of Theorem M6. From the assumptions of Theorem M7 we are given a GCM admissible spacetime M = M ( u ∗ ) ∈ ℵ ( u ∗ ) verifying the following improved bounds, for a universal constant C > N ( Dec ) k small +5 ( M ) ≤ C(cid:15) (8.2.1) .2. PROOF OF THEOREM M7 Step 1.
We extend M by a local existence argument, to a strictly larger spacetime M ( extend ) , with a naturally extended foliation and the following slightly increased bounds N ( Dec ) k small +5 ( M ( extend ) ) ≤ C(cid:15) . but which may not verify our admissibility criteria. Step 2.
We then invoke Theorem GCMH of section 3.7.4 to extend Σ ∗ in M ( extend ) \ M as a smooth spacelike hypersurface Σ ( extend ) ∗ , together with a scalar function u ( extend ) ,satisfying the same GCM conditions than Σ ∗ . Step 3.
We consider the outgoing geodesic foliation ( u ( extend ) , s ( extend ) ) initialized onΣ ( extend ) ∗ to the future of Σ ( extend ) ∗ in M ( extend ) . Note in particular that we have from thedefinition of Σ ∗ and Σ ( extend ) ∗ u ( extend ) + s ( extend ) = c Σ ∗ . We define the following spacetime region to the future of Σ ( extend ) ∗ (cid:101) R := (cid:110) u ∗ ≤ u ( extend ) ≤ u ∗ + δ ext , c Σ ∗ ≤ u ( extend ) + s ( extend ) ≤ c Σ ∗ + ∆ ext (cid:111) , where ∆ ext := 4 r ∗ u ∗ δ ext , r ∗ := r ( S ∗ ) , S ∗ := Σ ∗ ∩ C ∗ , and δ ext > (cid:101) R ⊂ M ( extend ) . From now on, forconvenience, we drop the index ( extend ) and simply denote u ( extend ) and s ( extend ) by u and s . Step 4.
Propagating the GCM quantities in the e direction from Σ ( extend ) ∗ , where theyall vanish, we obtain for all k ≤ k small + 4sup (cid:101) R (cid:18) r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r | d k − ( r d (cid:63) / d (cid:63) / κ ) | + r | d k − ( r d (cid:63) / d (cid:63) / µ ) | (cid:19) (cid:46) (cid:15) r ∆ ext . Similarly, we have for all k ≤ k small + 4sup (cid:101) R∩{ u ≥ u ∗ } (cid:18) r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) d k − (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) u ∗ δ ext + (cid:15) r ∆ ext (cid:46) (cid:15) r ∆ ext , CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) where we used the fact that u ( S ∗ ) = u ∗ and (cid:90) S ∗ βe Φ = (cid:90) S ∗ e θ ( κ ) e Φ = 0 . Also, recall that ν = e + a ∗ e denote the unique tangent vectorfield to Σ ∗ which isorthogonal to e θ and normalized by g ( ν, e ) = −
2. Then, one has, since u + r is constanton Σ ∗ and s = r on Σ ∗ ν ( u + s ) = e ( u ) + ae ( u ) + e ( s ) + ae ( s ) = 2 ς + Ω + a and hence a = − ς − Ω on Σ ∗ . Together with the GCM condition on a , we infer2 ς + Ω = 1 + 2 mr on Σ ∗ . As above, propagating forward in e , we infersup (cid:101) R (cid:12)(cid:12)(cid:12)(cid:12) ς + Ω − (cid:18) mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r ∆ ext . Step 5.
We fix the following sphere of the ( u ( extend ) , s ( extend ) ) foliation in (cid:101) R ∩ { u ≥ u ∗ } ◦ S := S ( ◦ u, ◦ s ) , ◦ u := u ∗ + δ ext , ◦ s := r ∗ + 3 r ∗ u ∗ δ ext . (8.2.2)Define ◦ δ := (cid:15) r ∆ ext = (cid:15) δ ext u ∗ , ◦ (cid:15) := (cid:15) u + δ dec and the small spacetime neighborhood of ◦ S R ( ◦ (cid:15), ◦ δ ) := (cid:110) | u − ◦ u | ≤ δ R , | s − ◦ s | ≤ δ R (cid:111) , δ R = ◦ δ (cid:0) ◦ (cid:15) (cid:1) − . Note that R ( ◦ (cid:15), ◦ δ ) ⊂ (cid:101) R . We are in position to apply Theorem GCMS II of section 3.7.4,with s max = k small + 4, which yields the existence of a unique sphere (cid:101) S ∗ , which is a .2. PROOF OF THEOREM M7 ◦ S , is included in R ( ◦ (cid:15), ◦ δ ), and is such that the following GCM conditionshold on it (cid:101) d (cid:63) / (cid:101) d (cid:63) / (cid:101) κ = (cid:101) d (cid:63) / (cid:101) d (cid:63) / (cid:101) µ = 0 , (cid:101) κ = 2 (cid:101) r , (cid:90) (cid:101) S ∗ (cid:101) βe Φ = (cid:90) (cid:101) S ∗ (cid:101) e θ ( (cid:101) κ ) e Φ = 0 , where the tilde refer to the quantities and tangential operators on (cid:101) S ∗ . Step 6.
Starting from (cid:101) S ∗ constructed in Step 5, we apply Theorem GCMH of section3.7.4, with s max = k small +4, which yields the existence of a smooth small piece of spacelike (cid:101) Σ ∗ starting from (cid:101) S ∗ towards the initial data layer, together with a scalar function (cid:101) u definedon (cid:101) Σ ∗ , whose level surfaces are topological spheres denoted by (cid:101) S , so that • the following GCM conditions are verified on (cid:101) Σ ∗ (cid:101) d (cid:63) / (cid:101) d (cid:63) / (cid:101) κ = (cid:101) d (cid:63) / (cid:101) d (cid:63) / (cid:101) µ = 0 , (cid:101) κ = 2 (cid:101) r , (cid:90) (cid:101) S (cid:101) ηe Φ = (cid:90) (cid:101) S (cid:101) ξe Φ = 0 , where the tilde refer to the quantities and tangential operators on (cid:101) Σ ∗ . • We have, for some constant c (cid:101) Σ ∗ , (cid:101) u + (cid:101) r = c (cid:101) Σ ∗ , along (cid:101) Σ ∗ . • The following normalization condition holds true at the South Pole SP of everysphere (cid:101) S , (cid:101) a (cid:12)(cid:12)(cid:12) SP = − − (cid:101) m (cid:101) r where (cid:101) a is such that we have (cid:101) ν = (cid:101) e + (cid:101) a (cid:101) e , with (cid:101) ν the unique vectorfield tangent to the hypersurface (cid:101) Σ ∗ , normal to (cid:101) S , andnormalized by g ( (cid:101) ν, (cid:101) e ) = − • The transition functions ( f, f , λ ) from the frame of M ( extend ) to the frame of (cid:101) Σ ∗ (cid:107) ( f, f , log( λ )) (cid:107) h ksmall +5 (cid:46) ◦ δ. CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Step 7.
The spacelike GCM hypersurface (cid:101) Σ ∗ has been constructed in Step 6 in a smallneighborhood of (cid:101) S ∗ . We now focus on proving that it in fact extends all the way to theinitial data layer. To this end, we denote by u with1 ≤ u < ◦ u, the minimal value of u such that: • We have (cid:101) Σ ∗ ∩ C u (cid:54) = ∅ for any u ≤ u ≤ ◦ u. (8.2.3) • There exists a large constant D ≥ (cid:101) S of (cid:101) Σ ∗ ( u ≥ u ) (cid:107) ( f, f , log( λ )) (cid:107) h ksmall +5 ( (cid:101) S ) ≤ Du ∗ ◦ δ. (8.2.4) • For the same large constant D ≥ (cid:101) Σ ∗ ( u ≥ u ) | ψ ( s ) | ≤ Du ∗ ◦ δ, (8.2.5)where the function ψ ( s ) is such that the curve (cid:16) u = − s + c (cid:101) Σ ∗ + ψ ( s ) , s, θ = 0 (cid:17) with ψ ( ◦ s ) = 0 , (8.2.6)coincides with the south poles of the sphere (cid:101) S of (cid:101) Σ ∗ and the constant c (cid:101) Σ ∗ is fixedby the condition ψ ( ◦ s ) = 0.The fact that ψ ( ◦ s ) = 0 together with the bounds of Step 6 implies that (8.2.3) (8.2.4)(8.2.5) hold for u < ◦ u with u close enough to ◦ u . By a continuity argument based onreapplying Theorem GCMH, it suffices to show that we may improve the bounds (8.2.4)(8.2.5) independently of the value of u . Step 8.
We now focus on improving the bounds (8.2.4) (8.2.5). We first prove that (cid:101) Σ ∗ ( u ≥ u ) is included in (cid:101) R . Indeed, (8.2.4) (8.2.5) implysup (cid:101) Σ ∗ ( u ≥ u ) | u + s − c (cid:101) Σ ∗ | (cid:46) sup (cid:101) Σ ∗ ( u ≥ u ) (cid:16) | ψ | + r | f | + r | f | ) (cid:46) Du ∗ ◦ δ (cid:46) Du ∗ r (cid:15) ∆ ext (cid:46) (cid:15) Du ∗ (cid:15) ∆ ext (cid:46) (cid:15) ∆ ext . .2. PROOF OF THEOREM M7 ψ ( ◦ s ) = 0 and the south pole of ◦ S and (cid:101) S ∗ coincide,so that we have c (cid:101) Σ ∗ = ◦ u + ◦ s = u ∗ + r ∗ + δ ext r ∗ u ∗ δ ext = c Σ ∗ + 34 (cid:18) u ∗ r ∗ (cid:19) ∆ ext and hence sup (cid:101) Σ ∗ ( u ≥ u ) (cid:12)(cid:12)(cid:12)(cid:12) u + s − c Σ ∗ −
34 ∆ ext (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:18) u ∗ r ∗ + (cid:15) (cid:19) ∆ ext (cid:46) (cid:15) ∆ ext . In view of the definition of (cid:101) R , we infer (cid:101) Σ ∗ ( u ≥ u ) ⊂ (cid:101) R (8.2.7)as claimed. Step 9.
Since (cid:101) Σ ∗ ( u ≥ u ) ⊂ (cid:101) R , the bound of Step 4 apply, and hence we havesup (cid:101) R (cid:12)(cid:12)(cid:12)(cid:12) ς + Ω − (cid:18) mr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) r ∆ ext (cid:46) ◦ δ, and for all k ≤ k small + 4sup (cid:101) R (cid:18) r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) κ − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r | d k − ( r d (cid:63) / d (cid:63) / κ ) | + r | d k − ( r d (cid:63) / d (cid:63) / µ ) | (cid:19) (cid:46) (cid:15) r ∆ ext (cid:46) ◦ δ, as well as sup (cid:101) R∩{ u ≥ u ∗ } (cid:18) r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18)(cid:90) S βe Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) d k − (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) (cid:15) r ∆ ext (cid:46) ◦ δ. Together with the a priori estimates of Chapter 9 on the GCM construction, this yields | ψ (cid:48) ( s ) | (cid:46) (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) m (cid:101) r + Ω + 2 ς (cid:12)(cid:12)(cid:12)(cid:12) + | λ − | (cid:46) (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) m (cid:101) r − mr (cid:12)(cid:12)(cid:12)(cid:12) + | λ − | + (cid:15) r ∆ ext . In view of (8.2.4), we have | (cid:101) r − r | + | (cid:101) m − m | (cid:46) sup (cid:101) S r ( | f | + | f | ) (cid:46) Du ∗ ◦ δ (8.2.8)26 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and we infer | ψ (cid:48) ( s ) | (cid:46) Du ∗ r ◦ δ + ◦ δ (cid:46) (cid:32) (cid:15) ( u ∗ ) D (cid:33) ◦ δ (cid:46) ◦ δ. Integrating from ◦ s where ψ ( ◦ s ) = 0, we infer | ψ ( s ) | (cid:46) | s − ◦ s | ◦ δ (cid:46) u ∗ ◦ δ which improves (8.2.5) for D ≥ (cid:107) ( f, f , log( λ )) (cid:107) h ksmall +5 ( (cid:101) S ) (cid:46) r − (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + ◦ δ and (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S f e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S f e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ◦ δ + 1 r (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . In view of (8.2.4), we infer (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S f e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18)(cid:90) S f e Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ◦ δ + rDu ∗ ◦ δ and integrating from (cid:101) S ∗ , we infer r − (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:46) u ∗ ◦ δ + D ( u ∗ ) r ◦ δ (cid:46) (cid:32) (cid:15) D ( u ∗ ) (cid:33) u ∗ ◦ δ (cid:46) u ∗ ◦ δ. This yields (cid:107) ( f, f , log( λ )) (cid:107) h ksmall +5 ( (cid:101) S ) (cid:46) u ∗ ◦ δ .2. PROOF OF THEOREM M7 D ≥ u = 1, (cid:101) Σ ∗ extendsall the way to the initial data layer, (cid:101) Σ ∗ ⊂ (cid:101) R , and we have the bounds (cid:107) ( f, f , log( λ )) (cid:107) h ksmall +5 ( (cid:101) S ) (cid:46) u ∗ ◦ δ, | ψ ( s ) | (cid:46) u ∗ ◦ δ. In view of the definition of ◦ δ , we infer in particular for any sphere (cid:101) S of (cid:101) Σ ∗ (cid:107) ( f, f , log( λ )) (cid:107) h ksmall +5 ( (cid:101) S ) (cid:46) (cid:15) δ ext , | ψ ( s ) | (cid:46) (cid:15) δ ext . (8.2.9) Step 10. As (cid:101) Σ ∗ extends all the way to the initial data layer, this allows us to calibrate˜ u along (cid:101) Σ ∗ by fixing the value (cid:101) u = 1 as in (3.1.5): (cid:101) S = (cid:101) Σ ∗ ∩ { (cid:101) u = 1 } is such that (cid:101) S ∩ C (1 , L ) ∩ SP (cid:54) = ∅ , (8.2.10)i.e. (cid:101) S is the unique sphere of (cid:101) Σ ∗ such that its south pole intersects the south pole of oneof the sphere of the outgoing null cone C (1 , L ) of the initial data layer.Now that (cid:101) u is calibrated, we define ˜ u ∗ := ˜ u ( (cid:101) S ∗ ) . (8.2.11)For the proof of Theorem M7, we need in particular to prove that ˜ u ∗ > u ∗ . First, notethat, since ˜ u + ˜ r is constant along (cid:101) Σ ∗ , we have (cid:101) Σ ∗ = (cid:110)(cid:101) u + (cid:101) r = 1 + (cid:101) r ( (cid:101) S ) (cid:111) . (8.2.12)Since (cid:101) S ∗ ⊂ (cid:101) Σ ∗ , and in view of (8.2.12), (8.2.2), (8.2.6), we infer, (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) u ( (cid:101) S ∗ ) − (cid:18) u ∗ + δ ext (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) u ( (cid:101) S ∗ ) − u ( ◦ S ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) r ( (cid:101) S ) − (cid:101) r ( (cid:101) S ∗ ) − (cid:18) − s ( ◦ S ) + c (cid:101) Σ ∗ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . Next, note from s = r on Σ ∗ , e ( r − s ) = r (cid:18) κ − r (cid:19) that we have sup (cid:101) R | r − s | (cid:46) (cid:15) r ∆ ext (cid:46) (cid:15) δ ext . (8.2.13)28 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Together with (8.2.8), this yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) u ( (cid:101) S ∗ ) − (cid:18) u ∗ + δ ext (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12) (cid:101) r ( (cid:101) S ) − c (cid:101) Σ ∗ (cid:12)(cid:12)(cid:12) + (cid:15) δ ext . Since c (cid:101) Σ ∗ in (8.2.6) is a constant, we have in particular c (cid:101) Σ ∗ = u ( (cid:101) S ) + r ( (cid:101) S ) − ψ ( s ( (cid:101) S ))and thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) u ( (cid:101) S ∗ ) − (cid:18) u ∗ + δ ext (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12) (cid:101) r ( (cid:101) S ) − u ( (cid:101) S ) − r ( (cid:101) S ) + ψ ( s ( (cid:101) S )) (cid:12)(cid:12)(cid:12) + (cid:15) δ ext (cid:46) (cid:12)(cid:12)(cid:12) − u ( (cid:101) S ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:101) r ( (cid:101) S ) − r ( (cid:101) S ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ ( s ( (cid:101) S )) (cid:12)(cid:12)(cid:12) + (cid:15) δ ext . In view of (8.2.9) and (8.2.8), we infer (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) u ( (cid:101) S ∗ ) − (cid:18) u ∗ + δ ext (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12) − u ( (cid:101) S ) (cid:12)(cid:12)(cid:12) + (cid:15) δ ext . Also, since (recall in particular (3.1.5)) u = 1 on S ∩ SP, e L ( u ) = O (cid:16) (cid:15) r (cid:17) , and since the south pole of S coincides with the one of the corresponding sphere of C L , ,we infer sup (cid:101) R∩C L , ∩ SP | u − | (cid:46) ∆ ext (cid:15) r (cid:46) (cid:15) δ ext . This yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) u ( (cid:101) S ∗ ) − (cid:18) u ∗ + δ ext (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) δ ext . (8.2.14)In particular, we deduce, for (cid:15) small enough, (cid:101) u ( (cid:101) S ∗ ) > u ∗ (8.2.15)as desired. Step 11.
We would like to check that the dominant condition (3.3.4) for r holds on (cid:101) Σ ∗ ,i.e. we need to prove (cid:101) r ( (cid:101) S ∗ ) ≥ (cid:15) − ( (cid:101) u ( (cid:101) S ∗ )) . .2. PROOF OF THEOREM M7 (cid:101) r ( (cid:101) S ∗ ) − (cid:15) − ( (cid:101) u ( (cid:101) S ∗ )) = s ( ◦ S ) + O ( (cid:15) δ ext ) − (cid:15) − (cid:18) u ∗ + δ ext O ( (cid:15) δ ext ) (cid:19) = s ( ◦ S ) − (cid:15) − ( u ∗ ) − (cid:15) − ( u ∗ ) δ ext + (cid:15) − ( u ∗ ) δ ext O (cid:18) δ ext u ∗ + (cid:15) (cid:19) + O ( (cid:15) δ ext ) . Together with (8.2.2), we infer (cid:101) r ( (cid:101) S ∗ ) − (cid:15) − ( (cid:101) u ( (cid:101) S ∗ )) = r ∗ + 3 r ∗ u ∗ δ ext − (cid:15) − ( u ∗ ) − (cid:15) − ( u ∗ ) δ ext + (cid:15) − ( u ∗ ) δ ext O (cid:18) δ ext u ∗ + (cid:15) (cid:19) + O ( (cid:15) δ ext )= r ∗ − (cid:15) − ( u ∗ ) + (cid:16) r ∗ − (cid:15) − ( u ∗ ) (cid:17) δ ext u ∗ + (cid:15) − ( u ∗ ) δ ext O (cid:18) δ ext u ∗ + (cid:15) (cid:19) + O ( (cid:15) δ ext ) . Since we have by the control (3.3.4) of r on Σ ∗ r ∗ ≥ (cid:15) − ( u ∗ ) , we deduce (cid:101) r ( (cid:101) S ∗ ) − (cid:15) − ( (cid:101) u ( (cid:101) S ∗ )) ≥ r ∗ δ ext u ∗ + (cid:15) − ( u ∗ ) δ ext O (cid:18) δ ext u ∗ + (cid:15) (cid:19) + O ( (cid:15) δ ext )= r ∗ δ ext u ∗ (1 + O ( (cid:15) ))so that, for (cid:15) small enough, we have (cid:101) r ( (cid:101) S ∗ ) ≥ (cid:15) − ( (cid:101) u ( (cid:101) S ∗ )) as desired. Step 12.
We summarize the properties of (cid:101) Σ ∗ obtained so far: • (cid:101) Σ ∗ is a spacelike hypersurface included in the spacetime region (cid:101) R . • The scalar function (cid:101) u is defined on (cid:101) Σ ∗ and it level sets are topological 2-spheresdenoted by (cid:101) S .30 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) • The following GCM conditions holds on (cid:101) Σ ∗ (cid:101) d (cid:63) / (cid:101) d (cid:63) / (cid:101) κ = (cid:101) d (cid:63) / (cid:101) d (cid:63) / (cid:101) µ = 0 , (cid:101) κ = 2 (cid:101) r , (cid:90) (cid:101) S (cid:101) ηe Φ = (cid:90) (cid:101) S (cid:101) ξe Φ = 0 . • In addition, the following GCM conditions holds on the sphere (cid:101) S ∗ of (cid:101) Σ ∗ (cid:90) (cid:101) S ∗ (cid:101) βe Φ = (cid:90) (cid:101) S ∗ (cid:101) e θ ( (cid:101) κ ) e Φ = 0 , • We have, for some constant c (cid:101) Σ ∗ , (cid:101) u + (cid:101) r = c (cid:101) Σ ∗ , along (cid:101) Σ ∗ . • The following normalization condition holds true at the South Pole SP of everysphere (cid:101) S , (cid:101) a (cid:12)(cid:12)(cid:12) SP = − − (cid:101) m (cid:101) r where (cid:101) a is such that we have (cid:101) ν = (cid:101) e + (cid:101) a (cid:101) e , with (cid:101) ν the unique vectorfield tangent to the hypersurface (cid:101) Σ ∗ , normal to (cid:101) S , andnormalized by g ( (cid:101) ν, (cid:101) e ) = − • The dominant condition (3.3.4) for r holds on (cid:101) Σ ∗ , i.e. we have (cid:101) r ( (cid:101) S ∗ ) ≥ (cid:15) − ( (cid:101) u ( (cid:101) S ∗ )) . • ˜ u is calibrated along (cid:101) Σ ∗ by fixing the value (cid:101) u = 1: (cid:101) S = (cid:101) Σ ∗ ∩ { (cid:101) u = 1 } is such that (cid:101) S ∩ C (1 , L ) ∩ SP (cid:54) = ∅ , (8.2.16)i.e. (cid:101) S is the unique sphere of (cid:101) Σ ∗ such that its south pole intersects the south poleof one of the sphere of the outgoing null cone C (1 , L ) of the initial data layer.Thus (cid:101) Σ ∗ satisfies all the required properties for the future spacelike boundary of a GCMadmissible spacetime, see item 3 of definition 3.1.2. Furthermore, we have on (cid:101) Σ ∗ (cid:101) u ( (cid:101) S ∗ ) > u ∗ , (8.2.17) .2. PROOF OF THEOREM M7 f, f , λ ) satisfy in view of (8.2.9) and Corollary Rigidity III of section 3.7.4sup (cid:101) Σ ∗ (cid:107) d ≤ k small +5 ( f, f , log( λ )) (cid:107) h ksmall +5 ( (cid:101) S ) (cid:46) (cid:15) δ ext . Together with the Sobolev embedding on the spheres (cid:101) S , and possibly reducing the size of δ ext >
0, we deduce sup (cid:101) Σ ∗ (cid:101) r (cid:101) u + δ dec | d ≤ k small +3 ( f, f , log( λ )) | (cid:46) (cid:15) . (8.2.18) Step 13.
We now control the outgoing geodesic foliation initialized on (cid:101) Σ ∗ . We denoteby ( ext ) (cid:102) M the region covered by this outgoing geodesic foliation. Let ( e , e , e θ ) of ( ext ) M extended to the spacetime M ( extend ) , and satisfying, as discussed in Step 1 to Step 3 N ( Dec ) k small +5 ( M ( extend ) ) (cid:46) (cid:15) . (8.2.19)Let ( f, f , λ ) the transition functions from the frame ( e , e , e θ ) to the frame ( (cid:101) e , (cid:101) e , (cid:101) e θ )of ( ext ) (cid:102) M . Since both frames are outgoing geodesic, we may apply Corollary 2.3.7 whichyields for ( f , f, log( λ )) the following transport equations λ − e (cid:48) ( rf ) = E (cid:48) ( f, Γ) ,λ − e (cid:48) (log( λ )) = E (cid:48) ( f, Γ) ,λ − e (cid:48) (cid:16) rf − r e (cid:48) θ (log( λ )) + rf Ω (cid:17) = E (cid:48) ( f, f , λ, Γ) , where E (cid:48) ( f, Γ) = − r κf − r ϑf + l.o.t. ,E (cid:48) ( f, Γ) = f ζ − f ω − ηf − f κ + l.o.t. ,E (cid:48) ( f, f , λ, Γ) = − r κf + r (cid:18) ˇ κ − (cid:18) κ − r (cid:19)(cid:19) e (cid:48) θ (log( λ )) + r (cid:16) d/ (cid:48) ( f ) + λ − ϑ (cid:48) (cid:17) e (cid:48) θ (log( λ )) − r κ Ω f + rE ( f, f , Γ) − r e (cid:48) θ ( E ( f, Γ)) + r Ω E ( f, Γ) , and where E , E and E are given in Lemma 2.3.6. Integrating these transport equationsfrom (cid:101) Σ ∗ , using the control (8.2.18) of ( f, f , λ ) on (cid:101) Σ ∗ , and together with the control (8.2.19)for the Ricci coefficients of the foliation of M ( extend ) , we obtainsup ( ext ) (cid:102) M (cid:16)(cid:101) r ≥ m (1+ δ H ) (cid:17) (cid:16)(cid:101) r (cid:101) u + δ dec + (cid:101) u δ dec (cid:17)(cid:16) | d ≤ k small +3 ( f, log( λ )) | + | d ≤ k small +2 f | (cid:17) (cid:46) (cid:15) . (8.2.20)32 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Then, for any r T in the interval2 m (cid:18) δ H (cid:19) ≤ r T ≤ m (cid:18) δ H (cid:19) , (8.2.21)we initialize the ingoing geodesic foliation of ( int ) (cid:102) M [ r T ] on (cid:101) r = r T using the outgoinggeodesic foliation of ( ext ) (cid:102) M as in item 4 of definition 3.1.2. Using the control of ( f, f , λ )induced on (cid:101) r = r T by (8.2.20), and using the analog of Corollary 2.3.7 in the e directionfor ingoing foliations, we obtain similarly, for any r T in the interval (8.2.21),sup ( int ) (cid:102) M [ r T ] (cid:101) u δ dec (cid:16) | d ≤ k small +2 ( f , log( λ )) | + | d ≤ k small +1 f | (cid:17) (cid:46) (cid:15) . (8.2.22)Let now, for any r T in the interval (8.2.21), M [ r T ] := ( ext ) (cid:102) M ( (cid:101) r ≥ r T ) ∪ ( int ) (cid:102) M [ r T ] . Then, in view of (8.2.20) (8.2.22), and (8.2.19), and using the transformation formulas ofProposition 2.3.4, we deduce N ( Dec ) k small ( M [ r T ]) (cid:46) (cid:15) which concludes the proof of Theorem M7. So far, we have only improved our bootstrap assumptions on decay estimates. We nowimprove our bootstrap assumptions on energies and weighted energies for ˇ R and ˇΓ relyingon an iterative procedure which recovers derivatives one by one .Let I m ,δ H the interval of R defined by I m ,δ H := (cid:20) m (cid:18) δ H (cid:19) , m (cid:18) δ H (cid:19)(cid:21) . (8.3.1) Remark 8.3.1.
Recall that the results of Theorems M0–M7 hold for any r T ∈ I m ,δ H , seeRemark 3.6.3. More precisely See also [29] for a related strategy to recover higher order derivatives from the control of lower orderones. .3. PROOF OF THEOREM M8 • they hold on ( ext ) M ( r ≥ m (1 + δ H )) , and hence on ( ext ) M ( r ≥ r T ) for any r T ∈ I m ,δ H , • they hold on ( int ) M [ r T ] for any r T ∈ I m ,δ H , where ( int ) M [ r T ] is initialized on T = { r = r T } using ( ext ) M ( r ≥ r T ) as in section 3.1.2. It is at this stage that we need to make a specific choice of r T in the context of a Lebesguepoint argument. More precisely, we choose r T such that we have (cid:90) { r = r T } | d ≤ k large ˇ R | = inf r ∈ I m ,δ H (cid:90) { r = r } | d ≤ k large ˇ R | . (8.3.2) Remark 8.3.2.
In case the above infimum is achieved for several values of r , we choose r T to be the largest of such values, so that r T is uniquely defined. Note also that theinfimum could a priori be infinite, and will only be shown to be finite - and more precisely O ( (cid:15) ) -, at the end of the proof of Theorem M8, see section 8.3.4. This could be maderigorous in the context of a continuity argument. In view of the definition of r T , and since T = { r = r T } , we have (cid:90) T | d ≤ k large ˇ R | ≤ m δ H (cid:90) I m ,δ H (cid:18)(cid:90) { r = r } | d ≤ k large ˇ R | (cid:19) dr and hence (cid:90) T | d ≤ k large ˇ R | (cid:46) (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d ≤ k large ˇ R | . (8.3.3)From now on, we may thus assume that the spacetime M satisfies • the conclusions of Theorem M0, i.e.max ≤ k ≤ k large (cid:40) sup C (cid:104) r + δ B (cid:0) | d k ( ext ) α | + | d k ( ext ) β | (cid:1) + r + δ B | d k − e ( ( ext ) α ) | (cid:105) (8.3.4)+ sup C (cid:20) r (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) ( ext ) ρ + 2 m r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + r | d k ( ext ) β | + r | d k ( ext ) α | (cid:21) (cid:41) (cid:46) (cid:15) We use the coarea formula, d M = √ g ( D r, D r ) d { r = r } dr and the fact that, for r ∈ I m ,δ H , g ( D r, D r ) = − e ( r ) e ( r ) = Υ + O ( (cid:15) ) ≥ δ H + O ( (cid:15) + δ H ) ≥ δ H . Note that (cid:46) here depends on δ − H , see theconvention for (cid:46) made at the end of section 3.3.1. CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and max ≤ k ≤ k large sup C (cid:34) | d k ( int ) α | + | d k ( int ) β | + (cid:12)(cid:12)(cid:12)(cid:12) d k (cid:18) ( int ) ρ + 2 m r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + | d k ( int ) β | + | d k ( int ) α | (cid:35) (cid:46) (cid:15) , (8.3.5) • the conclusions of Theorem M7, i.e. N ( Dec ) k small (cid:46) (cid:15) , (8.3.6)see section 3.2.3 for the definition of the combined norm on decay N ( Dec ) k , • the estimate (cid:90) T | d ≤ k large ˇ R | (cid:46) (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d ≤ k large ˇ R | . (8.3.7)The goal of this section is to prove Theorem M8, i.e. to prove that the following boundholds on M for the weighted energies N ( En ) k large (cid:46) (cid:15) , see section 3.2.3 for the definition of the combined norm on weighted energies N ( En ) k . We recall below our norms for measuring weighted energies for curvature components andRicci coefficients, see sections 3.2.1 and 3.2.2. Let r ≥ m . Then, we have for ( ext ) M (cid:16) ( ext ) R ≥ r [ ˇ R ] (cid:17) = sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ m ) (cid:16) r δ B α + r β (cid:17) + (cid:90) Σ ∗ (cid:16) r δ B ( α + β ) + r ( ˇ ρ ) + r β + α (cid:17) + (cid:90) ( ext ) M ( r ≥ m ) (cid:16) r δ B ( α + β ) + r − δ B ( ˇ ρ ) + r − δ B β + r − − δ B α (cid:17) , (cid:16) ( ext ) R ≤ r [ ˇ R ] (cid:17) = (cid:90) ( ext ) M ( r ≤ m ) (cid:18) − mr (cid:19) | ˇ R | , .3. PROOF OF THEOREM M8 ( ext ) R [ ˇ R ] = ( ext ) R ≥ m [ ˇ R ] + ( ext ) R ≤ m [ ˇ R ] , (cid:16) ( ext ) R k [ ˇ R ] (cid:17) = (cid:16) ( ext ) R [ d ≤ k ˇ R ] (cid:17) + (cid:90) ( ext ) M ( r ≤ m ) (cid:16) | d ≤ k − N ˇ R | + | d ≤ k − ˇ R | (cid:17) , for k ≥ , and (cid:16) ( ext ) G ≥ r k (cid:2) ˇΓ (cid:3) (cid:17) = (cid:90) Σ ∗ (cid:34) r (cid:16) ( d ≤ k ϑ ) + ( d ≤ k ˇ κ ) + ( d ≤ k ζ ) + ( d ≤ k ˇ κ ) (cid:17) + ( d ≤ k ϑ ) + ( d ≤ k η ) + ( d ≤ k ˇ ω ) + ( d ≤ k ξ ) (cid:35) + sup λ ≥ m (cid:32) (cid:90) { r = λ } (cid:34) λ (cid:16) ( d ≤ k ϑ ) + ( d ≤ k ˇ κ ) + ( d ≤ k ζ ) (cid:17) + λ − δ B ( d ≤ k ˇ κ ) + ( d ≤ k ϑ ) + ( d ≤ k η ) + ( d ≤ k ˇ ω ) + λ − δ B ( d ≤ k ξ ) (cid:35)(cid:33) , (cid:16) ( ext ) G ≤ r k (cid:2) ˇΓ (cid:3) (cid:17) = (cid:90) ( ext ) M ( ≤ m ) (cid:12)(cid:12) d ≤ k (cid:0) ˇΓ (cid:1)(cid:12)(cid:12) , ( ext ) G k (cid:2) ˇΓ (cid:3) = ( ext ) G ≤ m k (cid:2) ˇΓ (cid:3) + ( ext ) G ≥ m k (cid:2) ˇΓ (cid:3) . Also, we have for ( int ) M (cid:16) ( int ) R k [ ˇ R ] (cid:17) = (cid:90) ( int ) M | d ≤ k ˇ R | , and (cid:16) ( int ) G k [ˇΓ] (cid:17) = (cid:90) ( int ) M | d ≤ k ˇΓ | . Finally, we recall the following Morawetz type norms, see section 5.1.4. For δ >
0, wehave B δ [ ψ ]( τ , τ ) = (cid:90) ( trap ) M ( τ ,τ ) | Rψ | + r − | ψ | + (cid:18) − mr (cid:19) (cid:18) |∇ / ψ | + 1 r | T ψ | (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ − (cid:0) | d ψ | + | ψ | (cid:1) CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) where the scalar function τ and the spacetime region ( trap ) M have beed introduced insection 5.1.1, and where ( trap (cid:14) ) M denotes the complement of ( trap ) M . Also, we have E δ [ ψ ]( τ ) = (cid:90) Σ( τ ) (cid:18)
12 ( N Σ , e ) | e ψ | + 12 ( N Σ , e ) | e ψ | + |∇ / ψ | + r − | ψ | (cid:19) + (cid:90) Σ ≥ m ( τ ) r δ (cid:16) | e ψ | + r − | ψ | (cid:17) . Here Σ( τ ) denotes the level set of τ , see section 5.1.1, N Σ denotes a choice for the normalto Σ, and recall that we have N Σ = (cid:40) N Σ = e on ( int ) Σ ,N Σ = e on ( ext ) Σ , with ( int ) Σ and ( ext ) Σ defined in section 5.1.1, and( N Σ , e ) ≤ − N Σ , e ) ≤ − ( trap ) Σ . Moreover, we have F δ [ ψ ]( τ , τ ) = (cid:90) A ( τ ,τ ) (cid:16) δ − H | e Ψ | + δ H | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) + (cid:90) Σ ∗ ( τ ,τ ) (cid:16) | e Ψ | + r δ (cid:0) | e ψ | + |∇ / ψ | + r − | ψ | (cid:1)(cid:17) with A ( τ , τ ) = A ∩ M ( τ , τ ) and Σ ∗ ( τ , τ ) = Σ ∗ ∩ M ( τ , τ ). Some quantities will be controlled based on the wave equation they satisfy, and will thusneed to be defined w.r.t. a global frame, i.e. a smooth frame on M . To this end, wewill rely on the global frame of section 3.5.2. We recall below the main properties of thatglobal frame.From definition 3.5.1, the region where the frame of ( int ) M and a conformal renormaliza-tion of the frame of ( ext ) M are matched is given byMatch := (cid:18) ( ext ) M ∩ (cid:26) ( int ) r ≤ m (cid:18) δ H (cid:19)(cid:27)(cid:19) ∪ (cid:18) ( int ) M ∩ (cid:26) ( int ) r ≥ m (cid:18) δ H (cid:19)(cid:27)(cid:19) , .3. PROOF OF THEOREM M8 ( int ) r denotes the area radius of the ingoing geodesic foliation of ( int ) M and itsextension to ( ext ) M .The following proposition concerning the global frame is an immediate consequence ofProposition 3.5.2 and the decay estimates (8.3.6). Proposition 8.3.3.
Assume (8.3.6) . Then, there exists a global null frame defined on ( int ) M ∪ ( ext ) M and denoted by ( ( glo ) e , ( glo ) e , ( glo ) e θ ) such that(a) In ( ext ) M \
Match, we have ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( ext ) Υ ( ext ) e , ( ext ) Υ − ext ) e , ( ext ) e θ (cid:1) . (b) In ( int ) M \
Match, we have ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( int ) e , ( int ) e , ( int ) e θ (cid:1) . (c) In the matching region, we have max ≤ k ≤ k small − sup Match ∩ ( int ) M u δ dec (cid:12)(cid:12) d k ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12) (cid:46) (cid:15) , max ≤ k ≤ k small − sup Match ∩ ( ext ) M u δ dec (cid:12)(cid:12) d k ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12) (cid:46) (cid:15) , where ( glo ) ˇ R and ( glo ) ˇΓ are given by ( glo ) ˇ R = (cid:26) α, β, ρ + 2 mr , β, α (cid:27) , ( glo ) ˇΓ = (cid:26) ξ, ω + mr , κ − r , ϑ, ζ, η, η, κ + 2 r , ϑ, ω, ξ (cid:27) . (d) Furthermore, we may also choose the global frame such that, in addition, one of thefollowing two possibilities hold,i. We have on all ( ext ) M ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( ext ) Υ ( ext ) e , ( ext ) Υ − ext ) e , ( ext ) e θ (cid:1) . ii. We have on all ( int ) M ( ( glo ) e , ( glo ) e , ( glo ) e θ ) = (cid:0) ( int ) e , ( int ) e , ( int ) e θ (cid:1) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Recall our norms for measuring energies for curvature components and Ricci coefficientswhich are given respectively by ( int ) R k [ ˇ R ], ( ext ) R k [ ˇ R ] and ( int ) G k [ˇΓ], ( ext ) G k [ˇΓ], see sec-tions 3.2.1 and 3.2.2. Recall also our combined weighted energy norm N ( En ) k = ( ext ) R k [ ˇ R ] + ( ext ) G k [ˇΓ] + ( int ) R k [ ˇ R ] + ( int ) G k [ˇΓ] . We also introduce the following norm controlling on the matching region the Ricci coeffi-cients and curvature components of the global frame of Proposition 8.3.3 N ( match ) k := (cid:18)(cid:90) Match (cid:12)(cid:12) d ≤ k ( ( glo ) ˇΓ , ( glo ) ˇ R ) (cid:12)(cid:12) (cid:19) . (8.3.8)The estimate (8.3.6) and Proposition 8.3.3 imply in particular N ( En ) k small + N ( match ) k small − (cid:46) (cid:15) . (8.3.9)Next, for J such that k small − ≤ J ≤ k large −
1, consider the iteration assumption N ( En ) J + N ( match ) J (cid:46) (cid:15) B [ J ] , (8.3.10)where (cid:15) B [ J ] := J (cid:88) j = k small − ( (cid:15) ) (cid:96) ( j ) B − (cid:96) ( j ) + (cid:15) (cid:96) ( J )0 B , (cid:96) ( j ) := 2 k small − − j , (8.3.11) B := (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d ≤ k large ˇ R | . (8.3.12) Lemma 8.3.4.
The following estimate holds true for (cid:15) B [ J ] as defined above (cid:15) B [ J ] + B ( (cid:15) B [ J ]) + (cid:15) B (cid:46) (cid:15) B [ J + 1] . (8.3.13) Proof.
We clearly have (cid:15) B [ J ] + (cid:15) B (cid:46) (cid:15) B [ J + 1] . (8.3.14) .3. PROOF OF THEOREM M8 (cid:96) ( j ) = 2 (cid:96) ( j + 1), B (cid:15) B [ J ] (cid:46) J (cid:88) j = k small − ( (cid:15) ) (cid:96) ( j ) B − (cid:96) ( j ) + (cid:15) (cid:96) ( J )0 B (cid:46) J +1 (cid:88) j = k small − ( (cid:15) ) (cid:96) ( j ) B − (cid:96) ( j ) + (cid:15) (cid:96) ( J +1)0 B (cid:46) (cid:32) J +1 (cid:88) j = k small − ( (cid:15) ) (cid:96) ( j ) B − (cid:96) ( j ) + (cid:15) (cid:96) ( J +1)0 B (cid:33) = ( (cid:15) B [ J + 1]) which concludes the proof of the lemma.In view of (8.3.9), (8.3.10) holds for J = k small −
2. The propositions below will allow usto prove Theorem M8 in the next section.
Proposition 8.3.5.
Let J such that k small − ≤ J ≤ k large − . Consider the global frameconstructed in Proposition 8.3.3. In that frame, let ˜ ρ := r ρ + 2 mr − . (8.3.15) Then, under the iteration assumption (8.3.10) , we have sup τ ∈ [1 ,τ ∗ ] E Jδ [ ˜ ρ ]( τ ) + B Jδ [ ˜ ρ ](1 , τ ∗ ) + F Jδ [ ˜ ρ ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Proposition 8.3.6.
Let J such that k small − ≤ J ≤ k large − . Consider the global frameconstructed in Proposition 8.3.3. In that frame, under the iteration assumption (8.3.10) ,we have sup τ ∈ [1 ,τ ∗ ] E Jδ [ α + Υ α ]( τ ) + B Jδ [ α + Υ α ](1 , τ ∗ ) + F Jδ [ α + Υ α ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Proposition 8.3.7.
Let J such that k small − ≤ J ≤ k large − . Consider the global frameconstructed in Proposition 8.3.3. In that frame, under the iteration assumption (8.3.10) ,we have B J − (cid:104) ˇ ρ, α, α, β, β (cid:105) (1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Proposition 8.3.8.
Let J such that k small − ≤ J ≤ k large − . Under the iterationassumption (8.3.10) , we have for r ≥ m int ) R J +1 [ ˇ R ] + ( ext ) R J +1 [ ˇ R ] ≤ ( ext ) R ≥ r J +1 [ ˇ R ] + O (cid:16) r (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17)(cid:17) and ( ext ) R ≥ r J +1 [ ˇ R ] (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) . Proposition 8.3.9.
Let J such that k small − ≤ J ≤ k large − . Under the iterationassumption (8.3.10) , we have ( ext ) G J +1 [ˇΓ] + ( int ) R J +1 [ ˇ R ] + ( ext ) R J +1 [ ˇ R ] (cid:46) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Proposition 8.3.10.
Let J such that k small − ≤ J ≤ k large − . Under the iterationassumption (8.3.10) , we have ( int ) G J +1 [ˇΓ] (cid:46) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) + (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) . Proposition 8.3.11.
Let J such that k small − ≤ J ≤ k large − . Under the iterationassumption (8.3.10) , we have N ( match ) J +1 (cid:46) N ( En ) J +1 + (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) . The proof of Propositions 8.3.5, 8.3.6, 8.3.7, 8.3.8, 8.3.9, 8.3.10 and 8.3.11 are postponedrespectively to sections 8.4, 8.5, 8.6, 8.7, 8.8, 8.9 and 8.10.
To prove Theorem M8, we rely on Propositions 8.3.9, 8.3.10 and 8.3.11. Note that amongthem only the second two involve the dangerous boundary term (cid:0)(cid:82) T | d J +1 ( ( ext ) ˇ R ) | (cid:1) . Weproceed as follows. Step 1.
As mentioned earlier, the estimate (8.3.9) trivially implies the iteration assump-tion (8.3.10) with J = k small −
2. We assume that the iteration assumption (8.3.10) holdsfor any fixed J such that k small − ≤ J ≤ k large −
2. In view of Proposition 8.3.10, wehave ( int ) G J +1 [ˇΓ] (cid:46) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) + (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) . (8.3.16) .3. PROOF OF THEOREM M8 ( ext ) M ( r ∈ I m ,δ H ), and the fact that J + 2 ≤ k large , we obtain (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) (cid:46) (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | ( ( ext ) R J +1 [ ˇ R ]) + ( ext ) R J +1 [ ˇ R ] . (8.3.17)Proposition 8.3.9, (8.3.16) and (8.3.17) yield, for (cid:15) > N ( En ) J +1 (cid:46) (cid:15) B [ J ] + (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) + (cid:15) N ( match ) J +1 , and using also Proposition 8.3.11, N ( match ) J +1 (cid:46) N ( En ) J +1 + (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) (cid:46) (cid:15) B [ J ] + (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) + (cid:15) N ( match ) J +1 . For (cid:15) > N ( En ) J +1 + N ( match ) J +1 (cid:46) (cid:15) B [ J ] + (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) (cid:46) (cid:15) B [ J ] + (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | (cid:16) (cid:15) B [ J ] (cid:17) + (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | (cid:16) (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and hence N ( En ) J +1 + N ( match ) J +1 (cid:46) (cid:15) B [ J ] + (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | (cid:16) (cid:15) B [ J ] (cid:17) + (cid:15) (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d k large ˇ R | . In view of Lemma 8.3.4, we deduce N ( En ) J +1 + N ( match ) J +1 (cid:46) (cid:15) B [ J + 1]which is (8.3.10) for J + 1 derivatives. We deduce that (8.3.10) holds for all J ≤ k large − N ( En ) k large − + N ( match ) k large − (cid:46) (cid:15) B [ k large − . (8.3.18) Step 2.
Next, Proposition 8.3.9 implies in view of (8.3.18) ( ext ) G k large [ˇΓ] + ( int ) R k large [ ˇ R ] + ( ext ) R k large [ ˇ R ] (cid:46) (cid:15) B [ k large −
1] (8.3.19)+ (cid:15) (cid:16) N ( En ) k large + N ( match ) k large (cid:17) . In particular, we have (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d ≤ k large ˇ R | ≤ ( ext ) R k large [ ˇ R ] (cid:46) (cid:15) B [ k large −
1] + (cid:15) (cid:16) N ( En ) k large + N ( match ) k large (cid:17) . In view of the definition of (cid:15) B [ k large − (cid:15) > (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d ≤ k large ˇ R | (cid:46) (cid:15) + (cid:15) (cid:16) N ( En ) k large + N ( match ) k large (cid:17) and hence (cid:15) B [ k large − (cid:46) (cid:15) + (cid:15) (cid:16) N ( En ) k large + N ( match ) k large (cid:17) which yields, together with (8.3.19), ( ext ) G k large [ˇΓ] + ( int ) R k large [ ˇ R ] + ( ext ) R k large [ ˇ R ] (cid:46) (cid:15) + (cid:15) (cid:16) N ( En ) k large + N ( match ) k large (cid:17) . (8.3.20) .4. PROOF OF PROPOSITION 8.3.5 Step 3.
Next, Proposition 8.3.10 implies in view of (8.3.20), ( int ) G k large [ˇΓ] (cid:46) (cid:15) + (cid:15) (cid:16) N ( En ) k large + N ( match ) k large (cid:17) + (cid:18)(cid:90) T | d k large ( ( ext ) ˇ R ) | (cid:19) and hence, for (cid:15) > N ( En ) k large (cid:46) (cid:15) + (cid:15) N ( match ) k large + (cid:18)(cid:90) T | d k large ( ( ext ) ˇ R ) | (cid:19) . Together with Proposition 8.3.11, we infer for (cid:15) > N ( En ) k large + N ( match ) k large (cid:46) (cid:15) + (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) . Step 4.
It remains to estimate the last term of the RHS of the previous inequality. Now,in view of (8.3.7) and (8.3.20), we have (cid:18)(cid:90) T | d k large ( ( ext ) ˇ R ) | (cid:19) (cid:46) (cid:90) ( ext ) M (cid:16) r ∈ I m ,δ H (cid:17) | d ≤ k large ˇ R | (cid:46) ( ext ) R k large [ ˇ R ] (cid:46) (cid:15) + (cid:15) N ( En ) k large so that we finally obtain, for (cid:15) > N ( En ) k large (cid:46) (cid:15) . This concludes the proof of Theorem M8. ˜ ρ Proposition 8.4.1.
The following wave equations hold true.1. The curvature component ρ verifies the identity (cid:3) g ρ = κe ρ + κe ρ + 32 (cid:16) κ κ + 2 ρ (cid:17) ρ + Err [ (cid:3) g ρ ] , CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) whereErr [ (cid:3) g ρ ] := 32 ρ (cid:18) − ϑ ϑ + 2( ξ ξ + η η ) (cid:19) + (cid:18) κ − ω (cid:19) (cid:18) ϑ α − ζ β − η β + ξ β ) (cid:19) − ϑ d (cid:63) / β + ( ζ − η ) e β − ηe (Φ) β − ξ ( e β + e (Φ) β ) − ββ − e (cid:18) − ϑ α + ζ β + 2( η β + ξ β ) (cid:19) − d (cid:63) / ( κ ) β + 2 d (cid:63) / ( ω ) β + 3 η d (cid:63) / ( ρ ) − d/ (cid:16) − ϑβ + ξα (cid:17) − ηe θ ρ.
2. The small curvature quantity, ˜ ρ := r (cid:18) ρ + 2 mr (cid:19) verifies the wave equation, (cid:3) g ( ˜ ρ ) + 8 mr ˜ ρ = − m (cid:3) g ( r ) − (cid:0) r − mr (cid:1) r − mr (cid:18) κκ + 4Υ r (cid:19) − mr ( Aκ + Aκ ) + Err [ (cid:3) g ˜ ρ ] , whereErr [ (cid:3) g ˜ ρ ] := − mr AA + 3 r ˜ ρ + 32 (cid:32) A e ( r ) r + 43 A e ( r ) r (cid:33) ˜ ρ + (cid:18) (cid:16) κκ − mr + 23 r (cid:3) g ( r ) (cid:17) + 8 mr (cid:19) ˜ ρ − Ae ( ˜ ρ ) − Ae ( ˜ ρ ) + 2 r Ae ( m ) + 2 r Ae ( m )+4 D a ( m ) D a (cid:18) r (cid:19) + 2 r (cid:3) g ( m ) + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ ) + r Err [ (cid:3) g ρ ] , and where we recall that, A = 2 r e ( r ) − κ, A = 2 r e ( r ) − κ. Proof.
See appendix B.1. .4. PROOF OF PROPOSITION 8.3.5 (cid:3) g ( r ) Lemma 8.4.2.
Let r the function on M associated to the global frame constructed inProposition 8.3.3, see definition 4.6.4. Let J such that k small − ≤ J ≤ k large − . Underthe iteration assumption (8.3.10) , we have (cid:90) ( int ) M∪ ( ext ) M ( r ≤ m ) (cid:18) d J (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19)(cid:19) + sup r ≥ m (cid:90) { r = r } (cid:18) d J (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19)(cid:19) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) and (cid:90) ( trap ) M (cid:18) d J e (cid:18) (cid:3) g ( ( ext ) r ) − (cid:18) ( ext ) r − ( ext ) m ( ( ext ) r ) (cid:19)(cid:19)(cid:19) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Proof.
Recall that, according to definition 4.6.4, r is defined on ( ext ) M ∪ ( int ) M as follows • on ( ext ) M \
Match, we have ( glo ) r = ( ext ) r, • on ( int ) M \
Match, we have ( glo ) r = ( int ) r, • on the matching region, we have ( glo ) r = (1 − ψ m ,δ H ( ( int ) r )) ( int ) r + ψ m ,δ H ( ( int ) r ) ( ext ) r, where the matching region of Proposition 8.3.3 is given byMatch := (cid:18) ( ext ) M ∩ (cid:26) ( int ) r ≤ m (cid:18) δ H (cid:19)(cid:27)(cid:19) ∪ (cid:18) ( int ) M ∩ (cid:26) ( int ) r ≥ m (cid:18) δ H (cid:19)(cid:27)(cid:19) , and where ψ m ,δ H is given by ψ m ,δ H ( r ) = ψ (cid:32) r − m (cid:0) δ H (cid:1) m δ H (cid:33) on 2 m (cid:18) δ H (cid:19) ≤ r ≤ m (cid:18) δ H (cid:19) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) with ψ : R → R a smooth cut-off function such that 0 ≤ ψ ≤ ψ = 0 on ( −∞ ,
0] and ψ = 1 on [1 , + ∞ ).We have on ( ext ) M (cid:3) g ( ( ext ) r ) = − e e ( ( ext ) r ) + (cid:52) / ( ( ext ) r ) + (cid:18) ω − κ (cid:19) e ( ( ext ) r ) − κe ( ( ext ) r ) + 2 ηe θ ( ( ext ) r ) . Here, ( e , e , e θ ) denotes the frame of ( ext ) M and the Ricci coefficients are computed w.r.t.frame, so we have e ( ( ext ) r ) = ( ext ) r κ, e ( ( ext ) r ) = ( ext ) r κ + A ) , e θ ( ( ext ) r ) = 0and hence (cid:3) g ( ( ext ) r ) = − e (cid:18) ( ext ) r κ (cid:19) + (cid:18) ω − κ (cid:19) ( ext ) r κ − κ ( ext ) r κ + A )= − ( ext ) r e ( κ ) − e ( ( ext ) r )2 κ + (cid:18) ω − κ (cid:19) ( ext ) r κ − ( ext ) r κκ − ( ext ) r κA = − ( ext ) r e ( κ ) − κ ( ext ) r κ + A ) + (cid:18) ω − κ (cid:19) ( ext ) r κ − ( ext ) r κκ − ( ext ) r κA. Now, we have e ( κ ) = e ( κ ) + Err[ e κ ]= − κκ + 2 ωκ + 2 ρ + 2 d/ η − ϑϑ + 2 η + Err[ e κ ]= − κκ + 2 ωκ + 2 ρ − ϑϑ + 2 η + Err[ e κ ]and hence (cid:3) g ( ( ext ) r ) = − ( ext ) r (cid:18) − κκ + 2 ωκ + 2 ρ − ϑϑ + 2 η + Err[ e κ ] (cid:19) − κ ( ext ) r κ + A ) + (cid:18) ω − κ (cid:19) ( ext ) r κ − ( ext ) r κκ − ( ext ) r κA. .4. PROOF OF PROPOSITION 8.3.5 (cid:90) ( ext ) M ( r ≤ m ) (cid:18) d J (cid:18) (cid:3) g ( ( ext ) r ) − (cid:18) ( ext ) r − ( ext ) m ( ( ext ) r ) (cid:19)(cid:19)(cid:19) + sup r ≥ m (cid:90) { r = r } (cid:18) d J (cid:18) (cid:3) g ( ( ext ) r ) − (cid:18) ( ext ) r − ( ext ) m ( ( ext ) r ) (cid:19)(cid:19)(cid:19) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . (8.4.1)Also, using again (8.3.4) and the iteration assumption (8.3.10), we have (cid:90) ( trap ) M (cid:18) d J e (cid:18) (cid:3) g ( ( ext ) r ) − (cid:18) ( ext ) r − ( ext ) m ( ( ext ) r ) (cid:19)(cid:19)(cid:19) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) , (8.4.2)where we have used the null structure equations for e ( κ ), e ( κ ), e ( ω ), e ( ϑ ), e ( ϑ ), e ( η ),the equations for e (Ω), e ( ς ), e ( r ), and the Bianchi identity for e ( ρ ). Remark 8.4.3.
Note that we have used in the last estimate the following observations toavoid a potential loss of one derivative e ( κ ) = − d/ ζ + · · · = 2 (cid:18) ρ + µ − ϑϑ (cid:19) + · · · ,e ( ρ ) = d/ β + · · · = · · · ,e ( Err [ e κ ]) = 2 e ( ς − ˇ ς d/ η ) + · · · = 2 ς − ˇ ς d/ e η + · · · = − ς − e θ ( ς ) e η + · · · Note also that there is no term involving d J ρ (without average) as such a term appearsonly in the null structure equations for e ( κ ) , as well as e ( ω ) and vanishes due to thecancellation e (cid:18) ω − κ (cid:19) = 2 e ( ω ) − e ( κ )= 2 ρ + · · · −
12 ( − d/ ζ + 2 ρ ) + · · · = 2 µ + · · · This is important as such a term would otherwise violate (8.4.2) at r = 3 m . Remark 8.4.4.
Recall that the global frame constructed in Proposition 8.3.3 Recall in particular that ρ is under control in view of Lemma 3.4.1. CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) • concides with the frame of ( int ) M in ( int ) M \
Match, • concides with a conformal renormalization of the frame of ( ext ) M in ( ext ) M\ Match.Thus, J + 1 derivatives of its Ricci coefficients and curvature components are controlled • by N ( match ) J +1 in Match, • by N ( En ) J +1 in M \
Match,and hence by N ( En ) J +1 + N ( match ) J +1 on M . This explains the occurrence of N ( En ) J +1 + N ( match ) J +1 on the right-hand side of numerous estimates, see for example (8.4.1) (8.4.2) . Arguing similarly for ( int ) r , we obtain the following analog of (8.4.1) (cid:90) ( int ) M (cid:18) d J (cid:18) (cid:3) g ( ( int ) r ) − (cid:18) ( int ) r − ( int ) m ( ( int ) r ) (cid:19)(cid:19)(cid:19) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . (8.4.3)Then, since • on ( ext ) M \
Match, we have (cid:3) g ( r ) = (cid:3) g ( ( ext ) r ) , m = ( ext ) m, • on ( int ) M \
Match, we have (cid:3) g ( r ) = (cid:3) g ( ( int ) r ) , m = ( int ) m, we immediately infer from (8.4.1), (8.4.2) and (8.4.3) (cid:90)(cid:16) ( int ) M∪ ( ext ) M ( r ≤ m ) (cid:17) \ Match (cid:18) d J (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19)(cid:19) + sup r ≥ m (cid:90) { r = r } (cid:18) d J (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19)(cid:19) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) .4. PROOF OF PROPOSITION 8.3.5 (cid:90) ( trap ) M (cid:18) d J e (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19)(cid:19) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) which are the desired estimates outside of the matching region. Note that we have usedthe fact that ( trap ) M ∩
Match = ∅ .It remains to derive the desired estimates in the matching region. To this end, we needto estimate ( ext ) r − ( int ) r and ( int ) m − ( ext ) m in the matching region. Step 7 or the proofof Lemma 4.6.6 in section 4.6.2 yields (cid:90) ( int ) M (cid:0) d J +1 (cid:0) ( ext ) r − ( int ) r, ( ext ) m − ( int ) m (cid:1)(cid:1) (cid:46) ( N ( En ) J ) + ( N ( match ) J ) . We infer, in view of the iteration assumption (8.3.10), (cid:90) ( int ) M (cid:0) d J +1 (cid:0) ( ext ) r − ( int ) r, ( ext ) m − ( int ) m (cid:1)(cid:1) (cid:46) ( (cid:15) B [ J ]) . (8.4.4)Then, since we have on the matching region, r = (1 − ψ m ,δ H ( ( int ) r )) ( int ) r + ψ m ,δ H ( ( int ) r ) ( ext ) r,m = (1 − ψ m ,δ H ( ( int ) r )) ( int ) m + ψ m ,δ H ( ( int ) r ) ( ext ) m, (cid:3) g ( r ) = (1 − ψ m ,δ H ( ( int ) r )) (cid:3) g ( ( int ) r ) + ψ m ,δ H ( ( int ) r ) (cid:3) g ( ( ext ) r )+2 ψ (cid:48) m ,δ H ( ( int ) r ) D α ( ( int ) r ) D α ( ( ext ) r − ( int ) r )+( ( ext ) r − ( int ) r ) (cid:3) g ( ψ m ,δ H ) , we deduce there (cid:3) g ( r ) − (cid:18) r − mr (cid:19) = (1 − ψ m ,δ H ( ( int ) r )) (cid:18) (cid:3) g ( ( int ) r ) − (cid:18) ( int ) r − ( int ) m ( ( int ) r ) (cid:19)(cid:19) + ψ m ,δ H ( ( int ) r ) (cid:18) (cid:3) g ( ( ext ) r ) − (cid:18) ( ext ) r − ( ext ) m ( ( ext ) r ) (cid:19)(cid:19) +(1 − ψ m ,δ H ( ( int ) r )) (cid:18) ( int ) r − r − ( int ) m ( ( int ) r ) + 2 mr (cid:19) + ψ m ,δ H ( ( int ) r ) (cid:18) ( ext ) r − r − ( ext ) m ( ( ext ) r ) + 2 mr (cid:19) +2 ψ (cid:48) m ,δ H ( ( int ) r ) D α ( ( int ) r ) D α ( ( ext ) r − ( int ) r ) + ( ( ext ) r − ( int ) r ) (cid:3) g ( ψ m ,δ H ) The proof of Lemma 4.6.6 in section 4.6.2 is done in the particular case J = k large − k small − ≤ J ≤ k large − CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and thus, in view of (8.4.1), (8.4.3) and (8.4.4), we have on the matching region (cid:90)
Match (cid:18) d J (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19)(cid:19) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) as desired. This concludes the proof of the lemma. Corollary 8.4.5.
Let N the RHS of the wave equation for ˜ ρ provided by Proposition8.4.1, i.e. N = − m (cid:3) g ( r ) − (cid:0) r − mr (cid:1) r − mr (cid:18) κκ + 4Υ r (cid:19) − mr ( Aκ + Aκ ) + Err [ (cid:3) g ˜ ρ ] . Then, N − Err [ (cid:3) g ˜ ρ ] satisfies (cid:90) ( int ) M∪ ( ext ) M ( r ≤ m ) (cid:0) d J ( N − Err [ (cid:3) g ˜ ρ ]) (cid:1) + sup r ≥ m (cid:90) { r = r } (cid:0) d J ( N − Err [ (cid:3) g ˜ ρ ]) (cid:1) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) and d J e (cid:16) N − Err [ (cid:3) g ˜ ρ ] (cid:17) = − mκr d J ρ + a J on ( trap ) M where a J satisfies (cid:90) ( trap ) M (cid:0) d J e a J (cid:1) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Proof.
The first estimate is an immediate consequence of Lemma 8.4.2, (8.3.4) and theiteration assumption (8.3.10).Concerning the second estimate, note that the term d J ρ is due to the null structureequations for e ( κ ), i.e. e ( κ ) = − d/ ζ + 2 ρ + · · · = 4 ρ + · · · Then, the estimate for a J follows from Lemma 8.4.2, (8.3.4) and the iteration assumption(8.3.10). .4. PROOF OF PROPOSITION 8.3.5 In view of Proposition 8.4.1, ˜ ρ satisfies( (cid:3) + V ) ˜ ρ = N , V = 8 mr , where N := − m (cid:3) g ( r ) − (cid:0) r − mr (cid:1) r − mr (cid:18) κκ + 4Υ r (cid:19) − mr ( Aκ + Aκ ) + Err[ (cid:3) g ˜ ρ ] . We may thus apply the estimate (10.5.2) of Theorem 10.5.2 with φ = ˜ ρ and s = J toobtain for any k small ≤ J ≤ k large − τ ∈ [1 ,τ ∗ ] E Jδ [ ˜ ρ ]( τ ) + B Jδ [ ˜ ρ ](1 , τ ∗ ) + F Jδ [ ˜ ρ ](1 , τ ∗ ) (cid:46) E Jδ [ ˜ ρ ](1) + sup τ ∈ [1 ,τ ∗ ] E J − δ [ ˜ ρ ]( τ ) + B J − δ [ ˜ ρ ](1 , τ ∗ ) + F J − δ [ ˜ ρ ](1 , τ ∗ )+ D J [Γ] (cid:18) sup M ru + δ dec trap | d ≤ k small ˜ ρ | (cid:19) + (cid:90) Σ( τ ∗ ) ( d ≤ J ˜ ρ ) r + (cid:90) M r δ | d ≤ J N | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M T ( d J ˜ ρ ) d J N (cid:12)(cid:12)(cid:12)(cid:12) , where D J [Γ] is defined by D J [Γ] := (cid:90) ( int ) M∪ ( ext ) M ( r ≤ m ) ( d ≤ J ˇΓ) + sup r ≥ m (cid:18) r (cid:90) { r = r } | d ≤ J Γ g | + r − (cid:90) { r = r } | d ≤ J Γ b | (cid:19) . Next we use the iteration assumption (8.3.10) which yields in particular D J [Γ] (cid:46) ( (cid:15) B [ J ]) . Also, we have ˜ ρ = r (cid:18) ρ − mr (cid:19) + r ˇ ρ and hence, using again the iteration assumption (8.3.10), as well as the control on averagesprovided by Lemma 3.4.1, we infersup τ ∈ [1 ,τ ∗ ] E J − δ [ ˜ ρ ]( τ ) + B J − δ [ ˜ ρ ](1 , τ ∗ ) + F J − δ [ ˜ ρ ](1 , τ ∗ ) + (cid:90) Σ( τ ∗ ) ( d ≤ J ˜ ρ ) r (cid:46) ( (cid:15) B [ J ]) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Together with the control of d ≤ k small ˜ ρ provided by the decay estimate (8.3.6), we inferfrom the above estimatessup τ ∈ [1 ,τ ∗ ] E Jδ [ ˜ ρ ]( τ ) + B Jδ [ ˜ ρ ](1 , τ ∗ ) + F Jδ [ ˜ ρ ](1 , τ ∗ ) (cid:46) E Jδ [ ˜ ρ ](1) + ( (cid:15) B [ J ]) + (cid:90) M r δ | d ≤ J N | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M T ( d J ˜ ρ ) d J N (cid:12)(cid:12)(cid:12)(cid:12) . Next, using the form of N , as well as Corollary 8.4.5, we derive (cid:90) M r δ | d ≤ J N | (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Also, decomposing T as a combination of R and e , integrating e by parts, using againthe form of N , as well as Corollary 8.4.5, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M T ( d J ˜ ρ ) d J N (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M R ( d J ˜ ρ ) d J ( N − Err[ (cid:3) g ˜ ρ ]) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M e ( N − Err[ (cid:3) g ˜ ρ ]) d J N (cid:12)(cid:12)(cid:12)(cid:12) + (cid:90) ( trap ) M | T ( d J ˜ ρ ) || d J Err[ (cid:3) g ˜ ρ ] | (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M d J ˜ ρe ( d J ( N − Err[ (cid:3) g ˜ ρ ])) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) (cid:32) sup τ ∈ [1 ,τ ∗ ] E Jδ [ ˜ ρ ]( τ ) + B Jδ [ ˜ ρ ](1 , τ ∗ ) (cid:33) (cid:46) (cid:90) ( trap ) M ( d J ˜ ρ ) + (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) (cid:32) sup τ ∈ [1 ,τ ∗ ] E Jδ [ ˜ ρ ]( τ ) + B Jδ [ ˜ ρ ](1 , τ ∗ ) (cid:33) . In view of the above, we infersup τ ∈ [1 ,τ ∗ ] E Jδ [ ˜ ρ ]( τ ) + B Jδ [ ˜ ρ ](1 , τ ∗ ) + F Jδ [ ˜ ρ ](1 , τ ∗ ) (cid:46) (cid:90) ( trap ) M ( d J ˜ ρ ) + ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Next, note that we have on R ( r − m ) = 12 ( e ( r ) − Υ e ( r )) −
32 ( e ( m ) − Υ e ( m )) = Υ + O ( (cid:15) ) ≥
16 on ( trap ) M , .5. PROOF OF PROPOSITION 8.3.6 (cid:90) ( trap ) M ( d J ˜ ρ ) (cid:46) (cid:90) ( trap ) M R ( r − m )( d J ˜ ρ ) (cid:46) (cid:90) ( trap ) M (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | d J ˜ ρ || R d J ρ | (cid:46) (cid:15) B [ J ] (cid:32) sup τ ∈ [1 ,τ ∗ ] E Jδ [ ˜ ρ ]( τ ) + B Jδ [ ˜ ρ ](1 , τ ∗ ) (cid:33) . We deducesup τ ∈ [1 ,τ ∗ ] E Jδ [ ˜ ρ ]( τ ) + B Jδ [ ˜ ρ ](1 , τ ∗ ) + F Jδ [ ˜ ρ ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) as desired. This concludes the proof of Proposition 8.3.5. α + Υ α Lemma 8.5.1.
We have (cid:3) ( α + Υ α ) = 4 r (cid:18) − mr (cid:19) (cid:16) e ( α ) − Υ e ( α ) (cid:17) + (cid:18) − r + 16 mr (cid:19) α − r (cid:18) − mr − m r (cid:19) α + Err (cid:104) (cid:3) ( α + Υ α ) (cid:105) CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) whereErr (cid:104) (cid:3) ( α + Υ α ) (cid:105) = (cid:18) Υ V + 4 mr Υ (cid:3) g ( r ) − mr Υ D α ( r ) D α ( r ) + 8 m r D α ( r ) D α ( r ) (cid:19) α + (cid:18) (cid:16) ω + mr (cid:17) + 2 (cid:18) κ − r (cid:19)(cid:19) e ( α ) − ωe ( α ) − (cid:18) Υ (cid:16) ω + mr (cid:17) + mr ( e ( r ) − − e ( m ) r (cid:19) e ( α )+ (cid:18) ω + 2Υ (cid:18) κ + 2 r (cid:19) − m Υ ( e ( r ) + 1) r + 4Υ e ( m ) r (cid:19) e ( α )+ (cid:18)(cid:18) − ρ − mr (cid:19) + 2 (cid:18) ω κ − mr (cid:19) + 12 (cid:18) κ κ + 4Υ r (cid:19) − e ( ω ) − ωω − κ ω (cid:19) α + (cid:18) r D α ( m ) D α ( r ) − r (cid:3) g ( m ) − mr D α ( r ) D α ( m ) + 8 mr D α ( m ) D α (cid:16) mr (cid:17)(cid:19) α +Υ (cid:18)(cid:18) − ρ − mr (cid:19) − (cid:18) e ( ω ) − mr (cid:19) − (cid:18) κ ω − mr (cid:19) + 12 (cid:18) κ κ + 4Υ r (cid:19) − ωω + 2 κω (cid:19) α + 4 mr Υ (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19) α − mr (cid:18) − mr (cid:19) (cid:16) − e ( r ) e ( r ) − Υ + ( e θ ( r )) (cid:17) α +4Υ e θ (Υ) e θ ( α ) + Err [ (cid:3) g α ] + Υ Err [ (cid:3) g α ] . Proof.
Recall from Proposition 2.4.6 that the curvature components α and α verify thefollowing Teukolsky equations (cid:3) α = − ωe ( α ) + (4 ω + 2 κ ) e ( α ) + V α + Err[ (cid:3) g α ] ,V = − ρ − e ( ω ) − ωω + 2 ω κ − κ ω + 12 κ κ, whereErr( (cid:3) g α ) = 12 ϑe ( α ) + 34 ϑ ρ + e θ (Φ) ϑβ − κ ( ζ + 4 η ) β − ( ζ + η ) e ( β ) − ξe ( β )+ e θ (Φ)(2 ζ + η ) α + β + e (Φ) ηβ + e (Φ) ξβ − ( ζ + 4 η ) e ( β ) − ( e ( ζ ) + 4 e ( η )) β − κ + ω )( ζ + 4 η ) β + 2 e θ ( κ + ω ) β − e θ ((2 ζ + η ) α ) − ξe θ ( ρ ) + 2 ηe θ ( α ) + 32 ϑ d/ β + 3 ρ ( η + η + 2 ζ ) ξ + d/ ηα + 14 κϑα − ωϑα − ϑϑα + ξξα + η α + 32 ϑζβ + 3 ϑ ( ηβ + ξβ ) − ϑ ( ζ + 4 η ) β, .5. PROOF OF PROPOSITION 8.3.6 (cid:3) α = − ωe ( α ) + (4 ω + 2 κ ) e ( α ) + V α + Err[ (cid:3) g α ] ,V = − ρ − e ( ω ) − ωω + 2 ωκ − κ ω + 12 κ κ, whereErr( (cid:3) g α ) = 12 ϑe ( α ) + 34 ϑ ρ + e θ (Φ) ϑ β − κ ( − ζ + 4 η ) β − ( − ζ + η ) e ( β ) − ξe ( β )+ e θ (Φ)( − ζ + η ) α + β + e (Φ) ηβ + e (Φ) ξ β − ( − ζ + 4 η ) e ( β ) − ( − e ( ζ ) + 4 e ( η )) β − κ + ω )( − ζ + 4 η ) β + 2 e θ ( κ + ω ) β − e θ (( − ζ + η ) α ) − ξe θ ( ρ ) + 2 ηe θ ( α ) + 32 ϑ d/ β + 3 ρ ( η + η − ζ ) ξ + d/ ηα + 14 κϑ α − ωϑ α − ϑϑα + ξξα + η α − ϑζβ + 3 ϑ ( ηβ + ξβ ) − ϑ ( − ζ + 4 η ) β. We infer from the above wave equations (cid:3) ( α + Υ α ) = (cid:3) ( α ) + Υ (cid:3) ( α ) + 2 D µ (Υ ) D µ ( α ) + (cid:3) (Υ ) α = − ωe ( α ) + (4 ω + 2 κ ) e ( α )+Υ (cid:16) − ωe ( α ) + (4 ω + 2 κ ) e ( α ) (cid:17) − e (Υ) e ( α ) − e (Υ) e ( α )+ V α + (cid:16) Υ V + (cid:3) (Υ ) (cid:17) α + 4Υ e θ (Υ) e θ ( α ) + Err[ (cid:3) g α ] + Υ Err[ (cid:3) g α ]and hence (cid:3) ( α + Υ α ) = 4 r (cid:18) − mr (cid:19) (cid:16) e ( α ) − Υ e ( α ) (cid:17) + V α + (cid:18) Υ V + 4 mr Υ (cid:3) g ( r ) − mr Υ D α ( r ) D α ( r ) + 8 m r D α ( r ) D α ( r ) (cid:19) α + (cid:18) (cid:16) ω + mr (cid:17) + 2 (cid:18) κ − r (cid:19)(cid:19) e ( α ) − ωe ( α ) − (cid:18) Υ (cid:16) ω + mr (cid:17) + mr ( e ( r ) − − e ( m ) r (cid:19) e ( α )+ (cid:18) ω + 2Υ (cid:18) κ + 2 r (cid:19) − m Υ ( e ( r ) + 1) r + 4Υ e ( m ) r (cid:19) e ( α )+ (cid:18) r D α ( m ) D α ( r ) − r (cid:3) g ( m ) − mr D α ( r ) D α ( m ) + 8 mr D α ( m ) D α (cid:16) mr (cid:17)(cid:19) α +4Υ e θ (Υ) e θ ( α ) + Err[ (cid:3) g α ] + Υ Err[ (cid:3) g α ] . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Next, we have in view of the formula for VV = − ρ − e ( ω ) − ωω + 2 ω κ − κ ω + 12 κ κ = − r + 16 mr + (cid:18) − ρ − mr (cid:19) + 2 (cid:18) ω κ − mr (cid:19) + 12 (cid:18) κ κ + 4Υ r (cid:19) − e ( ω ) − ωω − κ ω. Also, we have in view of the formula for VV = − ρ − e ( ω ) − ωω + 2 ωκ − κ ω + 12 κ κ = − r + (cid:18) − ρ − mr (cid:19) − (cid:18) e ( ω ) + 2 mr (cid:19) − (cid:18) κ ω − mr (cid:19) + 12 (cid:18) κ κ + 4Υ r (cid:19) − ωω + 2 κω. Moreover, we have4 mr Υ (cid:3) g ( r ) − mr Υ D α ( r ) D α ( r ) + 8 m r D α ( r ) D α ( r )= 4 m Υ r (cid:18) r − mr (cid:19) + 4 mr Υ (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19) − mr (cid:18) − mr (cid:19) ( − e ( r ) e ( r ) + ( e θ ( r )) )= 16 m Υ r + 4 mr Υ (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19) − mr (cid:18) − mr (cid:19) (cid:16) − e ( r ) e ( r ) − Υ + ( e θ ( r )) (cid:17) and henceΥ V + 4 mr Υ (cid:3) g ( r ) − mr Υ D α ( r ) D α ( r ) + 8 m r D α ( r ) D α ( r )= − r (cid:18) − mr − m r (cid:19) +Υ (cid:18)(cid:18) − ρ − mr (cid:19) − (cid:18) e ( ω ) + 2 mr (cid:19) − (cid:18) κ ω − mr (cid:19) + 12 (cid:18) κ κ + 4Υ r (cid:19) − ωω + 2 κω (cid:19) + 4 mr Υ (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19) − mr (cid:18) − mr (cid:19) (cid:16) − e ( r ) e ( r ) − Υ + ( e θ ( r )) (cid:17) . We deduce (cid:3) ( α + Υ α ) = 4 r (cid:18) − mr (cid:19) (cid:16) e ( α ) − Υ e ( α ) (cid:17) + (cid:18) − r + 16 mr (cid:19) α − r (cid:18) − mr − m r (cid:19) α + Err (cid:104) (cid:3) ( α + Υ α ) (cid:105) .5. PROOF OF PROPOSITION 8.3.6 (cid:104) (cid:3) ( α + Υ α ) (cid:105) = (cid:18) Υ V + 4 mr Υ (cid:3) g ( r ) − mr Υ D α ( r ) D α ( r ) + 8 m r D α ( r ) D α ( r ) (cid:19) α + (cid:18) (cid:16) ω + mr (cid:17) + 2 (cid:18) κ − r (cid:19)(cid:19) e ( α ) − ωe ( α ) − (cid:18) Υ (cid:16) ω + mr (cid:17) + mr ( e ( r ) − − e ( m ) r (cid:19) e ( α )+ (cid:18) ω + 2Υ (cid:18) κ + 2 r (cid:19) − m Υ ( e ( r ) + 1) r + 4Υ e ( m ) r (cid:19) e ( α )+ (cid:18)(cid:18) − ρ − mr (cid:19) + 2 (cid:18) ω κ − mr (cid:19) + 12 (cid:18) κ κ + 4Υ r (cid:19) − e ( ω ) − ωω − κ ω (cid:19) α + (cid:18) r D α ( m ) D α ( r ) − r (cid:3) g ( m ) − mr D α ( r ) D α ( m ) + 8 mr D α ( m ) D α (cid:16) mr (cid:17)(cid:19) α +Υ (cid:18)(cid:18) − ρ − mr (cid:19) − (cid:18) e ( ω ) + 2 mr (cid:19) − (cid:18) κ ω − mr (cid:19) + 12 (cid:18) κ κ + 4Υ r (cid:19) − ωω + 2 κω (cid:19) α + 4 mr Υ (cid:18) (cid:3) g ( r ) − (cid:18) r − mr (cid:19)(cid:19) α − mr (cid:18) − mr (cid:19) (cid:16) − e ( r ) e ( r ) − Υ + ( e θ ( r )) (cid:17) α +4Υ e θ (Υ) e θ ( α ) + Err[ (cid:3) g α ] + Υ Err (cid:104) (cid:3) g α (cid:105) as desired. This concludes the proof of the lemma. Lemma 8.5.2.
We have e ( α ) = − κα − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ ) (cid:41) + 4 ωα − ϑρ + ( ζ + 4 η ) β. and e ( α ) = − κα − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:41) + 4 ωα − ϑρ + ( − ζ + 4 η ) β. CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Proof.
Recall that we have (cid:3) ( α + Υ α ) = 4 r (cid:18) − mr (cid:19) (cid:16) e ( α ) − Υ e ( α ) (cid:17) + (cid:18) − r + 16 mr (cid:19) α − r (cid:18) − mr − m r (cid:19) α + Err (cid:104) (cid:3) ( α + Υ α ) (cid:105) . We first express e ( α ) − Υ e ( α ) in terms of ˜ ρ , where we recall that ˜ ρ = r ρ + mr . UsingBianchi, we have e ( α ) = − κα − d (cid:63) / β + 4 ωα − ϑρ + ( ζ + 4 η ) β,d/ β = e ( ρ ) + 32 κρ + 12 ϑα − ζβ − ηβ + ξβ )= e (cid:18) ˜ ρr − mr (cid:19) + 32 κρ + 12 ϑα − ζβ − ηβ + ξβ )= e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ )and hence e ( α ) = − κα − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ ) (cid:41) + 4 ωα − ϑρ + ( ζ + 4 η ) β. Similarly, we have e ( α ) = − κα − d (cid:63) / β + 4 ωα − ϑρ + ( − ζ + 4 η ) β,d/ β = e ( ρ ) + 32 κρ + 12 ϑα + ζβ − ηβ + ξβ )= e (cid:18) ˜ ρr − mr (cid:19) + 32 κρ + 12 ϑα + ζβ − ηβ + ξβ )= e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) .5. PROOF OF PROPOSITION 8.3.6 e ( α ) = − κα − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:41) + 4 ωα − ϑρ + ( − ζ + 4 η ) β. This concludes the proof of the lemma.
Corollary 8.5.3.
We have (cid:3) ( α + Υ α ) − r (cid:18) mr (cid:19) ( α + Υ α )= − r (cid:18) − mr (cid:19) d (cid:63) / d/ − R (cid:18) ˜ ρr (cid:19) − r (cid:18) − mr (cid:19) ( ϑ − Υ ϑ ) ρ − r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r (cid:27) + 4Υ r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r (cid:27) + Err , whereErr := 4 r (cid:18) − mr (cid:19) (cid:34) ωα + ( ζ + 4 η ) β − Υ( − ζ + 4 η ) β + [Υ , d (cid:63) / d/ − ] e (cid:18) ˜ ρr (cid:19) − d (cid:63) / d/ − (cid:26) − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ ) (cid:27) +Υ d (cid:63) / d/ − (cid:26) − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:27) (cid:35) − r (cid:18) − mr (cid:19) (cid:18) κ + 2 r (cid:19) α + 4Υ r (cid:18) − mr (cid:19) (cid:18) (cid:18) κ − r (cid:19) − (cid:16) ω + mr (cid:17)(cid:19) α + Err (cid:104) (cid:3) ( α + Υ α ) (cid:105) . Proof.
Recall from Lemma 8.5.1 that we have (cid:3) ( α + Υ α ) = 4 r (cid:18) − mr (cid:19) (cid:16) e ( α ) − Υ e ( α ) (cid:17) + (cid:18) − r + 16 mr (cid:19) α − r (cid:18) − mr − m r (cid:19) α + Err (cid:104) (cid:3) ( α + Υ α ) (cid:105) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
In view of Lemma 8.5.2, we have e ( α ) − Υ e ( α ) = − d (cid:63) / d/ − R (cid:18) ˜ ρr (cid:19) − κα + Υ (cid:18) κ − ω (cid:19) α −
32 ( ϑ − Υ ϑ ) ρ − d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r (cid:27) +Υ d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r (cid:27) +4 ωα + ( ζ + 4 η ) β − Υ( − ζ + 4 η ) β + [Υ , d (cid:63) / d/ − ] e (cid:18) ˜ ρr (cid:19) − d (cid:63) / d/ − (cid:26) − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ ) (cid:27) +Υ d (cid:63) / d/ − (cid:26) − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:27) . We infer (cid:3) ( α + Υ α ) = − r (cid:18) − mr (cid:19) d (cid:63) / d/ − R (cid:18) ˜ ρr (cid:19) − r (cid:18) − mr (cid:19) κα + (cid:18) − r + 16 mr (cid:19) α + 4Υ r (cid:18) − mr (cid:19) (cid:18) κ − ω (cid:19) α − r (cid:18) − mr − m r (cid:19) α − r (cid:18) − mr (cid:19) ( ϑ − Υ ϑ ) ρ − r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r (cid:27) + 4Υ r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r (cid:27) + 4 r (cid:18) − mr (cid:19) (cid:34) ωα + ( ζ + 4 η ) β − Υ( − ζ + 4 η ) β + [Υ , d (cid:63) / d/ − ] e (cid:18) ˜ ρr (cid:19) − d (cid:63) / d/ − (cid:26) − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ ) (cid:27) +Υ d (cid:63) / d/ − (cid:26) − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:27) (cid:35) +Err (cid:104) (cid:3) ( α + Υ α ) (cid:105) . .5. PROOF OF PROPOSITION 8.3.6 − r (cid:18) − mr (cid:19) κα + 4Υ r (cid:18) − mr (cid:19) (cid:18) κ − ω (cid:19) α = 4 r (cid:18) − mr (cid:19) α + 4Υ r (cid:18) − mr (cid:19) (cid:18) mr (cid:19) α − r (cid:18) − mr (cid:19) (cid:18) κ + 2 r (cid:19) α + 4Υ r (cid:18) − mr (cid:19) (cid:18) (cid:18) κ − r (cid:19) − (cid:16) ω + mr (cid:17)(cid:19) α, this yields (cid:3) ( α + Υ α ) = − r (cid:18) − mr (cid:19) d (cid:63) / d/ − R (cid:18) ˜ ρr (cid:19) + 2 r (cid:18) mr (cid:19) α + 2Υ r (cid:18) − m r (cid:19) α − r (cid:18) − mr (cid:19) ( ϑ − Υ ϑ ) ρ − r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r (cid:27) + 4Υ r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r (cid:27) + Err , whereErr = 4 r (cid:18) − mr (cid:19) (cid:34) ωα + ( ζ + 4 η ) β − Υ( − ζ + 4 η ) β + [Υ , d (cid:63) / d/ − ] e (cid:18) ˜ ρr (cid:19) − d (cid:63) / d/ − (cid:26) − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ ) (cid:27) +Υ d (cid:63) / d/ − (cid:26) − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:27) (cid:35) − r (cid:18) − mr (cid:19) (cid:18) κ + 2 r (cid:19) α + 4Υ r (cid:18) − mr (cid:19) (cid:18) (cid:18) κ − r (cid:19) − (cid:16) ω + mr (cid:17)(cid:19) α +Err (cid:104) (cid:3) ( α + Υ α ) (cid:105) . Now, since we have2 r (cid:18) mr (cid:19) α + 2Υ r (cid:18) − m r (cid:19) α = 2 r (cid:18) mr (cid:19) ( α + Υ α ) , CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) we infer (cid:3) ( α + Υ α ) − r (cid:18) mr (cid:19) ( α + Υ α )= − r (cid:18) − mr (cid:19) d (cid:63) / d/ − R (cid:18) ˜ ρr (cid:19) − r (cid:18) − mr (cid:19) ( ϑ − Υ ϑ ) ρ − r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r (cid:27) + 4Υ r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r (cid:27) + Err , as desired. This concludes the proof of the corollary. In view of Corollary 8.5.3, α + Υ α satisfies( (cid:3) + V )( α + Υ α ) = N , V = − r (cid:18) mr (cid:19) , where N := − r (cid:18) − mr (cid:19) d (cid:63) / d/ − R (cid:18) ˜ ρr (cid:19) − r (cid:18) − mr (cid:19) ( ϑ − Υ ϑ ) ρ − r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r (cid:27) + 4Υ r (cid:18) − mr (cid:19) d (cid:63) / d/ − (cid:26) r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r (cid:27) + Err . We may thus apply the estimate (10.5.1) of Theorem 10.5.2 with ψ = α + Υ α and s = J to obtain for any k small ≤ J ≤ k large − τ ∈ [1 ,τ ∗ ] E Jδ [ α + Υ α ]( τ ) + B Jδ [ α + Υ α ](1 , τ ∗ ) + F Jδ [ α + Υ α ](1 , τ ∗ ) (cid:46) E Jδ [ α + Υ α ](1) + sup τ ∈ [1 ,τ ∗ ] E J − δ [ α + Υ α ]( τ ) + B J − δ [ α + Υ α ](1 , τ ∗ )+ F J − δ [ α + Υ α ](1 , τ ∗ ) + D J [Γ] (cid:18) sup M ru + δ dec trap | d ≤ k small ( α + Υ α ) | (cid:19) + (cid:90) M r δ | d ≤ J N | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M T ( d J ( α + Υ α )) d J N (cid:12)(cid:12)(cid:12)(cid:12) , .5. PROOF OF PROPOSITION 8.3.6 D J [Γ] is defined by D J [Γ] := (cid:90) ( int ) M∪ ( ext ) M ( r ≤ m ) ( d ≤ J ˇΓ) + sup r ≥ m (cid:18) r (cid:90) { r = r } | d ≤ J Γ g | + r − (cid:90) { r = r } | d ≤ J Γ b | (cid:19) . Next we use the iteration assumption (8.3.10) which yields in particular D J [Γ] (cid:46) ( (cid:15) B [ J ]) andsup τ ∈ [1 ,τ ∗ ] E J − δ [ α + Υ α ]( τ ) + B J − δ [ α + Υ α ](1 , τ ∗ ) + F J − δ [ α + Υ α ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) . Together with the control of d ≤ k small ( α + Υ α ) provided by the decay estimate (8.3.6), weinfer from the above estimatessup τ ∈ [1 ,τ ∗ ] E Jδ [ α + Υ α ]( τ ) + B Jδ [ α + Υ α ](1 , τ ∗ ) + F Jδ [ α + Υ α ](1 , τ ∗ ) (cid:46) E Jδ [ α + Υ α ](1) + ( (cid:15) B [ J ]) + (cid:90) M r δ | d ≤ J N | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M T ( d J ( α + Υ α )) d J N (cid:12)(cid:12)(cid:12)(cid:12) . Next, using the form of N , as well as the control of ˜ ρ provided by Proposition 8.3.5, wederive (cid:90) M r δ | d ≤ J N | (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M T ( d J ( α + Υ α )) d J N (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:90) ( trap ) M (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | T ( d J ( α + Υ α )) | (cid:16) | R ( d J ˜ ρ ) | + | d J ˜ ρ | + | d J ˇΓ | (cid:17) + (cid:90) ( trap ) M | T ( d J ( α + Υ α )) || Err | (cid:46) (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) (cid:32) sup τ ∈ [1 ,τ ∗ ] E Jδ [ ˜ ρ ]( τ ) + B Jδ [ ˜ ρ ](1 , τ ∗ ) (cid:33) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
In view of the above, we infersup τ ∈ [1 ,τ ∗ ] E Jδ [ α + Υ α ]( τ ) + B Jδ [ α + Υ α )](1 , τ ∗ ) + F Jδ [ α + Υ α )](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) as desired. This concludes the proof of Proposition 8.3.6. α and Υ α We initiate the proof of Proposition 8.3.7 by deriving a suitable control for α and Υ α .Recall from Lemma 8.5.2 that we have e ( α ) = − κα − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:41) + 4 ωα − ϑρ + ( − ζ + 4 η ) β. We infer e ( α − Υ α )= e ( α + Υ α ) − e (Υ α )= e ( α + Υ α ) − e ( α ) − e (Υ ) α = e ( α + Υ α ) + 2Υ d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:41) + Υ κα − ωα + 3Υ ϑρ − ( − ζ + 4 η ) β − m Υ e ( r ) r α + 8Υ e ( m ) r α. Also, recall from Lemma 8.5.2 that we have e ( α ) = − κα − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ ) (cid:41) + 4 ωα − ϑρ + ( ζ + 4 η ) β. .6. PROOF OF PROPOSITION 8.3.7 e ( α − Υ α )= − e ( α + Υ α ) + 2 e ( α )= − e ( α + Υ α ) − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ ) (cid:41) − κα + 8 ωα − ϑρ + 2( ζ + 4 η ) β. In view of the above identities for e ( α − Υ α ) and e ( α − Υ α ), and using the control for˜ ρ provided by Proposition 8.3.5 as well as the control for α + Υ α provided by Proposition8.3.6, and the iteration assumption (8.3.10), we obtain B J − δ [ e ( α − Υ α )](1 , τ ∗ ) + B J − δ [ re ( α − Υ α )](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Also, using the Bianchi identity for d/ α and d/ β , we have d/ d/ α = d/ (cid:16) e β + 2( κ + ω ) β − (2 ζ + η ) α − ξρ ) (cid:17) = e ( d/ β ) + [ d/ , e ] β + d/ (cid:16) κ + ω ) β − (2 ζ + η ) α − ξρ ) (cid:17) = e (cid:18) e ρ + 32 ρ + 12 ϑα − ζβ − ηβ + ξβ ) (cid:19) + [ d/ , e ] β + d/ (cid:16) κ + ω ) β − (2 ζ + η ) α − ξρ ) (cid:17) = e (cid:34) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ − r (cid:19) + 6 m ( e ( r ) − Υ) r − e ( m ) r + 12 ϑα − ζβ − ηβ + ξβ ) (cid:35) + [ d/ , e ] β + d/ (cid:16) κ + ω ) β − (2 ζ + η ) α − ξρ ) (cid:17) . Using the control for ˜ ρ provided by Proposition 8.3.5 as well as the iteration assumption(8.3.10), we obtain B J − δ [ r d/ d/ α ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Using the control for α + Υ α provided by Proposition 8.3.6, we infer B J − δ [ r d/ d/ ( α − Υ α )](1 , τ ∗ ) (cid:46) B J − δ [ r d/ d/ α ](1 , τ ∗ ) + B J − δ [ r d/ d/ ( α + Υ α )](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Using a Poincar´e inequality for d/ and for d/ , we deduce B J − δ [ d / ( α − Υ α )](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Together with the above estimate for e ( α − Υ α ) and re ( α − Υ α ), we deduce B Jδ [ α − Υ α ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Together with the control for α + Υ α provided by Proposition 8.3.6, we finally obtain B Jδ [ α ](1 , τ ∗ ) + B Jδ [Υ α ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . (8.6.1) α (8.6.1) provides in particular the control of Υ α . In this section, we infer a suitable controlfor α using the wave equation satisfied by α and the redshift vectorfield.Let Y (0) the vectorfield given by Y (0) := (cid:18) m ( r − m ) + Υ (cid:19) e + (cid:18) m ( r − m ) (cid:19) e , where Y (0) has been introduced in Proposition 10.1.29 in connection with the redshiftvectorfield. Lemma 8.6.1.
We have (cid:3) α = 4 mr (cid:0) m ( r − m ) + Υ (cid:1) Y (0) α + (cid:101) N where (cid:101) N is given by (cid:101) N := − r (cid:32) m (cid:0) m ( r − m ) (cid:1) r (cid:0) m ( r − m ) + Υ (cid:1) (cid:33) (cid:34) − κα − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:41) + 4 ωα − ϑρ + ( − ζ + 4 η ) β (cid:35) + V α − (cid:16) ω + mr (cid:17) e ( α ) + (cid:18) ω + 2 (cid:18) κ + 2 r (cid:19)(cid:19) e ( α ) + Err [ (cid:3) g α ] . .6. PROOF OF PROPOSITION 8.3.7 Proof.
Recall from Proposition 2.4.6 that α verifies the following Teukolsky equation (cid:3) α = − ωe ( α ) + (4 ω + 2 κ ) e ( α ) + V α + Err[ (cid:3) g α ] ,V = − ρ − e ( ω ) − ωω + 2 ωκ − κ ω + 12 κ κ, whereErr( (cid:3) g α ) = 12 ϑe ( α ) + 34 ϑ ρ + e θ (Φ) ϑ β − κ ( − ζ + 4 η ) β − ( − ζ + η ) e ( β ) − ξe ( β )+ e θ (Φ)( − ζ + η ) α + β + e (Φ) ηβ + e (Φ) ξ β − ( − ζ + 4 η ) e ( β ) − ( − e ( ζ ) + 4 e ( η )) β − κ + ω )( − ζ + 4 η ) β + 2 e θ ( κ + ω ) β − e θ (( − ζ + η ) α ) − ξe θ ( ρ ) + 2 ηe θ ( α ) + 32 ϑ d/ β + 3 ρ ( η + η − ζ ) ξ + d/ ηα + 14 κϑ α − ωϑ α − ϑϑα + ξξα + η α − ϑζβ + 3 ϑ ( ηβ + ξβ ) − ϑ ( − ζ + 4 η ) β. We deduce (cid:3) α = 4 mr e ( α ) − r e ( α ) + V α − (cid:16) ω + mr (cid:17) e ( α ) + (cid:18) ω + 2 (cid:18) κ + 2 r (cid:19)(cid:19) e ( α )+Err[ (cid:3) g α ] . In view of the definition of Y (0) , we infer (cid:3) α = 4 mr (cid:0) m ( r − m ) + Υ (cid:1) Y (0) α − r (cid:32) m (cid:0) m ( r − m ) (cid:1) r (cid:0) m ( r − m ) + Υ (cid:1) (cid:33) e ( α )+ V α − (cid:16) ω + mr (cid:17) e ( α ) + (cid:18) ω + 2 (cid:18) κ + 2 r (cid:19)(cid:19) e ( α ) + Err[ (cid:3) g α ] . Next, recall from Lemma 8.5.2 that we have e ( α ) = − κα − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:41) + 4 ωα − ϑρ + ( − ζ + 4 η ) β. We infer (cid:3) α = 4 mr (cid:0) m ( r − m ) + Υ (cid:1) Y (0) α + (cid:101) N CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) where (cid:101) N is given by (cid:101) N = − r (cid:32) m (cid:0) m ( r − m ) (cid:1) r (cid:0) m ( r − m ) + Υ (cid:1) (cid:33) (cid:34) − κα − d (cid:63) / d/ − (cid:40) e (cid:18) ˜ ρr (cid:19) + 32 r κ ˜ ρ − mr (cid:18) κ + 2 r (cid:19) + 6 m ( e ( r ) + 1) r − e ( m ) r + 12 ϑα + ζβ − ηβ + ξβ ) (cid:41) + 4 ωα − ϑρ + ( − ζ + 4 η ) β (cid:35) + V α − (cid:16) ω + mr (cid:17) e ( α ) + (cid:18) ω + 2 (cid:18) κ + 2 r (cid:19)(cid:19) e ( α ) + Err[ (cid:3) g α ] . This concludes the proof of the lemma.
Lemma 8.6.2. (cid:101) N , in the RHS of the wave equation for α introduced in Lemma 8.6.1,satisfies (cid:90) ( int ) M | d J (cid:101) N | (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Proof.
The proof of the lemma follows immediately from the form of (cid:101) N , see Lemma8.6.1, as well as the control for ˜ ρ provided by Proposition 8.3.5, (8.3.6), and the iterationassumption (8.3.10).In view of Lemma 8.6.1, we may apply Proposition 10.5.4 with ψ = α, f ( r, m ) = 4 mr (cid:0) m ( r − m ) + Υ (cid:1) . We infer (cid:90) ( int ) M (1 ,τ ∗ ) ( d J +1 α ) (cid:46) E Jδ [ α ]( τ = 1) + (cid:90) ( ext ) M r ≤ m (1 ,τ ∗ ) ( d J +1 α ) + D J [Γ] sup ( int ) M (1 ,τ ∗ ) ∪ ( ext ) M r ≤ m r | d ≤ k small α | + (cid:90) ( int ) M (1 ,τ ∗ ) ∪ ( ext ) M r ≤ m (cid:16) ( d ≤ s α ) + ( d ≤ J +1 (cid:101) N ) (cid:17) . .6. PROOF OF PROPOSITION 8.3.7 D J [Γ] (cid:46) ( (cid:15) B [ J ]) together with the control of d ≤ k small α provided by the decay estimate (8.3.6), as well asthe iteration assumption and the control for (cid:101) N provided by Lemma 8.6.2 to deduce (cid:90) ( int ) M (1 ,τ ∗ ) ( d J +1 α ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) + (cid:90) ( ext ) M r ≤ m (1 ,τ ∗ ) ( d J +1 α ) . Note that Υ (cid:38) δ H > ( ext ) M and hence (cid:90) ( ext ) M r ≤ m (1 ,τ ∗ ) ( d J +1 α ) (cid:46) (cid:90) ( ext ) M r ≤ m (1 ,τ ∗ ) ( d J +1 (Υ α )) which together with the control of Υ α provided by (8.6.1) yields (cid:90) ( ext ) M r ≤ m (1 ,τ ∗ ) ( d J +1 α ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) and hence (cid:90) ( int ) M (1 ,τ ∗ ) ( d J +1 α ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Since B Jδ [ α ](1 , τ ∗ ) (cid:46) (cid:90) ( int ) M (1 ,τ ∗ ) ( d ≤ J +1 α ) + B Jδ [Υ α ](1 , τ ∗ ) , using again (8.6.1), we finally obtain B Jδ [ α ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . (8.6.2) We have ˇ ρ = ˜ ρr − (cid:18) ρ − mr (cid:19) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Together with the control for ˜ ρ provided by Proposition 8.3.5, as well as the control onaverages provided by Lemma 3.4.1, we infer B Jδ [ ˇ ρ ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Together with the control for α provided by (8.6.1) and the control for α provided by(8.6.2), we infer B Jδ [ α, ˇ ρ, α ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . Together with the Bianchi identities for e ( β ), e ( β ), d/ β , e ( β ), e ( β ), d/ β , as well asthe iteration assumption (8.3.10), we infer B Jδ − [ β, β ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) and hence B J − [ α, β, ˇ ρ, β, α ](1 , τ ∗ ) (cid:46) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) as desired. This concludes the proof of Proposition 8.3.7. First, note that, by definition of the norms B J − , ( int ) R J +1 [ ˇ R ] and ( ext ) R J +1 [ ˇ R ], we havefor any r ≥ m int ) R J +1 [ ˇ R ] + ( ext ) R ≤ r J +1 [ ˇ R ] (cid:46) r B J − [ α, β, ˇ ρ, β, α ](1 , τ ∗ ) . Together with Proposition 8.3.7, this implies ( int ) R J +1 [ ˇ R ] + ( ext ) R ≤ r J +1 [ ˇ R ] (cid:46) r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) . Since we have ( int ) R J +1 [ ˇ R ] + ( ext ) R J +1 [ ˇ R ] = ( int ) R J +1 [ ˇ R ] + ( ext ) R ≤ r J +1 [ ˇ R ] + ( ext ) R ≥ r J +1 [ ˇ R ] , we deduce for any r ≥ m int ) R J +1 [ ˇ R ] + ( ext ) R J +1 [ ˇ R ] ≤ ( ext ) R ≥ r J +1 [ ˇ R ] + O (cid:16) r (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17)(cid:17) . Thus, to prove Proposition 8.3.8, it suffices to establish the following inequality ( ext ) R ≥ r J +1 [ ˇ R ] (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) . This will follow from r p weighted estimates for the curvature components. .7. PROOF OF PROPOSITION 8.3.8 r -weighted divergence identities for Bianchi pairs Lemma 8.7.1.
Let k ≥ , let a (1) and a (2) real numbers. We consider the followingequations. • If ψ (1) , h (1) ∈ s k , ψ (2) , h (2) ∈ s k − , let ( ψ (1) , ψ (2) ) such that (cid:40) e ( ψ (1) ) + a (1) κψ (1) = − d (cid:63) / k ψ (2) + h (1) ,e ( ψ (2) ) + a (2) κψ (2) = d/ k ψ (1) + h (2) , (8.7.1) • If ψ (1) , h (1) ∈ s k − , ψ (2) , h (2) ∈ s k , let ( ψ (1) , ψ (2) ) such that (cid:40) e ( ψ (1) ) + a (1) κψ (1) = d/ k ψ (2) + h (1) ,e ( ψ (2) ) + a (2) κψ (2) = − d (cid:63) / k ψ (1) + h (2) . (8.7.2) Then, the pair ( ψ (1) , ψ (2) ) satisfies for any real number b Div (cid:16) r b ψ e (cid:17) + Div (cid:16) r b ψ e (cid:17) − r b κ (cid:16) − a (1) + b + 2 (cid:17) ψ + 12 r b κ (cid:16) a (2) − b − (cid:17) ψ = 2 r b d/ ( ψ (1) ψ (2) ) − r b ωψ − r b ωψ + 2 r b ψ (1) h (1) + 2 r b ψ (2) h (2) + br b − (cid:16) e ( r ) − r κ (cid:17) ψ + br b − (cid:16) e ( r ) − r κ (cid:17) ψ . (8.7.3) Remark 8.7.2.
Note that the Bianchi identities can be written as systems of equationsof the type (8.7.1) (8.7.2) . In particular • the Bianchi pair ( α, β ) satisfies (8.7.1) with k = 2 , a (1) = , a (2) = 2 , • the Bianchi pair ( β, ρ ) satisfies (8.7.1) with k = 1 , a (1) = 1 , a (2) = , • the Bianchi pair ( ρ, β ) satisfies (8.7.2) with k = 1 , a (1) = , a (2) = 1 , • the Bianchi pair ( β, α ) satisfies (8.7.2) with k = 2 , a (1) = 2 , a (2) = .Proof of Lemma 8.7.1. The proof being identical for (8.7.1) and (8.7.2), it suffices to proveit in the case where ( ψ (1) , ψ (2) ) satisfies (8.7.1).We compute D γ e γ = − g ( D e , e ) − g ( D e , e ) + g ( D θ e , e θ ) + g ( D ϕ e , e ϕ )= κ − ω CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and D γ e γ = − g ( D e , e ) − g ( D e , e ) + g ( D θ e , e θ ) + g ( D ϕ e , e ϕ )= κ − ω. We infer in view of (8.7.1) D γ (cid:16) r b ψ e γ (cid:17) = 2 r b ψ (1) e ( ψ (1) ) + br b − e ( r ) ψ + r b ψ D γ e γ = 2 r b ψ (1) (cid:16) − a (1) κψ (1) − d (cid:63) / k ψ (2) + h (1) (cid:17) + br b − e ( r ) ψ + r b ψ ( κ − ω )= − r b ψ (1) d (cid:63) / k ψ (2) + r b (cid:16) − a (1) + b (cid:17) κψ + br b − (cid:16) e ( r ) − r κ (cid:17) ψ − ωr b ψ + 2 r b ψ (1) h (1) and D γ (cid:16) r b ψ e γ (cid:17) = 2 r b ψ (2) e ( ψ (2) ) + br b − e ( r ) ψ + r b ψ D γ e γ = 2 r b ψ (2) (cid:16) − a (2) κψ (2) + d/ k ψ (1) + h (2) (cid:17) + br b − e ( r ) ψ + r b ψ ( κ − ω )= 2 r b ψ (2) d/ k ψ (1) + r b (cid:16) − a (2) + b (cid:17) κψ + br b − (cid:16) e ( r ) − r κ (cid:17) ψ − r b ωψ + 2 r b ψ (2) h (2) . We sum the two identities D γ (cid:16) r b ψ e γ (cid:17) + D γ (cid:16) r b ψ e γ (cid:17) = − r b ψ (1) d (cid:63) / k ψ (2) + 2 r b ψ (2) d/ k ψ (1) + r b (cid:16) − a (1) + b (cid:17) κψ + r b (cid:16) − a (2) + b (cid:17) κψ + br b − (cid:16) e ( r ) − r κ (cid:17) ψ + br b − (cid:16) e ( r ) − r κ (cid:17) ψ − r b ωψ − r b ωψ +2 r b ψ (2) h (2) + 2 r b ψ (1) h (1) and hence D γ (cid:16) r b ψ e γ (cid:17) + D γ (cid:16) r b ψ e γ (cid:17) − r b κ (cid:16) − a (1) + b (cid:17) ψ + r b κ (cid:16) a (2) − b − (cid:17) ψ = 2 r b d/ ( ψ (1) ψ (2) ) + br b − (cid:16) e ( r ) − r κ (cid:17) ψ + br b − (cid:16) e ( r ) − r κ (cid:17) ψ − r b ωψ − r b ωψ +2 r b ψ (1) h (1) + 2 r b ψ (2) h (2) . This concludes the proof of Lemma 8.7.1. .7. PROOF OF PROPOSITION 8.3.8 r p weighted estimates for higher order derivatives of the curvature components,we will need several lemmas. Lemma 8.7.3.
Let k ≥ and s ≥ two integers. Let ψ (1) ∈ s k and ψ (2) ∈ s k − . Then,we have − d / s ψ (1) d / s d (cid:63) / k ψ (2) + d / s ψ (2) d / s d/ k ψ (1) = d/ (cid:16) d / s ψ (1) d / s ψ (2) (cid:17) + E [ d /, s, k, ψ (1) , ψ (2) ] where | E [ s, k, ψ (1) , ψ (2) ] | (cid:46) r | d / s ψ (1) | s − (cid:88) j =0 | d / s − − j ( ψ (2) ) || d / j ( K ) | + r | d / s ψ (2) | s − (cid:88) j =0 | d / s − − j ( ψ (1) ) || d / j ( K ) | . Proof.
Recall our definition d / s for higher angular derivatives. Given f a k -reduced scalarand s a positive integer we define, d / s f = (cid:40) r p (cid:52) / pk , if s = 2 p,r p +1 d/ k (cid:52) / pk , if s = 2 p + 1 . We start with the case s = 2 p , i.e. s is even. Since ψ (1) ∈ s k and ψ (2) ∈ s k − , we have − d / s ψ (1) d / s d (cid:63) / k ψ (2) + d / s ψ (2) d / s d/ k ψ (1) = r p (cid:16) − (cid:52) / pk ψ (1) (cid:52) / pk d (cid:63) / k ψ (2) + (cid:52) / pk − ψ (2) (cid:52) / pk − d/ k ψ (1) (cid:17) . Next, recall the commutation formulas − d/ k (cid:52) / k + (cid:52) / k − d/ k = K d/ k − ke θ ( K ) , − d (cid:63) / k (cid:52) / k − + (cid:52) / k d (cid:63) / k = (2 k − K d (cid:63) / k + ( k − e θ ( K ) . We infer (cid:52) / pk − d/ k = d/ k (cid:52) / pk + p (cid:88) j =1 (cid:52) / p − jk − (cid:16) (cid:52) / k − d/ k − d/ k (cid:52) / k (cid:17) (cid:52) / j − k = d/ k (cid:52) / pk + p (cid:88) j =1 (cid:52) / p − jk − (cid:16) K d/ k − ke θ ( K ) (cid:17) (cid:52) / j − k CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and (cid:52) / pk d (cid:63) / k = d (cid:63) / k (cid:52) / pk − + p (cid:88) j =1 (cid:52) / p − jk (cid:16) (cid:52) / k d (cid:63) / k − d (cid:63) / k (cid:52) / k − (cid:17) (cid:52) / j − k − = d (cid:63) / k (cid:52) / pk − + p (cid:88) j =1 (cid:52) / p − jk (cid:16) (2 k − K d (cid:63) / k + ( k − e θ ( K ) (cid:17) (cid:52) / j − k − This yields − d / s ψ (1) d / s d (cid:63) / k ψ (2) + d / s ψ (2) d / s d/ k ψ (1) = r p (cid:40) − (cid:52) / pk ψ (1) d (cid:63) / k (cid:52) / pk − ψ (2) − p (cid:88) j =1 (cid:52) / pk ψ (1) (cid:52) / p − jk (cid:16) (2 k − K d (cid:63) / k + ( k − e θ ( K ) (cid:17) (cid:52) / j − k − ψ (2) + (cid:52) / pk − ψ (2) d/ k (cid:52) / pk ψ (1) + p (cid:88) j =1 (cid:52) / pk − ψ (2) (cid:52) / p − jk − (cid:16) K d/ k − ke θ ( K ) (cid:17) (cid:52) / j − k ψ (1) (cid:41) = d/ (cid:16) d / s ψ (1) d / s ψ (2) (cid:17) − p (cid:88) j =1 d / s ψ (1) d / s − j (cid:16) (2 k − r K d (cid:63) / k + ( k − r e θ ( K ) (cid:17) d / j − ψ (2) + p (cid:88) j =1 d / s ψ (2) d / s − j (cid:16) r K d/ k − kr e θ ( K ) (cid:17) d / j − ψ (1) . Hence, we infer − d / s ψ (1) d / s d (cid:63) / k ψ (2) + d / s ψ (2) d / s d/ k ψ (1) = d/ (cid:16) d / s ψ (1) d / s ψ (2) (cid:17) + E [ s, k, ψ (1) , ψ (2) ]where | E [ s, k, ψ (1) , ψ (2) ] | (cid:46) r | d / s ψ (1) | s − (cid:88) j =0 | d / s − − j ( ψ (2) ) || d / j ( K ) | + r | d / s ψ (2) | s − (cid:88) j =0 | d / s − − j ( ψ (1) ) || d / j ( K ) | . Next, we deal with the case s = 2 p + 1, i.e. s odd. Since ψ (1) ∈ s k and ψ (2) ∈ s k − , wehave − d / s ψ (1) d / s d (cid:63) / k ψ (2) + d / s ψ (2) d / s d/ k ψ (1) = r p (cid:16) − d/ k (cid:52) / pk ψ (1) d/ k (cid:52) / pk d (cid:63) / k ψ (2) + d/ k − (cid:52) / pk − ψ (2) d/ k − (cid:52) / pk − d/ k ψ (1) (cid:17) .7. PROOF OF PROPOSITION 8.3.8 s = 2 p above, we infer − d / s ψ (1) d / s d (cid:63) / k ψ (2) + d / s ψ (2) d / s d/ k ψ (1) = r p +2 (cid:40) − d/ k (cid:52) / pk ψ (1) d/ k d (cid:63) / k (cid:52) / pk − ψ (2) − p (cid:88) j =1 d/ k (cid:52) / pk ψ (1) d/ k (cid:52) / p − jk (cid:16) (2 k − K d (cid:63) / k + ( k − e θ ( K ) (cid:17) (cid:52) / j − k − ψ (2) + d/ k − (cid:52) / pk − ψ (2) d/ k − d/ k (cid:52) / pk ψ (1) + p (cid:88) j =1 d/ k − (cid:52) / pk − ψ (2) d/ k − (cid:52) / p − jk − (cid:16) K d/ k − ke θ ( K ) (cid:17) (cid:52) / j − k ψ (1) (cid:41) . Next, recall the commutation formula d/ k d (cid:63) / k − d (cid:63) / k − d/ k − = − k − K. We infer − d / s ψ (1) d / s d (cid:63) / k ψ (2) + d / s ψ (2) d / s d/ k ψ (1) = r p +2 (cid:40) − d/ k (cid:52) / pk ψ (1) (cid:16) d (cid:63) / k − d/ k − − k − K (cid:17) (cid:52) / pk − ψ (2) − p (cid:88) j =1 d/ k (cid:52) / pk ψ (1) d/ k (cid:52) / p − jk (cid:16) (2 k − K d (cid:63) / k + ( k − e θ ( K ) (cid:17) (cid:52) / j − k − ψ (2) + d/ k − (cid:52) / pk − ψ (2) d/ k − d/ k (cid:52) / pk ψ (1) + p (cid:88) j =1 d/ k − (cid:52) / pk − ψ (2) d/ k − (cid:52) / p − jk − (cid:16) K d/ k − ke θ ( K ) (cid:17) (cid:52) / j − k ψ (1) (cid:41) = d/ (cid:16) d / s ψ (1) d / s ψ (2) (cid:17) +2( k − r K d / s ψ (1) d / s − ψ (2) − p (cid:88) j =1 d / s ψ (1) d / s − j (cid:16) (2 k − r K d (cid:63) / k + ( k − r e θ ( K ) (cid:17) d / j − ψ (2) + p (cid:88) j =1 d / s ψ (2) d / s − j (cid:16) r K d/ k − kr e θ ( K ) (cid:17) d / j − ψ (1) . Hence, we obtain − d / s ψ (1) d / s d (cid:63) / k ψ (2) + d / s ψ (2) d / s d/ k ψ (1) = d/ (cid:16) d / s ψ (1) d / s ψ (2) (cid:17) + E [ d /, s, k, ψ (1) , ψ (2) ]where | E [ s, k, ψ (1) , ψ (2) ] | (cid:46) r | d / s ψ (1) | s − (cid:88) j =0 | d / s − − j ( ψ (2) ) || d / j ( K ) | + r | d / s ψ (2) | s − (cid:88) j =0 | d / s − − j ( ψ (1) ) || d / j ( K ) | . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
This concludes the proof of the lemma.
Corollary 8.7.4.
Let k ≥ , let a (1) and a (2) real numbers and let ≤ s ≤ k large . Considerthe outgoing geodesic foliation of ( ext ) M . We consider the following equations. • If ψ (1) ∈ s k , ψ (2) ∈ s k − , let ( ψ (1) , ψ (2 ,s ) ) such that (cid:40) e ( d / s ψ (1) ) + a (1) κ d / s ψ (1) = − d / s d (cid:63) / k ψ (2) + h (1 ,s ) ,e ( d / s ψ (2) ) + a (2) κ d / s ψ (2) = d / s d/ k ψ (1) + h (2 ,s ) , • If ψ (1) ∈ s k − , ψ (2) , h (2) ∈ s k , let ( ψ (1) , ψ (2) ) such that (cid:40) e ( d / s ψ (1) ) + a (1) κ d / s ψ (1) = d / s d/ k ψ (2) + h (1 ,s ) ,e ( d / s ψ (2) ) + a (2) κ d / s ψ (2) = − d / s d (cid:63) / k ψ (1) + h (2 ,s ) . Then, the pair ( ψ (1) , ψ (2) ) satisfies for any real number b Div (cid:16) r b ( d / s ψ (1) ) e (cid:17) + Div (cid:16) r b ( d / s ψ (2) ) e (cid:17) − r b κ (cid:16) − a (1) + b + 2 (cid:17) ( d / s ψ (1) ) + 12 r b κ (cid:16) a (2) − b − (cid:17) ( d / s ψ (2) ) = 2 r b d/ (cid:16) d / s ψ (1) d / s ψ (2) (cid:17) + 2 r b E [ d /, s, k, ψ (1) , ψ (2) ] − r b ω ( d / s ψ (1) ) +2 r b d / s ψ (1) h (1 ,s ) + 2 r b d / s ψ (2) h (2 ,s ) + br b − (cid:16) e ( r ) − r κ (cid:17) ( d / s ψ (1) ) + br b − (cid:16) e ( r ) − r κ (cid:17) ( d / s ψ (2) ) . where E [ d /, s, k, ψ (1) , ψ (2) ] has been introduced in Lemma 8.7.3.Proof. The proof follows immediately from combining Lemma 8.7.1 and Lemma 8.7.3.
Lemma 8.7.5.
Let j, k, l three integers. Consider a Bianchi ( ψ (1) , ψ (2) ) satisfying (8.7.1) or (8.7.2) . Then, the pair ( ψ (1) , ψ (2) ) satisfies for any real number b Div (cid:16) r b ( d / j ( re ) k T l ψ (1) ) e (cid:17) + Div (cid:16) r b ( d / j ( re ) k T l ψ (2) ) e (cid:17) − r b κ (cid:16) − a (1) + 2 k + b + 2 (cid:17) ( d / j ( re ) k T l ψ (1) ) + 12 r b κ (cid:16) a (2) − k − b − (cid:17) ( d / j ( re ) k T l ψ (2) ) = 2 r b d/ (cid:16) d / j ( re ) k T l ψ (1) d / j ( re ) k T l ψ (2) (cid:17) + 2 r b E [ d /, j, k, ( re ) k T l ψ (1) , ( re ) k T l ψ (2) ] − r b ω ( d / j ( re ) k T l ψ (1) ) + 2 r b d / j ( re ) k T l ψ (1) h (1) ,j,k,l + 2 r b d / j ( re ) k T l ψ (2) h (2) ,j,k,l + br b − (cid:16) e ( r ) − r κ (cid:17) ( d / j ( re ) k T l ψ (1) ) + br b − (cid:16) e ( r ) − r κ (cid:17) ( d / j ( re ) k T l ψ (2) ) . .7. PROOF OF PROPOSITION 8.3.8 where E [ d /, s, k, ( re ) k T l ψ (1) , ( re ) k T l ψ (2) ] has been introduced in Lemma 8.7.3, and where h (1) ,j,k,l and h (2) ,j,k,l are given, schematically, by h (1) ,j,k,l = d / ≤ j + k + l ( h (1) ) + kr − d / j +1 ( re ) k − T l ψ (2) + r d j + k + l (cid:16) Γ g (cid:0) ψ (1) , ψ (2) (cid:1)(cid:17) + O ( r − ) d ≤ j + k + l − (cid:0) ψ (1) , ψ (2) (cid:1) and h (2) ,j,k,l = d / ≤ j + k + l ( h (2) ) + kr − d / j +1 ( re ) k − T l ψ (1) + r d j + k + l (cid:16) Γ g (cid:0) ψ (1) , ψ (2) (cid:1)(cid:17) + O ( r − ) d ≤ j + k + l − (cid:0) ψ (1) , ψ (2) (cid:1) . Proof.
We have the following simple schematic consequences of the commutator identities[
T, e ] , [ T, e ] = r − Γ b d , [ T, d/ k ] = − ηe + Γ g d , [ d /, e ] = Γ g d + Γ g , [ d /, e ] = − rηe + r Γ g d , [ re , e ] = − r κe + Γ g d , [ re , e ] = − r κe + Γ b d , [ re , d / k ] = r − d / + Γ g d + Γ g . Then, differentiating with d / j ( re ) k T l the equations (cid:40) e ( ψ (1) ) + a (1) κψ (1) = − d (cid:63) / k ψ (2) + h (1) ,e ( ψ (2) ) + a (2) κψ (2) = d/ k ψ (1) + h (2) , and using the above commutator identities we infer (cid:40) e ( d / j ( re ) k T l ψ (1) ) + (cid:0) a (1) − k (cid:1) κ d / j ( re ) k T l ψ (1) = − d / j d (cid:63) / k (( re ) k T l ψ (2) ) + h (1) ,j,k,l ,e ( d / j ( re ) k T l ψ (2) ) + (cid:0) a (2) − k (cid:1) κ d / j ( re ) k T l ψ (2) = d / j d/ k (( re ) k T l ψ (1) ) + h (2) ,j,k,l , were h (1) ,j,k,l = d / j ( re ) k T l ( h (1) ) + kr − d / j +1 ( re ) k − T l ψ (2) + jrη d j + k + l − e ψ (1) + r d j + k + l (cid:16) Γ g (cid:0) ψ (1) , ψ (2) (cid:1)(cid:17) + O ( r − ) d ≤ j + k + l − (cid:0) ψ (1) , ψ (2) (cid:1) and h (2) ,j,k,l = d / j ( re ) k T l ( h (2) ) + kr − d / j +1 ( re ) k − T l ψ (1) + r d j + k + l (cid:16) Γ g (cid:0) ψ (1) , ψ (2) (cid:1)(cid:17) + O ( r − ) d ≤ j + k + l − (cid:0) ψ (1) , ψ (2) (cid:1) . Also, using the equation e ( ψ (1) ) = − a (1) κψ (1) − d (cid:63) / k ψ (2) + h (1) , CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) we obtain jrη d j + k + l − e ψ (1) = r d j + k + l (cid:16) Γ g (cid:0) ψ (1) , ψ (2) (cid:1)(cid:17) + O ( r − ) d ≤ j + k + l − (cid:0) ψ (1) , ψ (2) (cid:1) + rη d k + j + l − ( h (1) )and hence, h (1) ,j,k,l = d / ≤ j + k + l ( h (1) ) + kr − d / j +1 ( re ) k − T l ψ (2) + r d j + k + l (cid:16) Γ g (cid:0) ψ (1) , ψ (2) (cid:1)(cid:17) + O ( r − ) d ≤ j + k + l − (cid:0) ψ (1) , ψ (2) (cid:1) . We have thus obtained the desired form for h (1) ,j,k,l and h (2) ,j,k,l .The divergence identity now follows from the equations (cid:40) e ( d / j ( re ) k T l ψ (1) ) + (cid:0) a (1) − k (cid:1) κ d / j ( re ) k T l ψ (1) = − d / j d (cid:63) / k (( re ) k T l ψ (2) ) + h (1) ,j,k,l ,e ( d / j ( re ) k T l ψ (2) ) + (cid:0) a (2) − k (cid:1) κ d / j ( re ) k T l ψ (2) = d / j d/ k (( re ) k T l ψ (1) ) + h (2) ,j,k,l , together with Corollary 8.7.4. This concludes the proof of the lemma. Corollary 8.7.6.
Let r ≥ m and ≤ u ≤ u ∗ . We introduce the spacetime region R u = ( ext ) M ∩ { r ≥ m } ∩ { ≤ u ≤ u } , . Let j, k, l three integers. Assume that the frame of ( ext ) M satisfies sup ( ext ) M (cid:16)(cid:12)(cid:12)(cid:12) e ( r ) − r κ (cid:12)(cid:12)(cid:12) + r (cid:16) | ω | + (cid:12)(cid:12)(cid:12) e ( r ) − r κ (cid:12)(cid:12)(cid:12)(cid:17)(cid:17) (cid:46) (cid:15) . Consider a pair ( ψ (1) , ψ (2) ) satisfying (8.7.1) or (8.7.2) . Then, ( ψ (1) , ψ (2) ) satisfies for anyreal number b (a) If − a (1) + 2 k + b + 2 > and a (2) − k − b − > , then, we have (cid:90) C u ( r ≥ r ) r b ( d / j ( re ) k T l ψ (1) ) + (cid:90) Σ ∗ ( ≤ u ) r b (cid:16) ( d / j ( re ) k T l ψ (1) ) + ( d / j ( re ) k T l ψ (2) ) (cid:17) + (cid:90) R u ( r ≥ r ) r b − (cid:16) ( d / j ( re ) k T l ψ (1) ) + ( d / j ( re ) k T l ψ (1) ) (cid:17) (cid:46) (cid:90) ( ext ) M ( r ≤ r ≤ r ) r b − (cid:16) ( d / j ( re ) k T l ψ (1) ) + ( d / j ( re ) k T l ψ (1) ) (cid:17) + (cid:90) R u ( r ≥ r ) r b +1 (cid:16) ( h (1) ,j,k,l ) + ( h (2) ,j,k,l ) (cid:17) + (cid:90) R u ( r ≥ r ) r b E [ d /, j, k, ( re ) k T l ψ (1) , ( re ) k T l ψ (2) ] . .7. PROOF OF PROPOSITION 8.3.8 (b) If − a (1) + 2 k + b + 2 > and a (2) − k − b − , then, we have (cid:90) C u ( r ≥ r ) r b ( d / j ( re ) k T l ψ (1) ) + (cid:90) Σ ∗ ( ≤ u ) r b (cid:16) ( d / j ( re ) k T l ψ (1) ) + ( d / j ( re ) k T l ψ (2) ) (cid:17) (cid:46) (cid:90) ( ext ) M ( r ≤ r ≤ r ) r b − (cid:16) ( d / j ( re ) k T l ψ (1) ) + ( d / j ( re ) k T l ψ (1) ) (cid:17) + (cid:90) R u ( r ≥ r ) r b +1 − δ B ( h (1) ,j,k,l ) + (cid:90) R u ( r ≥ r ) r b +1+ δ B ( h (2) ,j,k,l ) + (cid:90) R u ( r ≥ r ) r b − δ B ( d / j ( re ) k T l ψ (1) ) + (cid:90) R u ( r ≥ r ) r b − − δ B ( d / j ( re ) k T l ψ (2) ) + (cid:90) R u ( r ≥ r ) r b E [ d /, j, k, ( re ) k T l ψ (1) , ( re ) k T l ψ (2) ] . (c) If − a (1) + 2 k + b + 2 > and a (2) − k − b − < , then, we have (cid:90) C u ( r ≥ r ) r b ( d / j ( re ) k T l ψ (1) ) + (cid:90) Σ ∗ ( ≤ u ) r b (cid:16) ( d / j ( re ) k T l ψ (1) ) + ( d / j ( re ) k T l ψ (2) ) (cid:17) + (cid:90) R u ( r ≥ r ) r b − ( d / j ( re ) k T l ψ (1) ) (cid:46) (cid:90) ( ext ) M ( r ≤ r ≤ r ) r b − (cid:16) ( d / j ( re ) k T l ψ (1) ) + ( d / j ( re ) k T l ψ (1) ) (cid:17) + (cid:90) R u ( r ≥ r ) r b +1 ( h (1) ,j,k,l ) + (cid:90) R u ( r ≥ r ) r b +1 ( h (2) ,j,k,l ) + (cid:90) R u ( r ≥ r ) r b − (( d / j ( re ) k T l ψ (2) ) + (cid:90) R u ( r ≥ r ) r b E [ d /, j, k, ( re ) k T l ψ (1) , ( re ) k T l ψ (2) ] . Proof.
We multiply the pair ( ψ (1) , ψ (2) ) by a smooth cut-off function in r supported in r ≥ r and identically one for r ≥ r . We obtain again a solution to (8.7.1) or (8.7.2)up to error terms that are supported in the region r ≤ r ≤ r . We then integrate thedivergence identities of Lemma 8.7.5 on the region R u and the corollary follows.80 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Let r ≥ m . Recall that, to prove Proposition 8.3.8, it suffices to establish the followinginequality ( ext ) R ≥ r J +1 [ ˇ R ] (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) . To this end, we will rely on the r p -weighted estimates derived in Corollary 8.7.6 appliedto the Bianchi pairs, where we recall Remark 8.7.2. Remark 8.7.7.
For the Bianchi pair ( β, ρ ) , we replace the Bianchi identities for e ( ρ ) by its analog for e ( ˇ ρ ) , i.e. e ˇ ρ + 32 κ ˇ ρ = d/ β − ρ ˇ κ + Err [ e ˇ ρ ] , while for the Bianchi pair ( ρ, β ) , we replace the Bianchi identities for e ( ρ ) by its analogfor e ( ˇ ρ ) , i.e. e ˇ ρ + 32 κ ˇ ρ = d/ β − ρ ˇ κ − κ ρς − ˇ ς + 32 κ ρ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) + Err [ e ˇ ρ ] , see Proposition 2.2.18 for the derivation of these equations. Let j, k, l three integers such that j + k + l = J + 1 . To derive r p weighted curvature estimates for d / j ( re ) k T l derivatives in the region r ≥ r ,we proceed as follows. Step 1.
We start with the case k = 0, i.e. we derive r p weighted curvature estimates for d / j T l derivatives with j + l = J + 1. First, we apply Corollary 8.7.6 • to the Bianchi pair ( α, β ) with the choice b = 4 + δ B , • to the Bianchi pair ( β, ρ ) with the choice b = 4 − δ B , • to the Bianchi pair ( ρ, β ) with the choice b = 2 − δ B , • to the Bianchi pair ( β, α ) with the choice b = − δ B . .7. PROOF OF PROPOSITION 8.3.8 (cid:88) j + l = J +1 (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r δ B ( d / j T l α ) + r − δ B ( d / j T l β ) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r δ B (cid:16) ( d / j T l α ) + ( d / j T l β ) (cid:17) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − δ B ( d / j T l α ) (cid:17) + (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B (cid:16) ( d / j T l α ) + ( d / j T l β ) (cid:17) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − − δ B ( d / j T l α ) (cid:17)(cid:41) (cid:46) r δ B (cid:90) ( ext ) M ( r ≤ r ≤ r ) ( d J +1 ˇ R ) r + (cid:90) ( ext ) M ( r ≥ r ) (cid:40) r − δ B ( d / j T l ϑ ) + r − − δ B (cid:16) ( d / j T l η ) + ( d / j T l ˇ κ ) (cid:17) + r − − δ B (cid:16) ( d / j T l ˇ κ ) + ( d / j T l ζ ) (cid:17) + r − − δ B (cid:16) ( d / j T l ξ ) + ( d / j T l ϑ ) + ( d / j T l ˇ ς ) + ( d / j T l ˇΩ) (cid:17)(cid:41) + ( (cid:15) B [ J ]) + (cid:15) ( N ( En ) J +1 ) . Using Proposition 8.3.7 to bound the first term on the right-hand side, and using also thedefinition of the norm ( ext ) G ≥ r k [ˇΓ], we infer that (cid:88) j + l = J +1 (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r δ B ( d / j T l α ) + r − δ B ( d / j T l β ) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r δ B (cid:16) ( d / j T l α ) + ( d / j T l β ) (cid:17) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − δ B ( d / j T l α ) (cid:17) + (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B (cid:16) ( d / j T l α ) + ( d / j T l β ) (cid:17) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − − δ B ( d / j T l α ) (cid:17)(cid:41) (cid:46) (cid:18)(cid:90) + ∞ r drr δ B (cid:19) ( ext ) G ≥ r k [ˇΓ] + r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and hence (cid:88) j + l = J +1 (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r δ B ( d / j T l α ) + r − δ B ( d / j T l β ) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r δ B (cid:16) ( d / j T l α ) + ( d / j T l β ) (cid:17) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − δ B ( d / j T l α ) (cid:17) + (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B (cid:16) ( d / j T l α ) + ( d / j T l β ) (cid:17) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − − δ B ( d / j T l α ) (cid:17)(cid:41) (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) . (8.7.4) Step 2.
We derive additional r p weighted curvature estimates for d / j T l derivatives with j + l = J + 1. To this end, we apply Corollary 8.7.6 • to the Bianchi pair ( β, ρ ) with the choice b = 4, • to the Bianchi pair ( ρ, β ) with the choice b = 2, • to the Bianchi pair ( β, α ) with the choice b = 0.All the above choices are such that we have in case (b) of Corollary 8.7.6. In particular, .7. PROOF OF PROPOSITION 8.3.8 (cid:88) j + l = J +1 (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r ( d / j T l β ) + r ( d / j T l ˇ ρ ) + ( d / j T l β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r (cid:16) ( d / j T l β ) + ( d / j T l ˇ ρ ) (cid:17) + r ( d / j T l β ) + ( d / j T l α ) (cid:17)(cid:41) (cid:46) r (cid:90) ( ext ) M ( r ≤ r ≤ r ) ( d J +1 ˇ R ) r + (cid:88) j + l = J +1 (cid:40) (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B ( d / j T l β ) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − − δ B ( d / j T l α ) (cid:17)(cid:41) + (cid:90) ( ext ) M ( r ≥ r ) (cid:40) r − − δ B ( d / j T l η ) + r − δ B ( d / j T l ˇ κ ) + r − δ B (cid:16) ( d / j T l ˇ κ ) + ( d / j T l ζ ) (cid:17) + r − δ B (cid:16) ( d / j T l ξ ) + ( d / j T l ϑ ) + ( d / j T l ˇ ς ) + ( d / j T l ˇΩ) (cid:17)(cid:41) + ( (cid:15) B [ J ]) + (cid:15) ( N ( En ) J +1 ) . Using Proposition 8.3.7 to bound the first term on the right-hand side, and using also thedefinition of the norm ( ext ) G ≥ r k [ˇΓ], we infer that (cid:88) j + l = J +1 (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r ( d / j T l β ) + r ( d / j T l ˇ ρ ) + ( d / j T l β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r (cid:16) ( d / j T l β ) + ( d / j T l ˇ ρ ) (cid:17) + r ( d / j T l β ) + ( d / j T l α ) (cid:17)(cid:41) (cid:46) (cid:18)(cid:90) + ∞ r drr δ B (cid:19) ( ext ) G ≥ r k [ˇΓ] + r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) + (cid:88) j + l = J +1 (cid:40) (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B ( d / j T l β ) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − − δ B ( d / j T l α ) (cid:17)(cid:41) CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and hence (cid:88) j + l = J +1 (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r ( d / j T l β ) + r ( d / j T l ˇ ρ ) + ( d / j T l β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r (cid:16) ( d / j T l β ) + ( d / j T l ˇ ρ ) (cid:17) + r ( d / j T l β ) + ( d / j T l α ) (cid:17)(cid:41) (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) + (cid:88) j + l = J +1 (cid:40) (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B ( d / j T l β ) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − − δ B ( d / j T l α ) (cid:17)(cid:41) . Together with (8.7.4), we deduce (cid:88) j + l = J +1 (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r δ B ( d / j T l α ) + r ( d / j T l β ) + r ( d / j T l ˇ ρ ) +( d / j T l β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r δ B (cid:16) ( d / j T l α ) + ( d / j T l β ) (cid:17) + r ( d / j T l ˇ ρ ) + r ( d / j T l β ) + ( d / j T l α ) (cid:17) + (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B (cid:16) ( d / j T l α ) + ( d / j T l β ) (cid:17) + r − δ B ( d / j T l ˇ ρ ) + r − δ B ( d / j T l β ) + r − − δ B ( d / j T l α ) (cid:17)(cid:41) (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) . (8.7.5) Step 3.
We now argue by iteration on k . For 0 ≤ k ≤ J , we consider the following .7. PROOF OF PROPOSITION 8.3.8 (cid:88) j + l = J +1 − k (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r δ B ( d / j ( re ) k T l α ) + r ( d / j ( re ) k T l β ) + r ( d / j ( re ) k T l ˇ ρ ) +( d / j ( re ) k T l β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r δ B (cid:16) ( d / j ( re ) k T l α ) + ( d / j ( re ) k T l β ) (cid:17) + r ( d / j ( re ) k T l ˇ ρ ) + r ( d / j ( re ) k T l β ) + ( d / j ( re ) k T l α ) (cid:17) + (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B (cid:16) ( d / j ( re ) k T l α ) + ( d / j ( re ) k T l β ) (cid:17) + r − δ B ( d / j ( re ) k T l ˇ ρ ) + r − δ B ( d / j ( re ) k T l β ) + r − − δ B ( d / j ( re ) k T l α ) (cid:17)(cid:41) (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) . (8.7.6)(8.7.6) holds true for k = 0 in view of (8.7.5). We now assume that (8.7.6) holds true for k such that 0 ≤ k ≤ J , and our goal is to prove that it also holds for k + 1.First, note that the Bianchi identities for e ( β ), e ( ˇ ρ ), e ( β ) and e ( α ), together with(8.7.6), yields (cid:88) j + l = J +1 − ( k +1) (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r ( d / j ( re ) k +1 T l β ) + r ( d / j ( re ) k +1 T l ˇ ρ ) +( d / j ( re ) k +1 T l β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r δ B ( d / j ( re ) k +1 T l β ) + r ( d / j ( re ) k +1 T l ˇ ρ ) + r ( d / j ( re ) k +1 T l β ) + ( d / j ( re ) k +1 T l α ) (cid:17) + (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B ( d / j ( re ) k +1 T l β ) + r − δ B ( d / j ( re ) k +1 T l ˇ ρ ) + r − δ B ( d / j ( re ) k +1 T l β ) + r − − δ B ( d / j ( re ) k +1 T l α ) (cid:17)(cid:41) (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) . (8.7.7)We still need to estimate d / j ( re ) k +1 T l α . To this end, we apply Corollary 8.7.6 to theBianchi pair ( α, β ) with the choice b = 4 + δ B . Since k + 1 ≥
1, we are in case (c) of86
CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Corollary 8.7.6. In particular, we obtain, arguing similarly as above, (cid:88) j + l = J +1 − ( k +1) (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) r δ B ( d / j ( re ) k +1 T l α ) + (cid:90) Σ ∗ r δ B ( d / j ( re ) k +1 T l α ) + (cid:90) ( ext ) M ( r ≥ r ) r δ B ( d / j ( re ) k +1 T l α ) (cid:41) (cid:46) (cid:88) j + l = J +1 − ( k +1) (cid:26)(cid:90) ( ext ) M ( r ≥ r ) r δ B ( d / j ( re ) k +1 T l β ) (cid:27) + r − δ B ext ) G ≥ r k [ˇΓ]+ r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) . Together with (8.7.7), this implies (8.7.6) for k + 1. Hence, by iteration, (8.7.6) holds forany 0 ≤ k ≤ J + 1. This implies (cid:88) k ≤ J +1 (cid:40) sup ≤ u ≤ u ∗ (cid:90) C u ( r ≥ r ) (cid:16) r δ B ( d k α ) + r ( d k β ) + r ( d k ˇ ρ ) + ( d k β ) (cid:17) + (cid:90) Σ ∗ (cid:16) r δ B (cid:16) ( d k α ) + ( d k β ) (cid:17) + r ( d k ˇ ρ ) + r ( d k β ) + ( d k α ) (cid:17) + (cid:90) ( ext ) M ( r ≥ r ) (cid:16) r δ B (cid:16) ( d k α ) + ( d k β ) (cid:17) + r − δ B ( d k ˇ ρ ) + r − δ B ( d k β ) + r − − δ B ( d k α ) (cid:17)(cid:41) (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) ( (cid:15) B [ J ]) + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) (cid:17) . Hence, we have obtained ( ext ) R ≥ r J +1 [ ˇ R ] (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) which concludes the proof of Proposition 8.3.8. To prove Proposition 8.3.9, we rely on the following three propositions.
Proposition 8.8.1.
Let J such that k small − ≤ J ≤ k large − . Then, we have (Σ ∗ ) G J +1 [ˇΓ] + (Σ ∗ ) G (cid:48) J +1 [ˇΓ] (cid:46) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] , .8. PROOF OF PROPOSITION 8.3.9 where we have introduced the notations (Σ ∗ ) G k [ˇΓ] := (cid:90) Σ ∗ (cid:34) r (cid:16) ( d ≤ k ϑ ) + ( d ≤ k ˇ κ ) + ( d ≤ k ζ ) + ( d ≤ k ˇ κ ) (cid:17) + ( d ≤ k ϑ ) +( d ≤ k η ) + ( d ≤ k ˇ ω ) + ( d ≤ k ξ ) (cid:35) , (Σ ∗ ) G (cid:48) k [ˇΓ] := (cid:90) Σ ∗ (cid:104) r (cid:16) ( d k +1 d / ˇ κ ) + ( d k +1 ˇ κ ) + ( d ≤ k +1 ˇ µ ) + ( d k +1 ˇ κ ) + ( d k +1 ζ ) (cid:105) , and (Σ ∗ ) R k [ ˇ R ] := (cid:90) Σ ∗ (cid:16) r δ B (cid:0) ( d ≤ k α ) + ( d ≤ k β ) (cid:1) + r ( d ≤ k ˇ ρ ) + r ( d ≤ k β ) + ( d ≤ k α ) (cid:17) . Proposition 8.8.2.
Let J such that k small − ≤ J ≤ k large − . Then, we have ( ext ) G ≥ m J +1 [ˇΓ] + ( ext ) G ≥ m J +1 (cid:48) [ˇΓ] (cid:46) (Σ ∗ ) G J +1 [ˇΓ] + (Σ ∗ ) G (cid:48) J +1 [ˇΓ] + ( ext ) R J +1 [ ˇ R ] + ( ext ) G J [ˇΓ] , where we have introduced the notation ( ext ) G ≥ m k (cid:48) [ˇΓ] := sup λ ≥ m (cid:32) (cid:90) { r = λ } (cid:34) λ (cid:16) d k (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17)(cid:17) + λ ( d k +1 ˇ κ ) + λ (cid:0) d k (cid:0) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:1)(cid:1) + λ ( d ≤ k ˇ µ ) + λ (cid:0) d k (cid:0) e θ ( κ ) − β (cid:1)(cid:1) + λ (cid:0) d k (cid:0) e ( ζ ) + β (cid:1)(cid:1) (cid:35)(cid:33) . Proposition 8.8.3.
Let J such that k small − ≤ J ≤ k large − . Then, we have ( ext ) G ≤ m J +1 [ˇΓ] + ( ext ) G ≤ m J +1 (cid:48) [ˇΓ] (cid:46) ( ext ) G ≥ m J +1 [ˇΓ] + ( ext ) G ≥ m J +1 (cid:48) [ˇΓ] + ( ext ) R J +1 [ ˇ R ] + ( ext ) G J [ˇΓ] , where we have introduced the notation ( ext ) G ≤ m k (cid:48) [ˇΓ] := sup r T ≤ λ ≤ m (cid:32) (cid:90) { r = λ } (cid:34) λ (cid:16) d k (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17)(cid:17) + λ ( d k +1 ˇ κ ) + λ (cid:0) d k (cid:0) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:1)(cid:1) + λ ( d ≤ k ˇ µ ) + λ (cid:0) d k (cid:0) e θ ( κ ) − β (cid:1)(cid:1) + λ (cid:0) d k − N (cid:0) e ( ζ ) + β (cid:1)(cid:1) (cid:35)(cid:33) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
The proof of Proposition 8.8.1 is postponed to section 8.8.1, the proof of Proposition 8.8.2is postponed to section 8.8.4, and the proof of Proposition 8.8.3 is postponed to section8.8.5. The proof of the two latter propositions will rely in particular on basic weightedestimates for transport equations along e in ( ext ) M derived in section 8.8.2, as well asseveral renormalized identities derived in section 8.8.3We now conclude the proof of Proposition 8.3.9. In view of Propositions 8.8.1, 8.8.2 and8.8.3, we have, for J such that k small − ≤ J ≤ k large − ( ext ) G J +1 [ˇΓ] (cid:46) ( ext ) R J +1 [ ˇ R ] + ( ext ) G J [ˇΓ] , where we have used the fact that (Σ ∗ ) R J +1 [ ˇ R ] ≤ ( ext ) R J +1 [ ˇ R ] , (Σ ∗ ) G J [ˇΓ] ≤ ( ext ) G J [ˇΓ] . In view of the iteration assumption (8.3.10), we infer ( ext ) G J +1 [ˇΓ] (cid:46) ( ext ) R J +1 [ ˇ R ] + (cid:15) B [ J ] . Since the estimates in Proposition 8.8.2 are integrated from Σ ∗ , we obtain similarly, forany r ≥ m , ( ext ) G ≥ r J +1 [ˇΓ] (cid:46) ( ext ) R ≥ r J +1 [ ˇ R ] + (cid:15) B [ J ] . On the other hand, we have in view of Proposition 8.3.8, for any r ≥ m , ( ext ) R ≥ r J +1 [ ˇ R ] (cid:46) r − δ B ext ) G ≥ r k [ˇΓ] + r (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17) . and ( int ) R J +1 [ ˇ R ] + ( ext ) R J +1 [ ˇ R ] ≤ ( ext ) R ≥ r J +1 [ ˇ R ] + O (cid:16) r (cid:16) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17)(cid:17)(cid:17) . Choosing r ≥ m large enough, we infer from the above estimates ( ext ) G J +1 [ˇΓ] + ( int ) R J +1 [ ˇ R ] + ( ext ) R J +1 [ ˇ R ] (cid:46) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) . This concludes the proof of Proposition 8.3.9.
Step 1.
We control κ on Σ ∗ . Recall the GCM conditions κ = 2 /r on Σ ∗ . Since ν Σ ∗ and e θ are tangent, we infer ( d /, ν Σ ∗ ) k (cid:18) κ − r (cid:19) = 0 . .8. PROOF OF PROPOSITION 8.3.9 k ≤ J +2 (cid:90) Σ ∗ (cid:32) r (cid:18) d k (cid:18) κ − r (cid:19)(cid:19) + r (cid:0) d k e θ ( κ ) (cid:1) (cid:33) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) , where we have used the fact that e is in the span of e and ν Σ ∗ . Note that we have usedCodazzi for ϑ to control the term d J +1 e ( e θ ( κ )). Step 2.
We control the (cid:96) = 1 modes on Σ ∗ . In view of the GCM conditions for κ , andprojecting the Codazzi for ϑ on the (cid:96) = 1 mode, we infer on Σ ∗ (cid:90) S ζe Φ = r (cid:90) S βe Φ + r (cid:90) S ϑζe Φ . Since the vectorfield ν is tangent to Σ ∗ , we infer ν J +2 (cid:18)(cid:90) S ζe Φ (cid:19) = r (cid:90) S ν J +2 βe Φ + r (cid:90) S ν J +2 ( ϑζ ) e Φ + l.o.t.= r (cid:90) S ν J +2 βe Φ + r (cid:90) S ζν J +2 ( ϑ ) e Φ + r (cid:90) S ϑν J +2 ( ζ ) e Φ + l.o.t.where l.o.t. denote, here and below, terms that • either are linear and contain at most J + 1 derivatives of curvature components and J derivatives of Ricci coefficients, • or are quadratic and contain at most J + 1 derivatives of Ricci coefficients andcurvature components.Using Bianchi identities and the null structure equations, we deduce ν J +2 (cid:18)(cid:90) S ζe Φ (cid:19) = r (cid:90) S ν J +1 ( d/ α, d (cid:63) / ρ − ρη ) e Φ + r (cid:90) S ζν J +1 d (cid:63) / ηe Φ + r (cid:90) S ϑν J +1 d (cid:63) / ωe Φ + l.o.t.= r (cid:90) S ( d/ ν J +1 α, ν J d (cid:63) / d/ ( β, β )) e Φ − ρ (cid:90) S ν J +1 ηe Φ + r (cid:90) S ζ d (cid:63) / ν J +1 ηe Φ + r (cid:90) S ϑ d (cid:63) / ν J +1 ωe Φ + l.o.t.= r (cid:90) S ( d/ ν J +1 α, d (cid:63) / d/ ν J ( β, β )) e Φ + r (cid:90) S ζ d (cid:63) / ν J +1 ηe Φ + r (cid:90) S ϑ d (cid:63) / ν J +1 ωe Φ + l.o.t. , CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) where we have used, in the last equality, a cancellation due to the fact that ν is tangentto Σ ∗ and (cid:82) S ηe Φ = 0 on Σ ∗ . Using the identity d (cid:63) / d/ = d/ d (cid:63) / + 2 K , integration by partsfor all terms, and the fact that d (cid:63) / ( e Φ ) = 0 so that the top order linear term vanish, weinfer ν J +2 (cid:18)(cid:90) S ζe Φ (cid:19) = l.o.t.with the above convention for the lower order terms. Also, relying on the null equationfor e ( ζ ), i.e. e ( ζ ) = − κζ − β − ϑζ we obtain, with more ease since this estimate is at one lower level of derivatives( re , ν ) J +2 (cid:18)(cid:90) S ζe Φ (cid:19) = l.o.t.We infermax k ≤ J +2 (cid:90) u ∗ r − (cid:18) d k (cid:18)(cid:90) S ζe Φ (cid:19)(cid:19) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Next, we have in view of the definition of µ and the identity d (cid:63) / d/ = d/ d (cid:63) / + 2 K (cid:90) S e θ ( µ ) e Φ = (cid:90) S d (cid:63) / d/ ζe Φ − (cid:90) S e θ ( ρ ) e Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ = 2 (cid:90) S Kζe Φ − (cid:90) S e θ ( ρ ) e Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ = 2 r (cid:90) S ζe Φ − (cid:90) S e θ ( ρ ) e Φ + (cid:90) S (cid:18) K − r (cid:19) ζe Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ . To estimate the RHS, we use in particular .8. PROOF OF PROPOSITION 8.3.9 • for the second term, in view of Bianchi( e , re ) J +2 e θ ( ρ )= ( e , re ) J +1 d (cid:63) / d/ ( rβ, β ) −
32 ( e , re ) J +1 d (cid:63) / ( rκρ, κρ )+( e , re ) J +1 d (cid:63) / (cid:0) rϑα, ( rζ, ξ ) β, ( ζ, η ) β, ϑα (cid:1) + l.o.t.= ( e , re ) J +1 d/ d (cid:63) / ( rβ, β ) + 32 ρ ( e , re ) J +1 ( re θ ( κ ) , e θ ( κ ))+ (cid:0) rϑ ( e , re ) J +1 d (cid:63) / α, ( rζ, ξ )( e , re ) J +1 d (cid:63) / β, ( ζ, η )( e , re ) J +1 d (cid:63) / β, ϑ ( e , re ) J +1 d (cid:63) / α (cid:1) + (cid:0) rα ( e , re ) J +1 d (cid:63) / ϑ, β ( e , re ) J +1 d (cid:63) / ( rζ, ξ ) , β ( e , re ) J +1 d (cid:63) / ( ζ, η ) , α ( e , re ) J +1 d (cid:63) / ϑ (cid:1) + l.o.t.= d/ ( e , re ) J +1 d (cid:63) / ( rβ, β ) + [( e , re ) J +1 , d/ ] d (cid:63) / ( rβ, β ) + 32 ρ ( e , re ) J +1 ( re θ ( κ ) , e θ ( κ )) −
32 ˇ ρ d (cid:63) / ( e , re ) J +1 ( r ˇ κ, ˇ κ )+ (cid:0) rϑ d (cid:63) / ( e , re ) J +1 α, ( rζ, ξ ) d (cid:63) / ( e , re ) J +1 β, ( ζ, η ) d (cid:63) / ( e , re ) J +1 β, ϑ d (cid:63) / ( e , re ) J +1 α (cid:1) + (cid:0) rα d (cid:63) / ( e , re ) J +1 ϑ, β d (cid:63) / ( e , re ) J +1 ( rζ, ξ ) , β d (cid:63) / ( e , re ) J +1 ( ζ, η ) , α d (cid:63) / ( e , re ) J +1 ϑ (cid:1) + l.o.t. • for the third term( e , re ) J +2 (cid:16) (cid:18) K − r (cid:19) ζ (cid:17) = ζ ( e , re ) J +2 (cid:18) K − r (cid:19) + (cid:18) K − r (cid:19) ( e , re ) J +2 ζ + l.o.t.= ζ ( e , re ) J +1 (cid:16) d/ ( rβ, β, η, r − ξ ) (cid:17) + (cid:18) K − r (cid:19) ( e , re ) J +1 e θ ( ω ) + l.o.t.= ζ ( e , re ) J +1 d/ (cid:16) rβ, β, η, r − ξ (cid:17) + (cid:18) K − r (cid:19) ( e , re ) J +1 e θ ( ω ) + l.o.t.= ζ d/ ( e , re ) J +1 (cid:16) rβ, β, η, r − ξ (cid:17) + (cid:18) K − r (cid:19) d (cid:63) / ( e , re ) J +1 ˇ ω + l.o.t. , • for the fourth term( e , re ) J +2 e θ ( ϑϑ ) = ( e , re ) J +1 e θ ( ϑ d (cid:63) / ( ξ, rζ )) + ( e , re ) J +1 e θ ( ϑ d (cid:63) / η ) + l.o.t.= ϑ d (cid:63) / d (cid:63) / ( e , re ) J +1 ( ξ, rζ ) + ϑ d (cid:63) / d (cid:63) / ( e , re ) J +1 η + l.o.t.92 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
We infer( e , re ) J +2 (cid:18)(cid:90) S e θ ( µ ) e Φ (cid:19) = 2 r ( e , re ) J +2 (cid:18)(cid:90) S ζe Φ (cid:19) + (cid:90) S d/ ( e , re ) J +1 d (cid:63) / ( rβ, β ) e Φ + (cid:90) S [( e , re ) J +1 , d/ ] d (cid:63) / ( rβ, β ) e Φ + 32 ρ (cid:90) S ( e , re ) J +1 ( re θ ( κ ) , e θ ( κ )) e Φ − (cid:90) S ˇ ρ d (cid:63) / ( e , re ) J +1 ( r ˇ κ, ˇ κ ) e Φ + (cid:90) S (cid:0) rϑ d (cid:63) / ( e , re ) J +1 α, ( rζ, ξ ) d (cid:63) / ( e , re ) J +1 β, ( ζ, η ) d (cid:63) / ( e , re ) J +1 β, ϑ d (cid:63) / ( e , re ) J +1 α (cid:1) e Φ + (cid:90) S (cid:0) rα d (cid:63) / ( e , re ) J +1 ϑ, β d (cid:63) / ( e , re ) J +1 ( rζ, ξ ) , β d (cid:63) / ( e , re ) J +1 ( ζ, η ) , α d (cid:63) / ( e , re ) J +1 ϑ (cid:1) e Φ + (cid:90) S ζ d/ ( e , re ) J +1 (cid:16) rβ, β, ζ, r − ξ (cid:17) e Φ + (cid:90) S (cid:18) K − r (cid:19) d (cid:63) / ( e , re ) J +1 ˇ ωe Φ + (cid:90) S ϑ d (cid:63) / d (cid:63) / ( e , re ) J +1 ( ξ, rζ ) e Φ + (cid:90) S ϑ d (cid:63) / d (cid:63) / ( e , re ) J +1 ζe Φ + l.o.t.and after integrations by parts and the fact that d/ k ( F e Φ ) = d/ k +1 ( F ) e Φ , d (cid:63) / k ( F e Φ ) = d (cid:63) / k − ( F ) e Φ , we obtain( e , re ) J +2 (cid:18)(cid:90) S e θ ( µ ) e Φ (cid:19) = 2 r ( e , re ) J +2 (cid:18)(cid:90) S ζe Φ (cid:19) + (cid:90) S d (cid:63) / (cid:16) [( e , re ) J +1 , d/ ] (cid:17) ( rβ, β ) e Φ + 32 ρ (cid:90) S ( e , re ) J +1 ( re θ ( κ ) , e θ ( κ )) e Φ + 32 (cid:90) S d/ ρ ( e , re ) J +1 ( r ˇ κ, ˇ κ ) e Φ + (cid:90) S (cid:0) r d/ ϑ ( e , re ) J +1 α, d/ ( rζ, ξ )( e , re ) J +1 β, d/ ( ζ, η )( e , re ) J +1 β, d/ ϑ ( e , re ) J +1 α (cid:1) e Φ + (cid:90) S (cid:0) r d/ α ( e , re ) J +1 ϑ, d/ β ( e , re ) J +1 ( rζ, ξ ) , d/ β ( e , re ) J +1 ( ζ, η ) , d/ α ( e , re ) J +1 ϑ (cid:1) e Φ + (cid:90) S d/ ζ ( e , re ) J +1 (cid:16) rβ, β, ζ, r − ξ (cid:17) e Φ + (cid:90) S d/ (cid:18) K − r (cid:19) ( e , re ) J +1 ˇ ωe Φ + (cid:90) S d/ d/ ϑ ( e , re ) J +1 ( ξ, rζ ) e Φ + (cid:90) S d/ d/ ϑ ( e , re ) J +1 ζe Φ + l.o.t. .8. PROOF OF PROPOSITION 8.3.9 (cid:96) = 1 mode of ζ and the estimate of Step 1 for κ , we infermax k ≤ J +2 (cid:90) u ∗ r (cid:18) d k (cid:18)(cid:90) S e θ ( µ ) e Φ (cid:19)(cid:19) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) + max k ≤ J +1 (cid:90) u ∗ r − (cid:18) d k (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:19) . In view of the dominant condition (3.3.4) for r on Σ ∗ , we infermax k ≤ J +2 (cid:90) u ∗ r (cid:18) d k (cid:18)(cid:90) S e θ ( µ ) e Φ (cid:19)(cid:19) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) + (cid:15) max k ≤ J +1 (cid:90) u ∗ (cid:18) d k (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:19) . Next, in view of the remarkable identity for the (cid:96) = 1 mode of e θ ( K ), we have − (cid:90) S e θ ( ρ ) e Φ − (cid:90) S e θ ( κκ ) e Φ + 14 (cid:90) S e θ ( ϑϑ ) e Φ = 0and hence (cid:90) S e θ ( κ ) e Φ = − r (cid:90) S e θ ( ρ ) e Φ − r (cid:90) S (cid:18) κ − r (cid:19) e θ ( κ ) e Φ − r (cid:90) S κe θ ( κ ) e Φ + r (cid:90) S e θ ( ϑϑ ) e Φ . Arguing as for the estimate of the (cid:96) = 1 mode of e θ ( µ ), and using the smallness of (cid:15) , weinfer max k ≤ J +2 (cid:90) u ∗ r (cid:18) d k (cid:18)(cid:90) S e θ ( µ ) e Φ (cid:19)(cid:19) + max k ≤ J +2 (cid:90) u ∗ (cid:18) d k (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:19) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . We have thus obtainedmax k ≤ J +2 (cid:90) u ∗ (cid:32) r − (cid:18) d k (cid:18)(cid:90) S ζe Φ (cid:19)(cid:19) + r (cid:18) d k (cid:18)(cid:90) S e θ ( µ ) e Φ (cid:19)(cid:19) + (cid:18) d k (cid:18)(cid:90) S e θ ( κ ) e Φ (cid:19)(cid:19) (cid:33) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Step 3.
Recall the GCM conditions d (cid:63) / d (cid:63) / κ = d (cid:63) / d (cid:63) / µ = 0 on Σ ∗ . This yields on Σ ∗ e θ ( µ ) = (cid:82) S e θ ( µ ) e Φ (cid:82) S e e Φ , e θ ( κ ) = (cid:82) S e θ ( κ ) e Φ (cid:82) S e e Φ . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Together with Step 2, we infermax k ≤ J +2 (cid:90) Σ ∗ (cid:16) r (cid:0) ( d /, ν Σ ∗ ) k ˇ µ (cid:1) + r (cid:0) ( d /, ν Σ ∗ ) k ˇ κ (cid:1) (cid:17) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Then, in view of the null structure equations for e (ˇ µ ) and e (ˇ κ ), e (ˇ µ ) = − κ ˇ µ − µ ˇ κ + Err[ e ˇ µ ] e (ˇ κ ) = − κ ˇ κ − κ ˇ κ + 2ˇ µ + 4 ˇ ρ + Err[ e ˇ κ ] , we infer, together with the control of ˇ κ provided by Step 1,max k ≤ J +2 (cid:90) Σ ∗ (cid:16) r (cid:0) d k ˇ µ (cid:1) + r (cid:0) d k ˇ κ (cid:1) (cid:17) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Step 4.
Recall that we have d/ ζ = − ˇ µ − ˇ ρ + 14 ϑϑ. Differentiating, and using the Bianchi identities for e ( ˇ ρ ) and e ( ˇ ρ ), and the null structureequations for e ( ϑ ), e ( ϑ ), e ( ϑ ) and e ( ϑ ), we infer d/ d k ζ = − d k ˇ µ − d k − d / (cid:16) ˇ ρ, β, r − β (cid:17) + 14 d k − (cid:16) d / ( ϑϑ ) , r − ϑ d /η, ϑ d /ζ, r − ϑ d /ξ (cid:17) + l.o.t.= − d k ˇ µ − d / d k − (cid:16) ˇ ρ, β, r − β (cid:17) + 14 d / (cid:16) ϑ d k − ( ϑ, r − η ) (cid:17) + 14 d / (cid:16) ϑ d k − ( ϑ, ζ, r − ξ ) (cid:17) + l.o.t.We infer, since d/ is invertible in view of the corresponding Poincar´e inequality, d k ζ = − r d / − d k ˇ µ − d k − (cid:16) ˇ ρ, β, r − β (cid:17) + 14 (cid:16) ϑ d k − ( ϑ, r − η ) (cid:17) + 14 (cid:16) ϑ d k − ( ϑ, ζ, r − ξ ) (cid:17) + l.o.t.Together with the estimate for ˇ µ of Step 3, this yieldsmax k ≤ J +2 (cid:90) Σ ∗ r ( d k ζ ) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Step 5.
Recall from the GCM condition that we have on Σ ∗ (cid:90) S ηe Φ = 0 . .8. PROOF OF PROPOSITION 8.3.9 e ( η − ζ ) = − κ ( η − ζ ) − ϑ ( η − ζ ) , we infer in view of the the estimates for ζ of Step 4,max k ≤ J +1 (cid:90) u ∗ r − (cid:18) d k (cid:18)(cid:90) S ηe Φ (cid:19)(cid:19) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Next, recall from Proposition 2.2.19 that η verifies2 d/ d (cid:63) / η = κ (cid:0) − e ( ζ ) + β (cid:1) − e ( e θ ( κ )) − κ (cid:18) κζ − ωζ (cid:19) + 6 ρη − κe θ κ − κe θ ( κ ) + 2 ωe θ ( κ ) + 2 e θ ( ρ ) + Err[ d/ d (cid:63) / η ] , Err[ d/ d (cid:63) / η ] = (cid:18) d/ η − κϑ + 2 η (cid:19) η + 2 e θ ( η ) − κ (cid:18) ϑζ − ϑξ (cid:19) − ϑe θ ( κ ) − (cid:18) d/ η − ϑϑ + 2 η (cid:19) ζ − e θ ( ϑ ϑ ) − ϑ ξ − ϑϑη. Together with the estimates for κ of Step 1, the estimates for κ of Step 3, and the estimatesfor ζ of Step 4,max k ≤ J +1 (cid:90) Σ ∗ (cid:18) d k (cid:18) r d/ d (cid:63) / η − r e θ ( ρ ) − r d/ ( η ) − r e θ ( η ) + r d/ ( ζη ) + 14 r e θ ( ϑ ϑ ) (cid:19)(cid:19) (cid:46) max k ≤ J +1 (cid:90) Σ ∗ r − | d k η | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . In view of the dominant condition (3.3.4) for r on Σ ∗ , we infermax k ≤ J +1 (cid:90) Σ ∗ (cid:18) d k (cid:18) r d/ d (cid:63) / η − r e θ ( ρ ) − r d/ ( η ) − r e θ ( η ) + r d/ ( ζη ) + 14 r e θ ( ϑ ϑ ) (cid:19)(cid:19) (cid:46) (cid:15) max k ≤ J +1 (cid:90) Σ ∗ | d k η | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . This yields max k ≤ J +1 (cid:90) Σ ∗ (cid:32) r d/ d (cid:63) / d k η + r d (cid:63) / [ d k , r d/ ] η + r d/ [ d k , r d (cid:63) / ] η − r e θ ( d k ρ ) − r d/ d k ( η ) − r e θ d k ( η ) + r d/ d k ( ζη ) + 14 r e θ d k ( ϑ ϑ ) (cid:33) (cid:46) (cid:15) max k ≤ J +1 (cid:90) Σ ∗ | d k η | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
We deduce, using a Poincar´e inequality for d/ ,max k ≤ J +1 (cid:90) Σ ∗ (cid:0) r d (cid:63) / d k η (cid:1) (cid:46) (cid:15) max k ≤ J +1 (cid:90) Σ ∗ | d k η | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Together with a Poincar´e inequality for r d (cid:63) / and the above control of the (cid:96) = 1 mode of η , we infermax k ≤ J +1 (cid:90) Σ ∗ (cid:0) d k η (cid:1) (cid:46) (cid:15) max k ≤ J +1 (cid:90) Σ ∗ | d k η | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) , and hence, for (cid:15) small enough,max k ≤ J +1 (cid:90) Σ ∗ (cid:0) d k η (cid:1) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Step 6.
Recall from the GCM condition that we have on Σ ∗ (cid:90) S ξe Φ = 0 . Together with the transport equation e ( ξ ) = − e ( ζ ) + β − κζ − ζϑ, we infer in view of the the estimates for ζ of Step 4, the estimates for β , and the bootstrapassumptionsmax k ≤ J +1 (cid:90) u ∗ r − (cid:18) d k (cid:18)(cid:90) S ξe Φ (cid:19)(cid:19) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Next, from Proposition 2.2.19 that we have2 d/ d (cid:63) / ξ = − e ( e θ ( κ )) + κ (cid:0) e ( ζ ) − β (cid:1) + κ ζ − κe θ κ + 6 ρξ − ωe θ ( κ ) + Err[ d/ d (cid:63) / ξ ] , Err[ d/ d (cid:63) / ξ ] = (cid:18) d/ ξ + 12 κ ϑ + 2 ηξ − ϑ (cid:19) η + 2 e θ ( ηξ ) − e θ ( ϑ )+ κ (cid:18) ϑζ − ϑξ (cid:19) − ϑe θ κ − ϑϑξ − ζ (cid:18) d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) + ξ (cid:16) − ϑϑ − d/ ζ + 2 ζ (cid:17) − ηζξ − e θ ( ζξ ) . .8. PROOF OF PROPOSITION 8.3.9 κ of Step 1, the estimates for κ of Step 3, and the estimatesfor ζ of Step 4,max k ≤ J +1 (cid:90) Σ ∗ (cid:18) d k (cid:18) r d/ d (cid:63) / ξ + 12 e θ ( e (ˇ κ )) − η d/ ξ − e θ ( ηξ ) + 14 e θ ( ϑ ) + d/ ( ζξ ) + 3 e θ ( ζξ ) (cid:19)(cid:19) (cid:46) max k ≤ J +1 (cid:90) Σ ∗ r − | d k ξ | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . In view of the dominant condition (3.3.4) for r on Σ ∗ , we infermax k ≤ J +1 (cid:90) Σ ∗ (cid:18) d k (cid:18) r d/ d (cid:63) / ξ + 12 e θ ( e (ˇ κ )) − η d/ ξ − e θ ( ηξ ) + 14 e θ ( ϑ ) + d/ ( ζξ ) + 3 e θ ( ζξ ) (cid:19)(cid:19) (cid:46) (cid:15) max k ≤ J +1 (cid:90) Σ ∗ | d k ξ | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . This yieldsmax k ≤ J +1 (cid:90) Σ ∗ (cid:32) r d/ d (cid:63) / d k ξ + r d (cid:63) / [ d k , r d/ ] ξ + r d/ [ d k , r d (cid:63) / ] ξ + 12 e θ ( d k e (ˇ κ )) − d/ ( η d k ξ ) − e θ d k ( ηξ ) + 14 e θ d k ( ϑ ) + d/ d k ( ζξ ) + 3 e θ d k ( ζξ ) (cid:33) (cid:46) (cid:15) max k ≤ J +1 (cid:90) Σ ∗ | d k ξ | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . We deduce, using a Poincar´e inequality for d/ and the estimates for κ of Step 3,max k ≤ J +1 (cid:90) Σ ∗ (cid:0) r d (cid:63) / d k ξ (cid:1) (cid:46) (cid:15) max k ≤ J +1 (cid:90) Σ ∗ | d k ξ | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Together with a Poincar´e inequality for r d (cid:63) / and the above control of the (cid:96) = 1 mode of ξ , we infermax k ≤ J +1 (cid:90) Σ ∗ (cid:0) d k ξ (cid:1) (cid:46) (cid:15) max k ≤ J +1 (cid:90) Σ ∗ | d k ξ | + (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) , and hence, for (cid:15) small enough,max k ≤ J +1 (cid:90) Σ ∗ (cid:0) d k ξ (cid:1) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Step 7.
Using the Codazzi for ϑ and ϑ , the transport equation for ϑ and ϑ in the e and e direction, the control of ˇ κ of Step 1, the control of ˇ κ of Step 3, the control of ζ of Step98 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
4, the control of η of Step 5, the control of ξ of Step 6, and a Poincar´e inequality for d/ ,we infermax k ≤ J +1 (cid:90) Σ ∗ (cid:16) r ( d k ϑ ) + ( d k ϑ ) (cid:17) (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Step 8.
Recall form Proposition 2.2.19 that ω verifies2 d (cid:63) / ω = − κξ + (cid:18) κ + 2 ω + 12 ϑ (cid:19) η + e ( ζ ) − β + 12 κζ − ωζ + 12 ϑζ − ϑξ. Together with a Poincar´e inequality for d (cid:63) / , the control of ξ from Step 6, the control of η from Step 5, and the control of ζ from Step 4, we infermax k ≤ J +1 (cid:90) Σ ∗ | d k ˇ ω | (cid:46) (cid:16) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] (cid:17) . Finally, gathering the estimates of Step 1 to Step 8, we infer (Σ ∗ ) G J +1 [ˇΓ] + (Σ ∗ ) G (cid:48) J +1 [ˇΓ] (cid:46) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ] + (cid:15) ∗ ) G J +1 [ˇΓ] . and hence, for (cid:15) small enough, (Σ ∗ ) G J +1 [ˇΓ] + (Σ ∗ ) G (cid:48) J +1 [ˇΓ] (cid:46) (Σ ∗ ) R J +1 [ ˇ R ] + (Σ ∗ ) G J [ˇΓ]as desired. This concludes the proof of Proposition 8.8.1. e in ( ext ) M Lemma 8.8.4.
Let the following transport equation in ( ext ) M e ( f ) + a κf = h where a ∈ R is a given constant, and f and h are scalar functions. Also, let and δ B > .Then, f satisfies sup r ≥ m (cid:18) r a − (cid:90) { r = r } f (cid:19) (cid:46) (cid:90) Σ ∗ r a − f + (cid:90) ( ext ) M ( ≥ m ) r a − δ B h . .8. PROOF OF PROPOSITION 8.3.9 Proof.
Multiply by f to obtain 12 e ( f ) + a κf = hf. Next, integrate over S u,r to obtain12 e (cid:32)(cid:90) S u,r f (cid:33) = (cid:90) S u,r
12 ( e ( f ) + κf )= − (cid:90) S u,r a − κf + (cid:90) S u,r hf = − a − κ (cid:90) S u,r f − a − (cid:90) S u,r ˇ κf + (cid:90) S u,r hf and hence 12 e (cid:32)(cid:90) S u,r f (cid:33) + a − κ (cid:90) S u,r f = − a − (cid:90) S u,r ˇ κf + (cid:90) S u,r hf. Also, we multiply by r a − which yields12 e (cid:32) r a − (cid:90) S u,r f (cid:33) = − a − r a − (cid:90) S u,r ˇ κf + r a − (cid:90) S u,r hf where we used the fact that 2 e ( r ) = rκ . This yields − e (cid:32) r a − (cid:90) S u,r f (cid:33) ≤ r a − − δ B (cid:90) S u,r f + 14 r a − δ B (cid:90) S u,r h and hence − e (cid:32) e − δ − B r − δB r a − (cid:90) S u,r f (cid:33) (cid:46) e − δ − B r − δB r a − δ B (cid:90) S u,r h where we used the fact that 2 e ( r ) = rκ = 2 + O ( (cid:15) ). Integrating between r = r and r = r ∗ ( u ), where r ∗ ( u ) is such that S u,r ∗ ( u ) ⊂ Σ ∗ , we infer r a − (cid:90) S u,r f (cid:46) r ∗ ( u ) a − (cid:90) S u,r ∗ ( u ) f + (cid:90) r ∗ ( u ) r r a − δ B (cid:90) S u,r h . (8.8.1) Remark 8.8.5.
Note that we have the following consequences of the coarea formula d Σ ∗ = ς (cid:114) ς − Υ + r A dµ u, Σ ∗ du, d { r = r } = ς √− κ − A √ κ dµ u,r du, CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) where we used in particular that Σ ∗ = { u + r = c Σ ∗ } . Also, we have in ( ext ) M d M = 4 ς r κ dµ u,r dudr. We infer, in ( ext ) M , using in particular the dominant condition of r on Σ ∗ , d Σ ∗ = (cid:16) O (cid:16) (cid:15) (cid:17)(cid:17) dµ u, Σ ∗ du, d { r = r } = (cid:114) − m r (1 + O ( (cid:15) )) dµ u,r du, and d M = (1 + O ( (cid:15) )) dµ u,r dudr. Integrating (8.8.1) in u ∈ [1 , u ∗ ], and relying on Remark 8.8.5 we deduce for r ≥ m r a − (cid:90) { r = r } f (cid:46) (cid:90) Σ ∗ r a − f + (cid:90) ( ext ) M ( r ≥ m ) r a − h as desired. This concludes the proof of the lemma. Corollary 8.8.6.
Let the following transport equation in ( ext ) M e ( f ) + a κf = h where a ∈ R is a given constant, and f and h are scalar functions. Also, let and δ B > .Then, f satisfies for ≤ k ≤ k large + 1sup r ≥ m (cid:18) r a − (cid:90) { r = r } ( d k f ) (cid:19) (cid:46) (cid:90) Σ ∗ r a − ( d ≤ k f ) + sup r ≥ m (cid:18) r a − (cid:90) { r = r } ( d ≤ k − f ) (cid:19) + (cid:90) ( ext ) M ( ≥ m ) r a − δ B ( d ≤ k h ) + (cid:32) sup ( ext ) M ( ≥ m ) (cid:0) r a | d ≤ k − f | (cid:1)(cid:33) (cid:16) ( ext ) G ≥ m k − [ˇΓ] + ( ext ) G ≥ m k [ˇ κ ] (cid:17) . Proof.
We commute first differentiate the equation for f with ( d /, T ) l and obtain e (( d /, T ) l f ) + a κ ( d /, T ) l f = h l ,h l := ( d /, T ) l h − [( d /, T ) l , e ] f − a d /, T ) l , κ ] f. .8. PROOF OF PROPOSITION 8.3.9 r ≥ m (cid:18) r a − (cid:90) { r = r } (( d /, T ) l f ) (cid:19) (cid:46) (cid:90) Σ ∗ r a − ( d l f ) + (cid:90) ( ext ) M ( ≥ m ) r a − δ B h l . Now, we have the following schematic commutation formulas[ d /, e ] = Γ g d + Γ g , [ T , e ] = r − Γ b d , Together with the definition of h l and for 5 ≤ l ≤ k large + 1, we deduce (cid:90) ( ext ) M ( ≥ m ) r a − δ B h l (cid:46) (cid:90) ( ext ) M ( ≥ m ) r a − δ B ( d l h ) + (cid:15) (cid:90) ( ext ) M r a − δ B ( d ≤ l f ) + (cid:18) sup ( ext ) M (cid:0) r a | d ≤ l − f | (cid:1)(cid:19) (cid:16) ( ext ) G l − [ˇΓ] + ( ext ) G l [ˇ κ ] (cid:17) and hence sup r ≥ r T (cid:18) r a − (cid:90) { r = r } (( d /, T ) l f ) (cid:19) (cid:46) (cid:90) Σ ∗ r a − ( d l f ) + (cid:90) ( ext ) M r a − δ B ( d l h ) + (cid:15) (cid:90) ( ext ) M ( ≥ m ) r a − δ B ( d ≤ l f ) + (cid:32) sup ( ext ) M ( ≥ m ) (cid:0) r a | d ≤ l − f | (cid:1)(cid:33) (cid:16) ( ext ) G ≥ m l − [ˇΓ] + ( ext ) G ≥ m l [ˇ κ ] (cid:17) or, sup r ≥ m (cid:18) r a − (cid:90) { r = r } (( d /, T ) l f ) (cid:19) (cid:46) (cid:90) Σ ∗ r a − ( d l f ) + (cid:90) ( ext ) M ( ≥ m ) r a − δ B ( d l h ) + (cid:15) sup r ≥ m (cid:18) r a − (cid:90) { r = r } ( d ≤ l f ) (cid:19) + (cid:32) sup ( ext ) M ( ≥ m ) (cid:0) r a | d ≤ l − f | (cid:1)(cid:33) (cid:16) ( ext ) G ≥ m l − [ˇΓ] + ( ext ) G ≥ m l [ˇ κ ] (cid:17) . Together with the first equation which yields re (( d /, T ) l f ) + a rκ ( d /, T ) l f = rh l , and hence ( re ) j (( d /, T ) l f ) + a re ) j − (cid:16) rκ ( d /, T ) l f (cid:17) = ( re ) j − ( rh l ) , CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) we infer, for (cid:15) > ≤ k ≤ k large + 1,sup r ≥ m (cid:18) r a − (cid:90) { r = r } ( d k f ) (cid:19) (cid:46) (cid:90) Σ ∗ r a − ( d ≤ k f ) + (cid:90) ( ext ) M ( ≥ m ) r a − δ B ( d ≤ k h ) + sup r ≥ m (cid:18) r a − (cid:90) { r = r } ( d ≤ k − h ) (cid:19) + sup r ≥ m (cid:18) r a − (cid:90) { r = r } ( d ≤ k − f ) (cid:19) + (cid:32) sup ( ext ) M ( ≥ m ) (cid:0) r a | d ≤ k − f | (cid:1)(cid:33) (cid:16) ( ext ) G ≥ m k − [ˇΓ] + ( ext ) G ≥ m k [ˇ κ ] (cid:17) . Using a trace estimate, we infersup r ≥ m (cid:18) r a − (cid:90) { r = r } ( d k f ) (cid:19) (cid:46) (cid:90) Σ ∗ r a − ( d ≤ k f ) + (cid:90) ( ext ) M ( ≥ m ) r a − δ B ( d ≤ k h ) + sup r ≥ m (cid:18) r a − (cid:90) { r = r } ( d ≤ k − f ) (cid:19) + (cid:32) sup ( ext ) M ( ≥ m ) (cid:0) r a | d ≤ k − f | (cid:1)(cid:33) (cid:16) ( ext ) G ≥ m k − [ˇΓ] + ( ext ) G ≥ m k [ˇ κ ] (cid:17) as desired. This concludes the proof of the corollary. Lemma 8.8.7.
Let the following transport equation in ( ext ) M e ( f ) + a κf = h where a ∈ R is a given constant, and f and h are scalar functions. Let b > a − . Then, f satisfies sup r ≥ m (cid:18) r b (cid:90) { r = r } f (cid:19) + (cid:90) ( ext ) M ( ≥ m ) r b − f (cid:46) (cid:90) Σ ∗ r b f + (cid:90) ( ext ) M ( ≥ m ) r b +1 h . Proof.
Recall from Lemma 8.8.4 the following identity12 e (cid:32)(cid:90) S u,r f (cid:33) + a − κ (cid:90) S u,r f = − a − (cid:90) S u,r ˇ κf + (cid:90) S u,r hf. We multiply by r b which yields12 e (cid:32) r b (cid:90) S u,r f (cid:33) + 12 (cid:18) a − − b (cid:19) κ (cid:90) S u,r r b f = − a − r b (cid:90) S u,r ˇ κf + r b (cid:90) S u,r hf .8. PROOF OF PROPOSITION 8.3.9 e ( r ) = rκ . We choose b > a − r = r and r = r ∗ ( u ), where r ∗ ( u ) is such that S u,r ∗ ( u ) ⊂ Σ ∗ , which yields (cid:90) S u,r r b f + (cid:90) r ∗ r (cid:90) S u,r r b − f (cid:46) (cid:90) S u,r ∗ r b f + (cid:90) r ∗ r (cid:90) S u,r r b +1 h . Then, integrating in u in u ∈ [1 , u ∗ ], and relying on Remark 8.8.5 we deduce for r ≥ m , r b (cid:90) { r = r } f + (cid:90) ( ext ) M∩{ r ≥ r } r b − f (cid:46) (cid:90) Σ ∗ r b f + (cid:90) ( ext ) M ( ≥ m ) r b +1 h . This concludes the proof of the lemma.
Corollary 8.8.8.
Let the following transport equation in ( ext ) M e ( f ) + a κf = h where a ∈ R is a given constant, and f and h are scalar functions. Let b > a − . Then, f satisfies for ≤ l ≤ k large + 1sup r ≥ m (cid:18) r b (cid:90) { r = r } ( d k f ) (cid:19) + (cid:90) ( ext ) M ( ≥ m ) r b − ( d k f ) (cid:46) (cid:90) Σ ∗ r b ( d ≤ k f ) + sup r ≥ m (cid:18) r b (cid:90) { r = r } ( d ≤ k − f ) (cid:19) + (cid:90) ( ext ) M ( ≥ m ) r b − ( d ≤ k h ) + (cid:32) sup ( ext ) M ( ≥ m ) (cid:0) r b | d ≤ k − f | (cid:1)(cid:33) (cid:16) ( ext ) G ≥ m k − [ˇΓ] + ( ext ) G ≥ m k [ˇ κ ] (cid:17) . Proof.
The proof is based on Lemma 8.8.7. It is similar to the one of Corollary 8.8.6 andleft to the reader.
Lemma 8.8.9.
Let the following transport equation in ( ext ) M e ( f ) + a κf = h where a ∈ R is a given constant, and f and h are scalar functions. Then, f satisfies sup r T ≤ r ≤ m (cid:90) { r = r } f (cid:46) (cid:90) { r =4 m } f + (cid:90) ( ext ) M ( ≤ m ) h . Proof.
Let b > a −
2. Recall from Lemma 8.8.7 the following identity12 e (cid:32) r b (cid:90) S u,r f (cid:33) + 12 (cid:18) a − − b (cid:19) κ (cid:90) S u,r r b f = − a − r b (cid:90) S u,r ˇ κf + r b (cid:90) S u,r hf. CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Choosing b = 2 a , we obtain12 e (cid:32) r b (cid:90) S u,r f (cid:33) − κ (cid:90) S u,r r b f = − a − r b (cid:90) S u,r ˇ κf + r b (cid:90) S u,r hf. Next, let 1 ≤ u ≤ u ∗ and r T ≤ r ≤ m . We now integrate in r ≤ r ≤ m and along C u in ( ext ) M . Since r is bounded on ( ext ) M ( r ≤ m ) from above and below, we obtain,for (cid:15) > (cid:90) S u,r f (cid:46) (cid:90) S u, m f + (cid:90) m r T (cid:90) S u,r h . We may now integrate in u to deduce (cid:90) u ∗ (cid:90) S u,r f (cid:46) (cid:90) u ∗ (cid:90) S u, m f + (cid:90) u ∗ (cid:90) m r T (cid:90) S u,r h . Relying on Remark 8.8.5 we deducesup r T ≤ r ≤ m (cid:90) { r = r } f (cid:46) (cid:90) { r =4 m } f + (cid:90) ( ext ) M ( ≤ m ) h . as desired. This concludes the proof of the lemma. Corollary 8.8.10.
Let the following transport equation in ( ext ) M e ( f ) + a κf = h where a ∈ R is a given constant, and f and h are scalar functions. Then, f satisfies for ≤ l ≤ k large + 1sup r T ≤ r ≤ m (cid:90) { r = r } ( d k f ) (cid:46) (cid:90) { r =4 m } ( d ≤ k f ) + sup r T ≤ r ≤ m (cid:90) { r = r } ( d ≤ k − f ) + (cid:90) ( ext ) M ( ≤ m ) ( d ≤ k h ) + (cid:32) sup ( ext ) M ( r ≤ m ) (cid:0) | d ≤ k − f | (cid:1)(cid:33) (cid:16) ( ext ) G ≤ m k − [ˇΓ] + ( ext ) G ≤ m k [ˇ κ ] (cid:17) . Proof.
The proof is based on Lemma 8.8.9. It is similar to the one of Corollary 8.8.6 andleft to the reader. .8. PROOF OF PROPOSITION 8.3.9
The goal of this section is to prove the identities below that will be used to avoid loosingderivatives when controlling the weighted energies of the Ricci coefficients.
Lemma 8.8.11.
We have e (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) + 2 κ (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) = − (cid:18) − ϑ d (cid:63) / + ζe (Φ) − β (cid:19) d (cid:63) / κ − ϑ d/ d (cid:63) / κ + 12 d (cid:63) / ( κ + ϑ ) d (cid:63) / κ + ( d (cid:63) / κ ) − κϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ + ( κϑ + 2 α ) (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ + ϑ (cid:104)(cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − , e (cid:105) ˇ ρ + ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err [ e ˇ ρ ] (cid:19) − ϑ d/ d (cid:63) / d/ − ( − d (cid:63) / κ + κζ − ϑζ ) − ϑ d (cid:63) / ( − d (cid:63) / κ + κζ − ϑζ )+ 12 d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) d (cid:63) / ϑ + 12 ( − β − d (cid:63) / κ + κζ − ϑζ ) d/ ϑ,e (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) + 2 κ (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) = − µe θ ( κ ) − ϑe θ ( µ ) − ( κϑ + 2 α ) d/ d (cid:63) / ( d (cid:63) / d/ ) − β − ϑ (cid:104) d/ d (cid:63) / ( d (cid:63) / d/ ) − , e (cid:105) β − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − (cid:0) κβ + 3 ρζ + ϑβ (cid:1) + d (cid:63) / ϑ d (cid:63) / d/ − ˇ ρ − e θ (cid:18) ϑ d (cid:63) / d/ − (cid:18) − ˇ µ + 14 ϑϑ (cid:19)(cid:19) − e θ (cid:18) ϑ (cid:18) κϑ + ζ (cid:19)(cid:19) − ζ d/ d (cid:63) / κ − e θ ( κ ) d (cid:63) / ζ − κζ + β + ϑζ ) ˇ ρ − ζ (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err [ e ˇ ρ ] (cid:19) +2 β d (cid:63) / ζ + 32 e θ (cid:0) κζ (cid:1) + 2 κ (cid:16) ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) , and e ( e θ ( κ ) − β ) + κ ( e θ ( κ ) − β )= 2 e θ ( µ ) + 12 ρζ − κe θ ( κ ) + 4 ϑβ − ϑe θ ( κ ) − e θ ( ϑϑ ) + 2 e θ ( ζ )= 2 (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) + 12 ρζ − κe θ ( κ ) − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ + 4 ϑβ − ϑ ( e θ ( κ ) − β ) − ϑβ − e θ ( ϑϑ ) + 2 e θ ( ζ ) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Proof.
Recall Raychadhuri e ( κ ) + 12 κ = − ϑ . We commute with d/ d (cid:63) / which yields e ( d/ d (cid:63) / κ ) + 2 κ d/ d (cid:63) / κ = − (cid:18) − ϑ d (cid:63) / + ζe (Φ) − β (cid:19) d (cid:63) / κ − ϑ d/ d (cid:63) / κ + 12 d (cid:63) / ( κ + ϑ ) d (cid:63) / κ + ( d (cid:63) / κ ) − d/ d (cid:63) / ( ϑ ) . We have in view of Codazzi for ϑ d/ d (cid:63) / ( ϑ ) = d/ ( ϑ d (cid:63) / ϑ )= 12 d/ ( ϑ d (cid:63) / ϑ − ϑ d/ ϑ )= 12 d/ (cid:16) ϑ d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) − ϑ ( − β − d (cid:63) / κ + κζ − ϑζ ) (cid:17) = 12 ϑ d/ d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) + 12 ϑ d (cid:63) / ( − β − d (cid:63) / κ + κζ − ϑζ ) − d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) d (cid:63) / ϑ −
12 ( − β − d (cid:63) / κ + κζ − ϑζ ) d/ ϑ = − ϑ d/ d (cid:63) / d/ − β − ϑ d (cid:63) / β + 12 ϑ d/ d (cid:63) / d/ − ( − d (cid:63) / κ + κζ − ϑζ ) + 12 ϑ d (cid:63) / ( − d (cid:63) / κ + κζ − ϑζ ) − d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) d (cid:63) / ϑ −
12 ( − β − d (cid:63) / κ + κζ − ϑζ ) d/ ϑ. Together with Bianchi for e ( ˇ ρ ), we infer12 d/ d (cid:63) / ( ϑ ) = − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − (cid:18) e ( ˇ ρ ) + 32 κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) + 12 ϑ d/ d (cid:63) / d/ − ( − d (cid:63) / κ + κζ − ϑζ ) + 12 ϑ d (cid:63) / ( − d (cid:63) / κ + κζ − ϑζ ) − d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) d (cid:63) / ϑ −
12 ( − β − d (cid:63) / κ + κζ − ϑζ ) d/ ϑ = − ϑe (cid:16)(cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) − ϑ (cid:104)(cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − , e (cid:105) ˇ ρ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) + 12 ϑ d/ d (cid:63) / d/ − ( − d (cid:63) / κ + κζ − ϑζ ) + 12 ϑ d (cid:63) / ( − d (cid:63) / κ + κζ − ϑζ ) − d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) d (cid:63) / ϑ −
12 ( − β − d (cid:63) / κ + κζ − ϑζ ) d/ ϑ. .8. PROOF OF PROPOSITION 8.3.9 e ( ϑ ), we infer12 d/ d (cid:63) / ( ϑ ) = − e (cid:16) ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) − ( κϑ + 2 α ) (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ − ϑ (cid:104)(cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − , e (cid:105) ˇ ρ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) + 12 ϑ d/ d (cid:63) / d/ − ( − d (cid:63) / κ + κζ − ϑζ ) + 12 ϑ d (cid:63) / ( − d (cid:63) / κ + κζ − ϑζ ) − d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) d (cid:63) / ϑ −
12 ( − β − d (cid:63) / κ + κζ − ϑζ ) d/ ϑ. This yields e (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) + 2 κ (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) = − (cid:18) − ϑ d (cid:63) / + ζe (Φ) − β (cid:19) d (cid:63) / κ − ϑ d/ d (cid:63) / κ + 12 d (cid:63) / ( κ + ϑ ) d (cid:63) / κ + ( d (cid:63) / κ ) − κϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ + ( κϑ + 2 α ) (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ + ϑ (cid:104)(cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − , e (cid:105) ˇ ρ + ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) − ϑ d/ d (cid:63) / d/ − ( − d (cid:63) / κ + κζ − ϑζ ) − ϑ d (cid:63) / ( − d (cid:63) / κ + κζ − ϑζ )+ 12 d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) d (cid:63) / ϑ + 12 ( − β − d (cid:63) / κ + κζ − ϑζ ) d/ ϑ. Next, recall that we have e µ + 32 κµ = − ϑ d (cid:63) / ζ − ϑ (cid:18) κϑ + ζ (cid:19) + (cid:18) e θ ( κ ) − β + 32 κζ (cid:19) ζ We commute with e θ which yields e ( e θ ( µ )) + 2 κe θ ( µ )= − µe θ ( κ ) − ϑe θ ( µ ) − e θ (cid:18) ϑ d (cid:63) / d/ − (cid:18) − ˇ µ − ˇ ρ + 14 ϑϑ (cid:19)(cid:19) − e θ (cid:18) ϑ (cid:18) κϑ + ζ (cid:19)(cid:19) + e θ (cid:18)(cid:18) e θ ( κ ) − β + 32 κζ (cid:19) ζ (cid:19) = − µe θ ( κ ) − ϑe θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − d (cid:63) / ρ + d (cid:63) / ϑ d (cid:63) / d/ − ˇ ρ − e θ (cid:18) ϑ d (cid:63) / d/ − (cid:18) − ˇ µ + 14 ϑϑ (cid:19)(cid:19) − e θ (cid:18) ϑ (cid:18) κϑ + ζ (cid:19)(cid:19) − ζ d/ d (cid:63) / κ − e θ ( κ ) d (cid:63) / ζ − ζ d/ β + 2 β d (cid:63) / ζ + 32 e θ (cid:0) κζ (cid:1) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Now, using the Bianchi identities for e ( β ) and e ( ˇ ρ ), we have ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − d (cid:63) / ρ = − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − (cid:0) e β + κβ + 3 ρζ + ϑβ (cid:1) = − e (cid:16) ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β (cid:17) + e ( ϑ ) d/ d (cid:63) / ( d (cid:63) / d/ ) − β − ϑ (cid:104) d/ d (cid:63) / ( d (cid:63) / d/ ) − , e (cid:105) β − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − (cid:0) κβ + 3 ρζ + ϑβ (cid:1) = − e (cid:16) ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β (cid:17) − ( κϑ + 2 α ) d/ d (cid:63) / ( d (cid:63) / d/ ) − β − ϑ (cid:104) d/ d (cid:63) / ( d (cid:63) / d/ ) − , e (cid:105) β − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − (cid:0) κβ + 3 ρζ + ϑβ (cid:1) and ζ d/ β = ζ (cid:18) e ( ˇ ρ ) + 32 κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) = e ( ζ ˇ ρ ) − e ( ζ ) ˇ ρ + ζ (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) = e ( ζ ˇ ρ ) + ( κζ + β + ϑζ ) ˇ ρ + ζ (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) . We infer e (cid:16) e θ ( µ )) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) + 2 κe θ ( µ )= − µe θ ( κ ) − ϑe θ ( µ ) − ( κϑ + 2 α ) d/ d (cid:63) / ( d (cid:63) / d/ ) − β − ϑ (cid:104) d/ d (cid:63) / ( d (cid:63) / d/ ) − , e (cid:105) β − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − (cid:0) κβ + 3 ρζ + ϑβ (cid:1) + d (cid:63) / ϑ d (cid:63) / d/ − ˇ ρ − e θ (cid:18) ϑ d (cid:63) / d/ − (cid:18) − ˇ µ + 14 ϑϑ (cid:19)(cid:19) − e θ (cid:18) ϑ (cid:18) κϑ + ζ (cid:19)(cid:19) − ζ d/ d (cid:63) / κ − e θ ( κ ) d (cid:63) / ζ − κζ + β + ϑζ ) ˇ ρ − ζ (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) +2 β d (cid:63) / ζ + 32 e θ (cid:0) κζ (cid:1) .8. PROOF OF PROPOSITION 8.3.9 e (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) + 2 κ (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) = − µe θ ( κ ) − ϑe θ ( µ ) − ( κϑ + 2 α ) d/ d (cid:63) / ( d (cid:63) / d/ ) − β − ϑ (cid:104) d/ d (cid:63) / ( d (cid:63) / d/ ) − , e (cid:105) β − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − (cid:0) κβ + 3 ρζ + ϑβ (cid:1) + d (cid:63) / ϑ d (cid:63) / d/ − ˇ ρ − e θ (cid:18) ϑ d (cid:63) / d/ − (cid:18) − ˇ µ + 14 ϑϑ (cid:19)(cid:19) − e θ (cid:18) ϑ (cid:18) κϑ + ζ (cid:19)(cid:19) − ζ d/ d (cid:63) / κ − e θ ( κ ) d (cid:63) / ζ − κζ + β + ϑζ ) ˇ ρ − ζ (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) +2 β d (cid:63) / ζ + 32 e θ (cid:0) κζ (cid:1) + 2 κ (cid:16) ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) . Finally, recall that we have e ( κ ) + 12 κκ = − d/ ζ + 2 ρ − ϑϑ + 2 ζ = 2 µ + 4 ρ − ϑϑ + 2 ζ . We commute with e θ which yields e ( e θ ( κ )) + κe θ ( κ ) = 2 e θ ( µ ) + 4 e θ ( ρ ) − κe θ ( κ ) − ϑe θ ( κ ) − e θ ( ϑϑ ) + 2 e θ ( ζ ) . Together with Bianchi for e ( β ), we infer e ( e θ ( κ ) − β ) + κ ( e θ ( κ ) − β )= 2 e θ ( µ ) + 12 ρζ − κe θ ( κ ) + 4 ϑβ − ϑe θ ( κ ) − e θ ( ϑϑ ) + 2 e θ ( ζ )= 2 (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) + 12 ρζ − κe θ ( κ ) − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ + 4 ϑβ − ϑ ( e θ ( κ ) − β ) − ϑβ − e θ ( ϑϑ ) + 2 e θ ( ζ ) . This concludes the proof of the lemma.
We introduce the following notation which will constantly appear on the RHS of theequalities below N ≥ m [ J, ˇΓ , ˇ R ] := (Σ ∗ ) G J +1 [ˇΓ] + (Σ ∗ ) G (cid:48) J +1 [ˇΓ] + ( ext ) R J +1 [ ˇ R ] + ( ext ) G J [ˇΓ]+ (cid:15) (cid:16) ( ext ) G ≥ m J +1 [ˇΓ] + ( ext ) G ≥ m J +1 (cid:48) [ˇΓ] (cid:17) . (8.8.2)10 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Step 1.
Recall that e ( ϑ ) + κϑ = − α. In view of Corollary 8.8.6 with a = 2, we have for any r ≥ m max k ≤ J +1 sup r ≥ m r (cid:90) { r = r } ( d k ϑ ) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) . Step 2.
Next, recall that e (ˇ κ ) + κ ˇ κ = − ϑ + 14 ϑ − ˇ κ . In view of Corollary 8.8.6 with a = 2, we have for any r ≥ m max k ≤ J +2 sup r ≥ m r (cid:90) { r = r } ( d k ˇ κ ) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the null structure equations for e ( ϑ ), e ( ϑ ) and d/ ϑ to avoid a lossof one derivative for the RHS. Step 3.
Next, recall that e ( ζ ) + κζ = − β − ϑζ. In view of Corollary 8.8.6 with a = 2, we have for any r ≥ m max k ≤ J +1 sup r ≥ m r (cid:90) { r = r } ( d k ζ ) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) . Step 4.
Next, recall that e (ˇ µ ) + 32 κ ˇ µ = − µ ˇ κ + Err[ e ˇ µ ] . In view of Corollary 8.8.6 with a = 3, commuting with d / and T , we have for any r ≥ m max k ≤ J +1 sup r ≥ m r (cid:90) { r = r } ( d k ˇ µ ) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) where we used the estimates for ˇ κ on ( ext ) M derived in Step 2. Step 5.
Next, recall that e (ˇ κ ) + 12 κ ˇ κ = − κ ˇ κ + 2 ˇ ρ − d/ ζ + Err[ e ˇ κ ] . .8. PROOF OF PROPOSITION 8.3.9 a = 1 and b = 2 − δ B which satisfy the constraint b > a − r ≥ m max k ≤ J +1 sup r ≥ m (cid:18) r − δ B (cid:90) { r = r } ( d k ˇ κ ) (cid:19) + (cid:90) ( ext ) M ( r ≥ m ) r − δ B ( d k ˇ κ ) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) where we used the estimates for ˇ κ and ˇ µ on ( ext ) M derived respectively in Step 2 andStep 4. Step 6.
Next, recall that e ( ϑ ) + 12 κϑ = 2 d (cid:63) / ζ − κϑ + 2 ζ = 2 d (cid:63) / d/ − (cid:18) − µ − ρ + 14 ϑϑ (cid:19) − κϑ + 2 ζ . In view of Corollary 8.8.6 with a = 1, we have for any r ≥ m max k ≤ J +1 sup r ≥ m (cid:90) { r = r } ( d k ϑ ) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) where we used the estimates for ϑ and ˇ µ on ( ext ) M derived respectively in Step 1 andStep 4. Step 7.
Next, recall that e (ˇ ω ) = ˇ ρ + 3 ζ − ζ − ˇ κ ˇ ω. In view of Corollary 8.8.8 with a = 0 and b = 0 which satisfy the constraint b > a − r ≥ m max k ≤ J +1 sup r ≥ m (cid:18)(cid:90) { r = r } ( d k ˇ ω ) (cid:19) + (cid:90) ( ext ) M ( ≥ m ) r − ( d k ˇ ω ) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) . Step 8.
In order to estimate ξ in Step 9, we derive an estimate for e ( ζ ) + β . Recall thatwe have e ( ζ ) + κζ = − β − ϑζ. Commuting with e , we infer e ( e ( ζ )) + [ e , e ] ζ + κe ( ζ ) + e ( κ ) ζ = − e ( β ) − ϑζ. CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
In view of the null structure equation for e ( κ ), the Bianchi identity for e ( β ) and thecommutator identity for [ e , e ], we infer e ( e ( ζ )) + (cid:16) ωe + 4 ζe θ (cid:17) ζ + κe ( ζ ) + (cid:18) − κκ + 2 ωκ + 2 d/ ζ + 2 ρ − ϑϑ + 2 ζ (cid:19) ζ = ( κ − ω ) β + d (cid:63) / ρ − ζρ + ϑβ − ξα − ϑζ. Together with the null structure equation for e ( ζ ), the Bianchi identity for e ( β ) to getrid of the term d (cid:63) / ρ , and the definition of µ , we infer e ( e ( ζ )) + 2 ω ( − κζ − β − ϑζ ) + 4 ζ d (cid:63) / d/ − (cid:18) ˇ µ + ˇ ρ − ϑϑ + 14 ϑϑ (cid:19) + κe ( ζ ) + (cid:18) − κκ + 2 ωκ − µ + 2 ζ (cid:19) ζ = ( κ − ω ) β − e ( β ) − κβ − ζρ − ϑβ − ζρ + ϑβ − ξα − ϑζ. and hence e ( e ( ζ ) + β ) + κ ( e ( ζ ) + β )= κβ + (cid:18) κκ + 2 µ − ρ (cid:19) ζ − ϑβ + ϑβ − ξα − ϑζ + 2 ωϑζ − ζ d (cid:63) / d/ − (cid:18) ˇ µ + ˇ ρ − ϑϑ + 14 ϑϑ (cid:19) − ζ In view of Corollary 8.8.6 with a = 2, we have for any r ≥ m max k ≤ J +1 sup r ≥ m r (cid:90) { r = r } ( d k ( e ( ζ ) + β )) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) where we used the estimates for ζ derived in Step 3. Step 9.
Next, recall that we have e ( ξ ) = − e ( ζ ) + β − κζ − ζϑ = − ( e ( ζ ) + β ) + 2 β − κζ − ζϑ. In view of Corollary 8.8.8 with a = 0 and b = − δ B which satisfy the constraint b > a − r ≥ m max k ≤ J +1 sup r ≥ m r − δ B (cid:18)(cid:90) { r = r } ( d k ξ ) (cid:19) + (cid:90) ( ext ) M ( r ≥ m ) r − − δ B ( d k ξ ) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) where we used the estimates for ζ and e ( ζ ) + β on ( ext ) M derived respectively in Step 3and Step 8. .8. PROOF OF PROPOSITION 8.3.9 Step 10.
Recall that we have e (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) + 2 κ (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) = − (cid:18) − ϑ d (cid:63) / + ζe (Φ) − β (cid:19) d (cid:63) / κ − ϑ d/ d (cid:63) / κ + 12 d (cid:63) / ( κ + ϑ ) d (cid:63) / κ + ( d (cid:63) / κ ) − κϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ + ( κϑ + 2 α ) (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ + ϑ (cid:104)(cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − , e (cid:105) ˇ ρ + ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) − ϑ d/ d (cid:63) / d/ − ( − d (cid:63) / κ + κζ − ϑζ ) − ϑ d (cid:63) / ( − d (cid:63) / κ + κζ − ϑζ )+ 12 d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) d (cid:63) / ϑ + 12 ( − β − d (cid:63) / κ + κζ − ϑζ ) d/ ϑ. In view of Corollary 8.8.6 with a = 4, we havemax k ≤ J +1 sup r ≥ m (cid:90) { r = r } r (cid:16) d k (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17)(cid:17) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used • the fact that d /ϑ = d / d/ − d/ ϑ and Codazzi for ϑ to estimate the terms of the RHSwith one angular derivative of ϑ , • the estimates of Step 2 to estimate the terms of the RHS with one derivative of ˇ κ , • the fact that d /ζ = d / d/ − d/ ζ and the definition of µ to estimate terms of the RHSwith one angular derivative of ζ , • the identity d / d (cid:63) / κ = d / d/ − (cid:16) ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) + d / d/ − (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) to estimate the terms of the RHS with two angular derivatives of ˇ κ .14 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Step 11.
Recall that we have e (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) + 2 κ (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) = − µe θ ( κ ) − ϑe θ ( µ ) − ( κϑ + 2 α ) d/ d (cid:63) / ( d (cid:63) / d/ ) − β − ϑ (cid:104) d/ d (cid:63) / ( d (cid:63) / d/ ) − , e (cid:105) β − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − (cid:0) κβ + 3 ρζ + ϑβ (cid:1) + d (cid:63) / ϑ d (cid:63) / d/ − ˇ ρ − e θ (cid:18) ϑ d (cid:63) / d/ − (cid:18) − ˇ µ + 14 ϑϑ (cid:19)(cid:19) − e θ (cid:18) ϑ (cid:18) κϑ + ζ (cid:19)(cid:19) − ζ d/ d (cid:63) / κ − e θ ( κ ) d (cid:63) / ζ − κζ + β + ϑζ ) ˇ ρ − ζ (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) +2 β d (cid:63) / ζ + 32 e θ (cid:0) κζ (cid:1) + 2 κ (cid:16) ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) . In view of Corollary 8.8.6 with a = 4, we havemax k ≤ J +1 sup r ≥ m r (cid:90) { r = r } (cid:0) d k (cid:0) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:1)(cid:1) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) + (cid:15) (cid:90) ( ext ) M ( ≥ m ) (cid:0) d ≤ J +1 (cid:0) e θ ( κ ) − β (cid:1)(cid:1) , where we have used • the fact that d /ϑ = d / d/ − d/ ϑ and Codazzi for ϑ to estimate the terms of the RHSwith one angular derivative of ϑ , • the estimates of Step 2 to estimate the terms of the RHS with one derivative of ˇ κ , • the fact that d /ζ = d / d/ − d/ ζ and the definition of µ to estimate terms of the RHSwith one angular derivative of ζ , • the fact that e θ ( κ ) = ( e θ ( κ ) − β ) + 4 β to estimate the term with one angularderivative of κ , • the identity d / d (cid:63) / κ = d / d/ − (cid:16) ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) + d / d/ − (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) and the estimates of Step 10 to estimate the terms of the RHS with two angularderivatives of ˇ κ . .8. PROOF OF PROPOSITION 8.3.9 Step 12.
Recall that we have e ( e θ ( κ ) − β ) + κ ( e θ ( κ ) − β )= 2 e θ ( µ ) + 12 ρζ − κe θ ( κ ) + 4 ϑβ − ϑe θ ( κ ) − e θ ( ϑϑ ) + 2 e θ ( ζ )= 2 (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) + 12 ρζ − κe θ ( κ ) − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ + 4 ϑβ − ϑ ( e θ ( κ ) − β ) − ϑβ − e θ ( ϑϑ ) + 2 e θ ( ζ ) . In view of Corollary 8.8.6 with a = 2, we havemax k ≤ J +1 sup r ≥ m r (cid:90) { r = r } (cid:0) d k (cid:0) e θ ( κ ) − β (cid:1)(cid:1) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) + (cid:90) ( ext ) M ( ≥ m ) r (cid:0) d ≤ J +1 (cid:0) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:1)(cid:1) , where we have used • the fact that d /ϑ = d / d/ − d/ ϑ and Codazzi for ϑ to estimate the terms of the RHSwith one angular derivative of ϑ , • the fact that d /ϑ = d / d/ − d/ ϑ and Codazzi for ϑ to estimate the terms of the RHSwith one angular derivative of ϑ , • the estimates of Step 2 to estimate the terms of the RHS with one derivative of ˇ κ , • the fact that d /ζ = d / d/ − d/ ζ and the definition of µ to estimate terms of the RHSwith one angular derivative of ζ , • the estimate for ζ of Step 3.Together with the estimate of Step 11, we infermax k ≤ J +1 sup r ≥ m r (cid:90) { r = r } (cid:0) d k (cid:0) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:1)(cid:1) + max k ≤ J +1 sup r ≥ m r (cid:90) { r = r } (cid:0) d k (cid:0) e θ ( κ ) − β (cid:1)(cid:1) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) . Finally, we have obtainedmax k ≤ J +1 sup r ≥ m (cid:32) (cid:90) { r = r } (cid:32) r ( d k ˇ µ ) + r ( d k ϑ ) + r ( d k ζ ) + r ( d k ( e ( ζ ) + β )) + r − δ B ( d k ˇ κ ) + ( d k ϑ ) + ( d k ˇ ω ) + r − δ B ( d k ξ ) (cid:33)(cid:33) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) , CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) max k ≤ J +2 sup r ≥ m (cid:32) r (cid:90) { r = r } (cid:18) d k (cid:18) κ − r (cid:19)(cid:19) (cid:33) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) , andmax k ≤ J +1 sup r T ≤ r ≤ m (cid:32) (cid:90) { r = r } (cid:40) r (cid:16) d k (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17)(cid:17) + r (cid:0) d k (cid:0) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:1)(cid:1) + r (cid:0) d k (cid:0) e θ ( κ ) − β (cid:1)(cid:1) (cid:41)(cid:33) (cid:46) (cid:16) N ≥ m [ J, ˇΓ , ˇ R ] (cid:17) . In view of the definition (8.8.2) of N ≥ m [ J, ˇΓ , ˇ R ], and of the various norms, we infer ( ext ) G ≥ m J +1 [ˇΓ] + ( ext ) G ≥ m J +1 (cid:48) [ˇΓ] (cid:46) (Σ ∗ ) G J +1 [ˇΓ] + (Σ ∗ ) G (cid:48) J +1 [ˇΓ] + ( ext ) R J +1 [ ˇ R ] + ( ext ) G J [ˇΓ]+ (cid:15) (cid:16) ( ext ) G ≥ m J +1 [ˇΓ] + ( ext ) G ≥ m J +1 (cid:48) [ˇΓ] (cid:17) and hence, for (cid:15) small enough, ( ext ) G ≥ m J +1 [ˇΓ] + ( ext ) G ≥ m J +1 (cid:48) [ˇΓ] (cid:46) (Σ ∗ ) G J +1 [ˇΓ] + (Σ ∗ ) G (cid:48) J +1 [ˇΓ] + ( ext ) R J +1 [ ˇ R ] + ( ext ) G J [ˇΓ] . This concludes the proof of Proposition 8.8.2.
In the proof below, we will repeatedly use the following estimatemax k ≤ J +1 (cid:90) ( ext ) M ( r ≤ m ) ( d k f ) (cid:46) max k ≤ J (cid:90) ( ext ) M ( r ≤ m ) (cid:16) ( d k f ) + ( d k N f ) + ( d k e f ) + ( d k d /f ) (cid:17) (8.8.3)which follows from the fact that d = ( d /, re , e ) and e = Υ e − N , where we recall that N = 12 (cid:16) Υ e − e (cid:17) . Also, we introduce the following notation which will constantly appear on the RHS of theequalities below N ≤ m [ J, ˇΓ , ˇ R ] := ( ext ) G ≥ m J +1 [ˇΓ] + ( ext ) G ≥ m J +1 (cid:48) [ˇΓ] + ( ext ) R J +1 [ ˇ R ] + ( ext ) G J [ˇΓ]+ (cid:15) (cid:16) ( ext ) G ≤ m J +1 [ˇΓ] + ( ext ) G ≤ m J +1 (cid:48) [ˇΓ] (cid:17) . (8.8.4) .8. PROOF OF PROPOSITION 8.3.9 Step 1.
Recall that e (ˇ κ ) + κ ˇ κ = Err[ e ˇ κ ] . In view of Corollary 8.8.10, we havemax k ≤ J +2 sup r T ≤ r ≤ m (cid:90) { r = r } ( d k ˇ κ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the null structure equations for e ( ϑ ), e ( ϑ ) and d/ ϑ to avoid loosingone derivative. Step 2.
Next, recall that e (ˇ µ ) + 32 κ ˇ µ = − µ ˇ κ + Err[ e ˇ µ ] . In view of Corollary 8.8.10, we havemax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } ( d k ˇ µ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimates for ˇ κ of Step 1. Step 3.
Next, recall that e ( ζ ) + κζ = − β − ϑζ. In view of Corollary 8.8.10, we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } (cid:16) ( d k e ζ ) + ( d k ζ ) (cid:17) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . Also, commuting first with N , and proceeding analogously, we infermax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k N ζ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) Furthermore, in view of the definition of µ and a Poincar´e inequality for d/ , we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k d /ζ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used a trace estimate and the estimate for ˇ µ of Step 2. The above estimates,together with (8.8.3), implymax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } ( d k ζ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Step 4.
Recall that e ( ϑ ) + κϑ = − α. In view of Corollary 8.8.10, we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } (cid:16) ( d k e ϑ ) + ( d k ϑ ) (cid:17) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . Also, commuting first one time with N , and proceeding analogously, we infermax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k N ϑ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . Furthermore, in view of Codazzi for ϑ , and a Poincar´e inequality for d/ , we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k d /ϑ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used a trace estimate, and the estimate for ˇ κ and ζ respectively in Step 1and Step 3. The above estimates, together with (8.8.3), implymax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } ( d k ϑ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . Step 5.
Recall that we have e (ˇ κ ) + 12 κ ˇ κ = − κ ˇ κ − d/ ζ + 2 ˇ ρ + Err[ e ˇ κ ]= − κ ˇ κ + 2ˇ µ + 4 ˇ ρ − ϑϑ + Err[ e ˇ κ ] . In view of Corollary 8.8.10, we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } (cid:16) ( d k e ˇ κ ) + ( d k ˇ κ ) (cid:17) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimates for ˇ κ and ˇ µ derived respectively in Step 1 and Step 2.Also, commuting first one time with N , and proceeding analogously, we infermax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k N ˇ κ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimates for ˇ κ and ˇ µ derived respectively in Step 1 and Step 2.Furthermore, commuting the equation for e ( κ ) once with e θ , we have e ( e θ ( κ )) + κe θ ( κ ) = − κe θ ( κ ) + 2 e θ ( µ ) + 4 e θ ( ρ ) − e θ ( ϑϑ ) + 2 e θ ( ζ ) − ϑe θ ( κ ) . .8. PROOF OF PROPOSITION 8.3.9 e ( β ), we infer e ( e θ ( κ ) − β ) + κ ( e θ ( κ ) − β ) = − κe θ ( κ ) + 2 e θ ( µ ) + 12 ρζ +4 ϑβ − e θ ( ϑϑ ) + 2 e θ ( ζ ) − ϑe θ ( κ ) . In view of Corollary 8.8.10, we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } (cid:16) ( d k ( e ( e θ ( κ ) − β )) + ( d k ( e θ ( κ ) − β )) (cid:17) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimates for ˇ κ , ˇ µ and ζ derived respectively in Step 1, Step 2and Step 3.The above estimates, together with (8.8.3), implymax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } ( d k ˇ κ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) + max k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k β ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used a trace estimate on { r = r } for r T ≤ r ≤ m . Step 6.
Recall that we have e (ˇ ω ) = ˇ ρ + Err[ e ˇ ω ] . In view of Corollary 8.8.10, we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } (cid:16) ( d k e ˇ ω ) + ( d k ˇ ω ) (cid:17) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . Also, commuting first one time with N , and proceeding analogously, we infermax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k N ˇ ω ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . Step 7.
Recall that we have e ( e ( ζ ) + β ) + κ ( e ( ζ ) + β )= κβ + (cid:18) κκ + 2 µ − ρ (cid:19) ζ − ϑβ + ϑβ − ξα − ϑζ + 2 ωϑζ − ζ d (cid:63) / d/ − (cid:18) ˇ µ + ˇ ρ − ϑϑ + 14 ϑϑ (cid:19) − ζ CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Commuting first one time with N , and in view of Corollary 8.8.10, we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k N ( e ( ζ ) + β )) (cid:17) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimate for ζ in Step 3. Step 8.
Recall that we have e ( ξ ) = − e ( ζ ) + β − κζ − ζϑ. In view of Corollary 8.8.10, we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } (cid:16) ( d k e ξ ) + ( d k ξ ) (cid:17) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimates for ζ derived in Step 3. Also, commuting first one timewith N , and proceeding analogously, we infermax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k N ξ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimates for e ( ζ ) + β derived in Step 7. Step 9.
Recall 2 d (cid:63) / ω = e ζ + κζ − β − κξ + ϑζ − ϑξ. Using a Poincar´e inequality for d (cid:63) / , we infermax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k d / ˇ ω ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used a trace estimate and the estimate for ζ and ξ respectively in Step 3and Step 8. The above estimates, together with the estimates for ˇ ω of Step 6 and (8.8.3),imply max k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } ( d k ˇ ω ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . Step 10.
Recall that we have e ( ˇΩ) = − ω + ˇ κ ˇΩ . .8. PROOF OF PROPOSITION 8.3.9 k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } ( d k ˇΩ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimates for ˇ ω derived in Step 9. Step 11.
Recall2 d/ ξ = e (ˇ κ ) + κ ˇ κ + 2 ω ˇ κ + 2 κ ˇ ω − (cid:18) κκ − ρ (cid:19) ˇΩ − Err[ e ˇ κ ] . Using a Poincar´e inequality for d/ , we infermax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k d /ξ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimates for ˇ κ , ˇ ω and ˇΩ respectively in Step 5, Step 9 and Step10. The above estimates, together with the estimates for ξ of Step 8 and (8.8.3), implymax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } ( d k ξ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . Step 12.
Recall that e ( ϑ ) + 12 κϑ = 2 d (cid:63) / ζ − κϑ + 2 ζ = 2 d (cid:63) / d/ − (cid:18) − ˇ µ − ˇ ρ + 14 ϑϑ − ϑϑ (cid:19) − κϑ + 2 ζ . In view of Corollary 8.8.10, we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } (cid:16) ( d k e ϑ ) + ( d k ϑ ) (cid:17) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimate for ˇ µ and ϑ respectively in Step 2 and Step 4. Also,commuting first one time with N , and proceeding analogously, we infermax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k N ϑ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used the estimate for ˇ µ and ϑ respectively in Step 2 and Step 4. Further-more, in view of Codazzi for ϑ , and a Poincar´e inequality for d/ , we havemax k ≤ J sup r T ≤ r ≤ m (cid:90) { r = r } ( d k d /ϑ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) where we have used a trace estimate and the estimate for ˇ κ and ζ respectively in Step 5and Step 3. The above estimates, together with (8.8.3), implymax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } ( d k ϑ ) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . Step 13.
Recall that we have e (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) + 2 κ (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) = − (cid:18) − ϑ d (cid:63) / + ζe (Φ) − β (cid:19) d (cid:63) / κ − ϑ d/ d (cid:63) / κ + 12 d (cid:63) / ( κ + ϑ ) d (cid:63) / κ + ( d (cid:63) / κ ) − κϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ + ( κϑ + 2 α ) (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ + ϑ (cid:104)(cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − , e (cid:105) ˇ ρ + ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) − ϑ d/ d (cid:63) / d/ − ( − d (cid:63) / κ + κζ − ϑζ ) − ϑ d (cid:63) / ( − d (cid:63) / κ + κζ − ϑζ )+ 12 d (cid:63) / d/ − ( − β − d (cid:63) / κ + κζ − ϑζ ) d (cid:63) / ϑ + 12 ( − β − d (cid:63) / κ + κζ − ϑζ ) d/ ϑ. In view of Corollary 8.8.10, we havemax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } (cid:16) d k (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17)(cid:17) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) where we have used • the fact that d /ϑ = d / d/ − d/ ϑ and Codazzi for ϑ to estimate the terms of the RHSwith one angular derivative of ϑ , • the estimates of Step 1 to estimate the terms of the RHS with one derivative of ˇ κ , • the fact that d /ζ = d / d/ − d/ ζ and the definition of µ to estimate terms of the RHSwith one angular derivative of ζ , • the identity d / d (cid:63) / κ = d / d/ − (cid:16) ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) + d / d/ − (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) to estimate the terms of the RHS with two angular derivatives of ˇ κ . .8. PROOF OF PROPOSITION 8.3.9 Step 14.
Recall that we have e (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) + 2 κ (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) = − µe θ ( κ ) − ϑe θ ( µ ) − ( κϑ + 2 α ) d/ d (cid:63) / ( d (cid:63) / d/ ) − β − ϑ (cid:104) d/ d (cid:63) / ( d (cid:63) / d/ ) − , e (cid:105) β − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − (cid:0) κβ + 3 ρζ + ϑβ (cid:1) + d (cid:63) / ϑ d (cid:63) / d/ − ˇ ρ − e θ (cid:18) ϑ d (cid:63) / d/ − (cid:18) − ˇ µ + 14 ϑϑ (cid:19)(cid:19) − e θ (cid:18) ϑ (cid:18) κϑ + ζ (cid:19)(cid:19) − ζ d/ d (cid:63) / κ − e θ ( κ ) d (cid:63) / ζ − κζ + β + ϑζ ) ˇ ρ − ζ (cid:18) κ ˇ ρ + 32 ρ ˇ κ − Err[ e ˇ ρ ] (cid:19) +2 β d (cid:63) / ζ + 32 e θ (cid:0) κζ (cid:1) + 2 κ (cid:16) ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) . In view of Corollary 8.8.10, we havemax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } (cid:0) d k (cid:0) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:1)(cid:1) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) + (cid:15) (cid:90) ( ext ) M ( ≤ m ) (cid:0) d ≤ J +1 (cid:0) e θ ( κ ) − β (cid:1)(cid:1) , where we have used • the fact that d /ϑ = d / d/ − d/ ϑ and Codazzi for ϑ to estimate the terms of the RHSwith one angular derivative of ϑ , • the estimates of Step 1 to estimate the terms of the RHS with one derivative of ˇ κ , • the fact that d /ζ = d / d/ − d/ ζ and the definition of µ to estimate terms of the RHSwith one angular derivative of ζ , • the fact that e θ ( κ ) = ( e θ ( κ ) − β ) + 4 β to estimate the term with one angularderivative of κ , • the identity d / d (cid:63) / κ = d / d/ − (cid:16) ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) + d / d/ − (cid:16) d/ d (cid:63) / κ − ϑ (cid:16) d/ d (cid:63) / d/ − + d (cid:63) / (cid:17) d/ − ˇ ρ (cid:17) and the estimates of Step13 to estimate the terms of the RHS with two angularderivatives of ˇ κ .24 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Step 15.
Recall that we have e ( e θ ( κ ) − β ) + κ ( e θ ( κ ) − β )= 2 e θ ( µ ) + 12 ρζ − κe θ ( κ ) + 4 ϑβ − ϑe θ ( κ ) − e θ ( ϑϑ ) + 2 e θ ( ζ )= 2 (cid:16) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:17) + 12 ρζ − κe θ ( κ ) − ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ + 4 ϑβ − ϑ ( e θ ( κ ) − β ) − ϑβ − e θ ( ϑϑ ) + 2 e θ ( ζ ) . In view of Corollary 8.8.10, we havemax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } (cid:0) d k (cid:0) e θ ( κ ) − β (cid:1)(cid:1) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) + (cid:90) ( ext ) M ( ≤ m ) (cid:0) d ≤ J +1 (cid:0) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:1)(cid:1) . where we have used • the fact that d /ϑ = d / d/ − d/ ϑ and Codazzi for ϑ to estimate the terms of the RHSwith one angular derivative of ϑ , • the fact that d /ϑ = d / d/ − d/ ϑ and Codazzi for ϑ to estimate the terms of the RHSwith one angular derivative of ϑ , • the estimates of Step 1 to estimate the terms of the RHS with one derivative of ˇ κ , • the fact that d /ζ = d / d/ − d/ ζ and the definition of µ to estimate terms of the RHSwith one angular derivative of ζ , • the estimate for ζ of Step 3.Together with the estimate of Step 14, we infermax k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } (cid:0) d k (cid:0) e θ ( µ ) + ϑ d/ d (cid:63) / ( d (cid:63) / d/ ) − β + 2 ζ ˇ ρ (cid:1)(cid:1) + max k ≤ J +1 sup r T ≤ r ≤ m (cid:90) { r = r } (cid:0) d k (cid:0) e θ ( κ ) − β (cid:1)(cid:1) (cid:46) (cid:16) N ≤ m [ J, ˇΓ , ˇ R ] (cid:17) . In view of Step 1 to Step 15, of the definition (8.8.4) of N ≤ m [ J, ˇΓ , ˇ R ], and of the variousnorms, we infer ( ext ) G ≤ m J +1 [ˇΓ] + ( ext ) G ≤ m J +1 (cid:48) [ˇΓ] (cid:46) ( ext ) G ≥ m J +1 [ˇΓ] + ( ext ) G ≥ m J +1 (cid:48) [ˇΓ] + ( ext ) R J +1 [ ˇ R ] + ( ext ) G J [ˇΓ]+ (cid:15) (cid:16) ( ext ) G ≤ m J +1 [ˇΓ] + ( ext ) G ≤ m J +1 (cid:48) [ˇΓ] (cid:17) . .9. PROOF OF PROPOSITION 8.3.10 (cid:15) small enough, ( ext ) G ≤ m J +1 [ˇΓ] + ( ext ) G ≤ m J +1 (cid:48) [ˇΓ] (cid:46) ( ext ) G ≥ m J +1 [ˇΓ] + ( ext ) G ≥ m J +1 (cid:48) [ˇΓ] + ( ext ) R J +1 [ ˇ R ] + ( ext ) G J [ˇΓ] . This concludes the proof of Proposition 8.8.3.
To prove Proposition 8.3.10, we rely on the following proposition.
Proposition 8.9.1.
Let J such that k small − ≤ J ≤ k large − . Then, we have ( int ) G J +1 [ˇΓ] + ( int ) G (cid:48) J +1 [ˇΓ] (cid:46) ( ext ) G J +1 [ˇΓ] + ( ext ) G (cid:48) J +1 [ˇΓ] + ( int ) R J +1 [ ˇ R ]+ (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) , where the notation ( ext ) G (cid:48) J +1 [ˇΓ] has been introduced in Proposition 8.8.2, and where wehave introduced the notation ( int ) G (cid:48) k [ˇΓ] := (cid:90) ( int ) M (cid:104) (cid:0) d k e θ ( κ ) (cid:1) + ( d ≤ k ˇ µ ) + (cid:0) d k ( e ( ζ ) − β ) (cid:1) (cid:105) . The proof of Proposition 8.9.1 is postponed to section 8.9.2. It will rely in particular onbasic weighted estimates for transport equations along e in ( int ) M derived in section 8.9.1.We now conclude the proof of Proposition 8.3.10. In view of Proposition 8.9.1, we have ( int ) G J +1 [ˇΓ] (cid:46) ( ext ) G J +1 [ˇΓ] + ( ext ) G (cid:48) J +1 [ˇΓ] + ( int ) R J +1 [ ˇ R ]+ (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) . Also, we have in view of Proposition 8.8.1, Proposition 8.8.2 and the iteration assumption(8.3.10) ( ext ) G J +1 [ˇΓ] + ( ext ) G (cid:48) J +1 [ˇΓ] (cid:46) ( ext ) R J +1 [ ˇ R ] + (cid:15) B [ J ] . We infer ( int ) G J +1 [ˇΓ] (cid:46) ( int ) R J +1 [ ˇ R ] + ( ext ) R J +1 [ ˇ R ] + (cid:15) B [ J ] + (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Together with Proposition 8.3.9, we deduce ( int ) G J +1 [ˇΓ] (cid:46) (cid:15) B [ J ] + (cid:15) (cid:16) N ( En ) J +1 + N ( match ) J +1 (cid:17) + (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) which concludes the proof of Proposition 8.3.10.The rest of this section is dedicated to the proof of Proposition 8.9.1. e in ( int ) M Lemma 8.9.2.
Let the following transport equation in ( int ) M e ( f ) + a κf = h where a ∈ R is a given constant, and f and h are scalar functions. Then, f satisfies (cid:90) ( int ) M f (cid:46) (cid:90) T f + (cid:90) ( int ) M h . Proof.
Multiply by f to obtain 12 e ( f ) + ar f = hf. Next, integrate over S u,r to obtain12 e (cid:32)(cid:90) S u,r f (cid:33) = (cid:90) S u,r
12 ( e ( f ) + κf )= − (cid:90) S u,r a − κf + (cid:90) S u,r hf = − a − κ (cid:90) S u,r f − a − (cid:90) S u,r ˇ κf + (cid:90) S u,r hf and hence12 e (cid:32) r b (cid:90) S u,r f (cid:33) + 12 (cid:18) a + 1 + b (cid:19) κr b (cid:90) S u,r f = − a − r b (cid:90) S u,r ˇ κf + r b (cid:90) S u,r hf .9. PROOF OF PROPOSITION 8.3.10 e ( r ) = rκ . Also, choosing b = − a , we obtain12 e (cid:32) r − a (cid:90) S u,r f (cid:33) + 12 κr − a (cid:90) S u,r f = − a − r − a (cid:90) S u,r ˇ κf + r − a (cid:90) S u,r hf. Next, let 1 ≤ u ≤ u ∗ . We now integrate in r and along C u in ( int ) M . Since r is boundedon ( int ) M from above and below, we obtain, for (cid:15) > (cid:90) r T m − m δ (cid:90) S u,r f (cid:46) (cid:90) S u,r T f + (cid:90) r T m − m δ (cid:90) S u,r h . We may now integrate in u to deduce (cid:90) u ∗ (cid:90) r T m − m δ (cid:90) S u,r f (cid:46) (cid:90) u ∗ (cid:90) S u,r T f + (cid:90) u ∗ (cid:90) r T m − m δ (cid:90) S u,r h . (8.9.1) Remark 8.9.3.
Note that we have the following consequence of the coarea formula d T = ς √ κ + A √− κ dµ ur T du, where we used that T = { r = r T } . Also, we have in ( int ) M d M = 4 ς r κ dµ u,r dudr. We infer, in ( int ) M , d T = (cid:114) − m r T (1 + O ( (cid:15) )) dµ u,r du, and d M = (1 + O ( (cid:15) )) dµ u,r dudr. Relying on Remark 8.9.3 we deduce from (8.9.1) (cid:90) ( int ) M f (cid:46) (cid:90) T f + (cid:90) ( int ) M h as desired. This concludes the proof of the lemma.28 CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)
Corollary 8.9.4.
Let the following transport equation in ( int ) M e ( f ) + a κf = h where a ∈ R is a given constant, and f and h are scalar functions. Then, f satisfies for ≤ l ≤ k large + 1 (cid:90) ( int ) M ( d k f ) (cid:46) (cid:90) T ( d ≤ k f ) + (cid:90) ( int ) M ( d ≤ k − f ) + (cid:90) ( int ) M ( d ≤ k h ) + (cid:18) sup ( int ) M | d ≤ k − f | (cid:19) (cid:16) ( int ) G k − [ˇΓ] + ( int ) G k [ˇ κ ] (cid:17) . Proof.
The proof is based on Lemma 8.9.2. It is similar to the one of Corollary 8.8.6 andleft to the reader.
We introduce the following notation which will constantly appear on the RHS of theequalities below N ( int ) [ J, ˇΓ , ˇ R ] := ( ext ) G J +1 [ˇΓ] + ( ext ) G (cid:48) J +1 [ˇΓ] + ( int ) R J +1 [ ˇ R ]+ (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) + (cid:15) (cid:16) ( int ) G J +1 [ˇΓ] + ( int ) G (cid:48) J +1 [ˇΓ] (cid:17) . (8.9.2) Step 1.
In view of Lemma 7.7.1 relating the Ricci coefficients and curvature componentsof ( int ) M to the ones of ( ext ) M on the timelike hypersurface T , we have (cid:90) T (cid:12)(cid:12) d J +1 ( ( int ) ˇΓ) (cid:12)(cid:12) (cid:46) (cid:90) T | d J +1 ( ( ext ) ˇΓ) | . Also, using again Lemma 7.7.1, we have (cid:90) T (cid:12)(cid:12)(cid:12) d J +1 (cid:16) ( int ) e θ ( ( int ) κ ) , ( int ) ˇ µ, ( int ) e ( ( int ) ζ − ( int ) β ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:90) T (cid:12)(cid:12)(cid:12) d J +1 (cid:16) ( ext ) e θ ( ( int ) κ ) − ( ext ) β, ( int ) ˇ µ, ( int ) e ( ( int ) ζ ) + ( int ) β (cid:17)(cid:12)(cid:12)(cid:12) + (cid:90) T | d J +1 ( ( ext ) ˇ R ) | We deduce, using that T = { r = r T } and the definitions of the various norms on ( ext ) M , (cid:90) T (cid:12)(cid:12) d J +1 ( ( int ) ˇΓ) (cid:12)(cid:12) + (cid:90) T (cid:12)(cid:12)(cid:12) d J +1 (cid:16) ( int ) e θ ( ( int ) κ ) , ( int ) ˇ µ, ( int ) e ( ( int ) ζ − ( int ) β ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) ( ext ) G J +1 [ˇΓ] + ( ext ) G (cid:48) J +1 [ˇΓ] + (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) . .9. PROOF OF PROPOSITION 8.3.10 (cid:90) T (cid:12)(cid:12) d J +1 ( ( int ) ˇΓ) (cid:12)(cid:12) + (cid:90) T (cid:12)(cid:12)(cid:12) d J +1 (cid:16) ( int ) e θ ( ( int ) κ ) , ( int ) ˇ µ, ( int ) e ( ( int ) ζ − ( int ) β ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . From now on, we only consider the frame of ( int ) M . The previous estimate can be writtenasmax k ≤ J +1 (cid:32) (cid:90) T (cid:16) ( d k ˇ µ ) + ( d k ζ ) + ( d k ˇ κ ) + ( d k ϑ ) + ( d k ˇ κ ) + ( d k ϑ ) +( d k ( e ( ζ ) − β )) + ( d k ξ ) + ( d k ˇ ω ) + ( d k ˇΩ) (cid:17)(cid:33) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) and max k ≤ J +1 (cid:90) T ( d k e θ ( κ )) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . Step 2.
We have obtained all the desired estimates on T for the foliation of ( int ) M inStep 1. We now derive the desired estimates on ( int ) M . To this end, we rely on thetransport equations in the e directions which we estimate thanks to Corollary 8.9.4. Theinitial data on T is estimated thanks to Step 1. In particular, we proceed in the followingorder • From e (ˇ κ ) + κ ˇ κ = Err[ e ˇ κ ]and the bootstrap assumptions, we infermax k ≤ J +1 (cid:90) ( int ) M ( d k ˇ κ ) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . • From e ( e θ ( κ )) + 32 κe θ ( κ ) = − ϑe θ ( κ ) − e θ ( ϑ )and the bootstrap assumptions, we infermax k ≤ J +1 (cid:90) ( int ) M ( d k e θ ( κ )) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) • From e (ˇ µ ) + 32 κ ˇ µ = − µ ˇ κ + Err[ e ˇ µ ] , the above control of ˇ κ and e θ ( κ ) (the control of e θ ( κ ) is needed to estimate Err[ e ˇ µ ]),and the bootstrap assumptions, we infermax k ≤ J +1 (cid:90) ( int ) M ( d k ˇ µ ) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . • From e ( ϑ ) + κ ϑ = − α and the control of α , we infermax k ≤ J +1 (cid:90) ( int ) M ( d k ϑ ) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . • From e ( ζ ) + κζ = β − ϑζ the control of β , and the bootstrap assumptions, we infermax k ≤ J +1 (cid:90) ( int ) M ( d k ζ ) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . • From e (ˇ κ ) + 12 κ ˇ κ = − κ ˇ κ + 2 d/ ζ + 2 ˇ ρ + Err[ e ˇ κ ]= − κ ˇ κ + 2ˇ µ + 4 ˇ ρ − ϑϑ + 12 ϑϑ + Err[ e ˇ κ ] , the control of ˇ ρ , the above control of ˇ κ and ˇ µ , and the bootstrap assumptions, weinfer max k ≤ J +1 (cid:90) ( int ) M ( d k ˇ κ ) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . • From e ( ϑ ) + 12 κϑ = 2 d (cid:63) / ζ − κϑ + 2 ζ = 2 d (cid:63) / d/ − (cid:18) ˇ µ + ˇ ρ − ϑϑ + 14 ϑϑ (cid:19) − κϑ + 2 ζ , .9. PROOF OF PROPOSITION 8.3.10 ρ , the above control of ϑ and ˇ µ , and the bootstrap assumptions, weinfer max k ≤ J +1 (cid:90) ( int ) M ( d k ϑ ) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . • From e (ˇ ω ) = ˇ ρ + Err[ e ˇ ω ] , the control of ˇ ρ , and the bootstrap assumptions, we infermax k ≤ J +1 (cid:90) ( int ) M ( d k ˇ ω ) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . • From e ( e ( ζ ) − β ) + κ ( e ( ζ ) − β )= − κβ + (cid:18) κκ + 2 µ − ρ (cid:19) ζ + ϑβ − ϑβ + ξα − ϑζ + 2 ωϑζ − ζ d (cid:63) / d/ − (cid:18) ˇ µ + ˇ ρ − ϑϑ + 14 ϑϑ (cid:19) − ζ , the control of β , the above control of ζ , and the bootstrap assumptions, we infermax k ≤ J +1 (cid:90) ( int ) M ( d k ( e ( ζ ) − β )) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . • From e ( ξ ) = ( e ( ζ ) − β ) + 2 β + κζ + ϑζ, the control of β , the above control of e ( ζ ) − β and ζ , and the bootstrap assumptions,we infer max k ≤ J +1 (cid:90) ( int ) M ( d k ξ ) (cid:46) (cid:16) N ( int ) [ J, ˇΓ , ˇ R ] (cid:17) . In view of the above estimates, of the definition (8.9.2) of N ≤ m [ J, ˇΓ , ˇ R ], and of thevarious norms, we infer ( int ) G J +1 [ˇΓ] + ( int ) G (cid:48) J +1 [ˇΓ] (cid:46) ( ext ) G J +1 [ˇΓ] + ( ext ) G (cid:48) J +1 [ˇΓ] + ( int ) R J +1 [ ˇ R ]+ (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) + (cid:15) (cid:16) ( int ) G J +1 [ˇΓ] + ( int ) G (cid:48) J +1 [ˇΓ] (cid:17) CHAPTER 8. INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8) and hence, for (cid:15) small enough, ( int ) G J +1 [ˇΓ] + ( int ) G (cid:48) J +1 [ˇΓ] (cid:46) ( ext ) G J +1 [ˇΓ] + ( ext ) G (cid:48) J +1 [ˇΓ] + ( int ) R J +1 [ ˇ R ]+ (cid:18)(cid:90) T | d J +1 ( ( ext ) ˇ R ) | (cid:19) . This concludes the proof of Proposition 8.9.1.
Lemma 4.6.6 corresponds to the particular case J = k large − k small − ≤ J ≤ k large − hapter 9GCM PROCEDURE We consider an axially symmetric polarized spacetime regions R foliated by two functions( u, s ) such that • On R , ( u, s ) defines an outgoing geodesic foliation as in section 2.2.4. • We denote by ( e , e , e θ ) the null frame adapted to the outgoing geodesic foliation( u, s ) on R . • Let ◦ S := S ( ◦ u, ◦ s ) (9.1.1)and ◦ r the area radius of ◦ S , where S ( u, s ) denote the 2-spheres of the outgoinggeodesic foliation ( u, s ) on R . • In adapted coordinates ( u, s, θ, ϕ ) with b = 0, see Proposition 2.2.20, the spacetimemetric g in R takes the form, with Ω = e ( s ) , b = e ( θ ), g = − ςduds + ς Ω du + γ (cid:18) dθ − ςbdu (cid:19) + e dϕ , (9.1.2)where θ is chosen such that b = e ( θ ) = 0.53334 CHAPTER 9. GCM PROCEDURE • The spacetime metric induced on S ( u, s ) is given by, g/ = γdθ + e dϕ . (9.1.3) • The relation between the null frame and coordinate system is given by e = ∂ s , e = 2 ς ∂ u + Ω ∂ s + b∂ θ , e θ = γ − / ∂ θ . (9.1.4) • We denote the induced metric on ◦ S by ◦ g/ = ◦ γ dθ + e dϕ . Definition 9.1.1.
Let < ◦ δ ≤ ◦ (cid:15) two sufficiently small constants. Let ( ◦ u, ◦ s ) real numbersso that ≤ ◦ u < + ∞ , m ≤ ◦ s < + ∞ . (9.1.5) We define R = R ( ◦ δ, ◦ (cid:15) ) to be the region R := (cid:110) | u − ◦ u | ≤ δ R , | s − ◦ s | ≤ δ R (cid:111) , δ R := ◦ δ (cid:0) ◦ (cid:15) (cid:1) − , (9.1.6) such that assumption A1-A3 below with constant ◦ (cid:15) on the background foliation of R , areverified. The smaller constant ◦ δ controls the size of the GCMS quantities as it will bemade precise below. In this section we define the renormalized Ricci and curvature components,ˇΓ : = (cid:26) ˇ κ, ϑ, ζ, η, κ − r , κ + 2Υ r , ˇ κ, ϑ, ξ, ˇ ω, ω − mr , ˇΩ , (cid:0) Ω + Υ (cid:1) , (cid:0) ς + 1 (cid:1)(cid:27) , ˇ R : = (cid:26) α, β, ˇ ρ, ρ + 2 mr , β, α (cid:27) . Since our foliation is outgoing geodesic we also have, ξ = ω = 0 , η + ζ = 0 . (9.1.7)We decompose ˇΓ = Γ g ∪ Γ b where,Γ g = (cid:26) ˇ κ, ϑ, ζ, ˇ κ, κ − r , κ + 2Υ r , (cid:27) , Γ b = (cid:110) η, ϑ, ξ, ˇ ω, ω − mr , r − ˇΩ , r − ˇ ς, r − (cid:0) Ω + Υ (cid:1) , r − (cid:0) ς + 1 (cid:1)(cid:111) . (9.1.8) .1. PRELIMINARIES p -reduced scalar f ∈ s p ( M ), with respect to the given geodesic foliation on R ,we consider the following norms on spheres S = S ( u, r ) ⊂ R , (cid:107) f (cid:107) ∞ ( u, r ) : = (cid:107) f (cid:107) L ∞ (cid:0) S ( u,r ) (cid:1) , (cid:107) f (cid:107) ( u, r ) := (cid:107) f (cid:107) L (cid:0) S ( u,r ) (cid:1) , (cid:107) f (cid:107) ∞ ,k ( u, r ) = k (cid:88) i =0 (cid:107) d i f (cid:107) ∞ ( u, r ) , (cid:107) f (cid:107) ,k ( u, r ) = k (cid:88) i =0 (cid:107) d i f (cid:107) ( u, r ) . (9.1.9)where, we recall, that d i stands for any combination of length i of operators of the from e , re , d / . Recall that, d / s f = (cid:40) r p (cid:52) / pk , if s = 2 p,r p +1 d/ k (cid:52) / pk , if s = 2 p + 1 . (9.1.10)On a given polarized surface S ⊂ R , not necessarily a leaf S of the given foliation, wedefine (cid:107) f (cid:107) h qs ( S ) : = s (cid:88) i =0 (cid:107) (cid:0) d / S (cid:1) i f (cid:107) L q ( S ) . (9.1.11)where d / S is defined as above with respect to the intrinsic metric on S . In the particularcase when q = 2 we omit the upper index i.e., h s ( S ) = h s ( S ). Given an integer s max , we assume the following A1.
For all k ≤ s max , we have on R(cid:107) Γ g (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − , (cid:107) Γ b (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − , (9.1.12)and, (cid:107) α, β, ˇ ρ, ˇ µ (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − , (cid:107) e ( α, β ) (cid:107) k − , ∞ (cid:46) ◦ (cid:15)r − , (cid:107) β (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − , (cid:107) α (cid:107) k, ∞ (cid:46) ◦ (cid:15)r − . (9.1.13) In applications, s max = k small + 4 in Theorem M7, and s max = k large + 5 in Theorem M0 andTheorem M6. CHAPTER 9. GCM PROCEDURE
A2.
We have, with m denoting the mass of the unperturbed spacetime,sup R (cid:12)(cid:12)(cid:12)(cid:12) mm − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15). (9.1.14) A3.
The metric coefficients are assumed to satisfy the following assumptions in R , forall k ≤ s max r (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) γr − , b, e Φ r sin θ − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ,k + (cid:107) Ω + Υ (cid:107) ∞ ,k + (cid:107) ς − (cid:107) ∞ ,k (cid:46) ◦ (cid:15). (9.1.15) Remark 9.1.2.
The above assumptions imply in particular the following | e ( r ) | , | e ( r ) | (cid:46) , e ( s ) = 1 + O ( ◦ (cid:15) ) , e ( u ) = 2 + O ( ◦ (cid:15) ) , e ( u ) = 0 . Hence, since r = ◦ r at ( ◦ u, ◦ s ) , we infer | r − ◦ r | (cid:46) | s − ◦ s | + | u − ◦ u | , and thus, in view of the definition (9.1.6) of R , sup R | r − ◦ r | (cid:46) ◦ δ (cid:0) ◦ (cid:15) (cid:1) − . (9.1.16)We will make use of the following lemma, see Lemmas 4.3.3 and 4.3.4. Lemma 9.1.3.
Under the assumption A3 for the metric coefficients we have, r (cid:12)(cid:12) e θ (Φ) (cid:12)(cid:12) ≤ sinθ , θ ≤ r | e θ Φ | + 1) . (9.1.17) Moreover, for any reduced 1-scalar h , we have sup S | h | e Φ (cid:46) r − sup S ( | h | + | d /h | ) , (cid:13)(cid:13) he Φ (cid:13)(cid:13) L ( S ) (cid:46) r − (cid:107) h (cid:107) h ( S ) . (9.1.18) We shall often make use of the results of Proposition 2.1.30 and Lemma 2.1.35 which werewrite as follows.
Lemma 9.1.4.
Under the assumptions A1 , A3 the following elliptic estimates hold truefor the Hodge operators d/ , d/ , d (cid:63) / , d (cid:63) / , for all k ≤ s max .2. DEFORMATIONS OF S SURFACES
1. If f ∈ s ( S ) (cid:107) d /f (cid:107) h k ( S ) + (cid:107) f (cid:107) h k ( S ) (cid:46) r (cid:107) d/ f (cid:107) h k ( S ) .
2. If f ∈ s ( S ) (cid:107) d /f (cid:107) h k ( S ) + (cid:107) f (cid:107) h k ( S ) (cid:46) r (cid:107) d/ f (cid:107) h k ( S ) .
3. If f ∈ s ( S ) (cid:107) d /f (cid:107) h k ( S ) (cid:46) r (cid:107) d (cid:63) / f (cid:107) h k ( S ) .
4. If f ∈ s ( S ) (cid:107) f (cid:107) h k +1 ( S ) (cid:46) r (cid:107) d (cid:63) / f (cid:107) h k ( S ) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e Φ f (cid:12)(cid:12)(cid:12)(cid:12) .
5. If f ∈ s ( S ) (cid:13)(cid:13)(cid:13)(cid:13) f − (cid:82) S f e Φ (cid:82) S e e Φ (cid:13)(cid:13)(cid:13)(cid:13) h k +1 ( S ) (cid:46) r (cid:107) d (cid:63) / f (cid:107) h k ( S ) . We shall often make use fo the following non-sharp product estimate on S , see Proposition2.1.40. Lemma 9.1.5.
The following estimates hold true on a given polarized surface S ⊂ R ,for any contraction between two reduced scalars ψ , ψ , k ≥ , (cid:107) ψ · ψ (cid:107) h k ( S ) (cid:46) r − (cid:107) ψ (cid:107) h k ( S ) (cid:107) ψ (cid:107) h k ( S ) . S surfaces Recall that ◦ S = S ( ◦ u, ◦ s ) is a fixed sphere of the ( u, s ) outgoing geodesic foliation of a fixedspacetime region R = R ( ◦ (cid:15), ◦ δ ).38 CHAPTER 9. GCM PROCEDURE
Definition 9.2.1.
We say that S is an O ( ◦ (cid:15) ) Z -polarized deformation of ◦ S if there existsa map Ψ : ◦ S −→ S of the form, Ψ( ◦ u, ◦ s, θ, ϕ ) = (cid:16) ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ, ϕ (cid:17) (9.2.1) where U, S are smooth functions defined on the interval [0 , π ] of amplitude at most ◦ (cid:15) . Wedenote by ψ the reduce map defined on the interval [0 , π ] , ψ ( θ ) = ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) . (9.2.2) We restrict ourselves to deformations which fix the South Pole, i.e. U (0) = S (0) = 0 . (9.2.3) We recall that given a scalar function f on S one defines its pull-back on ◦ S to be thefunction, f := Ψ f = f ◦ Ψ . On the other hand, given a vectorfield X on ◦ S one defines its push-forward Ψ X to bethe vectorfield on S defined by,Ψ X ( f ) = X (Ψ f ) = X ( f ◦ Ψ) . Given a covariant tensor U on S , one defines its pull back to ◦ S to be the tensorΨ U ( X , . . . , X k ) = U (Ψ X , . . . , Ψ X k ) . Lemma 9.2.2.
Given a Z -invariant deformation Ψ : ◦ S −→ S , we have,1. Let g/ S the induced metric on S and g/ S , = γ S , dθ + e dϕ its pull-back to ◦ S .The metric coefficients γ S and γ S , are related by, γ S , ( θ ) = γ S ( ψ ( θ )) = γ S ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) (9.2.4) where γ S is defined implicitly by, ( γ S ) = γ + (cid:0) ς (cid:1) (cid:18) Ω + 14 b γ (cid:19) ( U (cid:48) ) − ς U (cid:48) S (cid:48) − ( γςb ) U (cid:48) , (9.2.5) .2. DEFORMATIONS OF S SURFACES that is, γ S ( ψ ( θ )) = γ ( ψ ( θ )) + ς ( ψ ( θ )) (cid:18) Ω( ψ ( θ )) + 14 ( b ( ψ ( θ ))) γ ( ψ ( θ )) (cid:19) ( U (cid:48) ( θ )) − ς ( ψ ( θ )) U (cid:48) ( θ ) S (cid:48) ( θ ) − γ ( ψ ( θ )) ς ( ψ ( θ )) b ( ψ ( θ )) U (cid:48) ( θ ) .
2. The Z -invariant vectorfield ∂ S θ := Ψ ( ∂ θ ) is tangent to S and ∂ S θ | Ψ( p ) = (cid:104) (cid:16) ∂ θ S − ς ∂ θ U (cid:17) e + ς ∂ θ U e + √ γ (cid:16) − ς b∂ θ U (cid:17) e θ (cid:105)(cid:12)(cid:12)(cid:12) Ψ( p ) . (9.2.6)
3. If f ∈ s k ( S ) and P S is a geometric operator acting on f then, ( P S [ f ]) = P S , [ f ] (9.2.7) where, P S , is the corresponding geometric operator on ◦ S with respect to the metric g/ S , and f = ψ f .4. The L norm of f = ψ f with respect to the metric g/ S , is the same as as the L norm of f with respect to the metric g/ S , i.e., (cid:90) ◦ S | f | da g/ S , = (cid:90) S | f | da g/ S .
5. If f ∈ h k ( S ) and f is its pull-back by ψ then, (cid:107) f (cid:107) h k ( ◦ S, g/ S , ) = (cid:107) f (cid:107) h k ( S ) . Proof. If ∂ θ denotes the coordinate derivative ∂ θ = ∂∂θ then, at every point p ∈ ◦ S ,Ψ ( ∂ θ ) | Ψ( p ) = ∂ θ U ∂ u | Ψ( p ) + ∂ θ S∂ s | Ψ( p ) + ∂ θ | Ψ( p ) , Ψ ( ∂ ϕ ) = ∂ ϕ . In view of (9.1.4) we have ∂ s = e , ∂ u = ς (cid:0) e − Ω e − bγ / e θ (cid:1) , ∂ θ = √ γe θ . Hence, at a point Ψ( p ) on S we have,Ψ ( ∂ θ ) = (cid:16) ∂ θ S − ς ∂ θ U (cid:17) e + ς ∂ θ U e + √ γ (cid:16) − ς b∂ θ U (cid:17) e θ . CHAPTER 9. GCM PROCEDURE
We denote by g/ = Ψ ( g/ S ) the pull back to ◦ S of the metric g/ S on S , i.e. at any point p ∈ ◦ S , g/ ( ∂ θ , ∂ θ ) = g/ S (Ψ ∂ θ , Ψ ∂ θ ) = g ( ∂ θ U ∂ u + ∂ θ S∂ s + ∂ θ , ∂ θ U ∂ u + ∂ θ S∂ s + ∂ θ )= ( ∂ θ U ) g uu + 2 ∂ θ U ∂ θ S g us + 2 ∂ θ U g uθ + g θθ ,g/ ( ∂ θ , ∂ ϕ ) = 0 ,g/ ( ∂ ϕ , ∂ ϕ ) = e , where, g uu = ς (cid:0) Ω + 14 γb (cid:1) , g us = − ς, g uθ = − ς γb, g ss = g sθ = 0 , g θθ = γ. Hence the pull-back metric Ψ ( g/ S ) on ◦ S is given by, γ S , dθ + e dϕ where γ S , = ( γ S ) , (9.2.8)with γ S is defined by,( γ S ) = γ + ( ς ) (cid:18) Ω + 14 b γ (cid:19) ( U (cid:48) ) − ς U (cid:48) S (cid:48) − ( γςb ) U (cid:48) . (9.2.9)Note that the vectorfield, e S θ := 1( γ S ) / ψ ( ∂ θ )is tangent, Z invariant and forms together with e ϕ an orthonormal frame on S . Note thatwe can also write, e S θ := ( ◦ γ ) / ( γ S ) / Ψ ( e θ )where ◦ γ is the coefficient in front of dθ of the metric induced by g on ◦ S , ◦ g/ = ◦ γ dθ + e dϕ . In general, any geometric calculation on S can be reduced to a geometric calculation on ◦ S with respect to the metric g/ S , . Moreover the L norm on S with respect to the metric g/ S is the same as the L norm of f = ψ f with respect to the norm g/ S , . This concludesthe proof of the lemma. .2. DEFORMATIONS OF S SURFACES
Lemma 9.2.3.
Let
Ψ : ◦ S −→ S a Z -invariant deformation in R ( ◦ (cid:15), ◦ δ ) with U, V verifyingthe bounds sup ≤ θ ≤ π (cid:16) | U (cid:48) ( θ ) | + | S (cid:48) ( θ ) | (cid:17) (cid:46) ◦ δ, (9.2.10) as well as the bound (9.1.15) for the coordinates system ( u, s, θ, ϕ ) of R . The followinghold true1. We have, (cid:12)(cid:12) γ S , − ◦ γ (cid:12)(cid:12) (cid:46) ◦ δ ◦ r. (9.2.11)
2. For every f ∈ s k ( S ) we have, (cid:107) f (cid:107) L ( ◦ S,g/ S , ) = (cid:107) f (cid:107) L ( ◦ S, ◦ g/ ) (cid:16) O ( r − ◦ δ ) (cid:17) . (9.2.12)
3. As a corollary of (9.2.12) (choosing f = 1 ) we deduce , r S ◦ r = 1 + O ( ◦ r − ◦ δ ) (9.2.13) where r S is the area radius of S and ◦ r that of ◦ S .Proof. Recall, γ S , ( ◦ u, ◦ s, θ ) = γ ( ψ ( θ )) + ς ( ψ ( θ )) (cid:18) Ω( ψ ( θ )) + 14 ( b ( ψ ( θ ))) γ ( ψ ( θ )) (cid:19) ( U (cid:48) ( θ )) − ς ( ψ ( θ )) U (cid:48) ( θ ) S (cid:48) ( θ ) − γ ( ψ ( θ )) ς ( ψ ( θ ) b ( ψ ( θ )) U (cid:48) ( θ ) . In view of our assumptions on U (cid:48) and S (cid:48) as well as our estimates (9.1.15) for γ , Ω and b and ς , we infer | γ S , − γ | (cid:46) | γ − γ | + ◦ r ◦ (cid:15) / ◦ δ. Recall also from (9.1.16) that r − ◦ r = O ( ◦ δ ◦ (cid:15) − / ). CHAPTER 9. GCM PROCEDURE
Also, we have γ ( ◦ u, ◦ s, θ ) − γ ( ◦ u, ◦ s, θ ) = γ ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) − γ ( ◦ u, ◦ s, θ )= (cid:90) ddλ (cid:104) γ ( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) (cid:105) dλ = U ( θ ) (cid:90) ∂ u γ ( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) dλ + S ( θ ) (cid:90) ∂ s γ ( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) dλ. In view of our estimates (9.1.15) for γ , the assumption (9.2.10) on ( U (cid:48) , V (cid:48) ) and the factthat ∂ s = e , ∂ u = ς (cid:0) e − Ω e − bγ / e θ (cid:1) , we infer | γ − γ | (cid:46) ◦ r ◦ δ. We have finally, obtained | γ S , − γ | (cid:46) | γ − γ | + ◦ r ◦ δ (cid:46) ◦ r ◦ δ. To prove the second part of the lemma we write, (cid:90) ◦ S | f | da g/ S , = (cid:90) ◦ S | f | (cid:112) γ S , (cid:113) ◦ γ da ◦ g/ = (cid:90) ◦ S | f | da ◦ g/ + (cid:90) ◦ S | f | (cid:112) γ S , (cid:113) ◦ γ − da ◦ g/ which yields, in view of the first part, (cid:90) ◦ S | f | da g/ S , = (cid:90) ◦ S | f | da ◦ g/ (cid:16) O ( ◦ r − ◦ δ ) (cid:17) . This concludes the proof of the lemma.
Remark 9.2.4.
In view of (9.2.13) and (9.1.16) , ◦ r , r S and the value of r along S are allcomparable. Corollary 9.2.5.
Under the assumptions of Lemma 9.2.3 the following estimate holdstrue for an arbitrary scalar f ∈ s ( R ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f − (cid:90) ◦ S f (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ ◦ r (cid:18) sup R | d ≤ (cid:37) f | + sup R r | e f | (cid:19) . Note that we also use the assumption U (0) = S (0) = 0 to estimate ( U, S ) from ( U (cid:48) , S (cid:48) ). Recall that R := {| u − ◦ u | ≤ δ R , | s − ◦ s | ≤ δ R } , see (9.1.6). .2. DEFORMATIONS OF S SURFACES
Proof.
We have, (cid:90) S f − (cid:90) ◦ S f = (cid:90) ◦ S f (cid:112) γ S , (cid:113) ◦ γ − (cid:90) ◦ S f = (cid:90) ◦ S f (cid:112) γ S , (cid:113) ◦ γ − + (cid:90) ◦ S ( f − f ) . Hence, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f − (cid:90) ◦ S f (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ ◦ r sup S | f | + (cid:90) ◦ S (cid:12)(cid:12) f − f (cid:12)(cid:12) . Now, proceeding as in the proof of (9.2.11), f ( ◦ u + U ( θ ) , ◦ s + S ( θ )) − f ( ◦ u, ◦ s ) (cid:46) (cid:90) ddλ (cid:104) f ( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) (cid:105) dλ = U ( θ ) (cid:90) ∂ u f ( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) dλ + S ( θ ) (cid:90) ∂ s f ( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) dλ. Therefore , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f − (cid:90) ◦ S f (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ r ◦ δ sup S | f | + ◦ δ ◦ r (cid:18) sup R ◦ r | d (cid:37) f | + sup R r | e f | (cid:19) (cid:46) ◦ δ ◦ r (cid:18) sup R | d ≤ (cid:37) f | + sup R r | e f | (cid:19) as stated.To compare higher order Sobolev spaces, we will need the following lemma. Lemma 9.2.6.
Let ◦ S ⊂ R = R ( ◦ (cid:15), ◦ δ ) as in Definition 9.1.1 verifying the assumptions A1-A3 . Let
Ψ : ◦ S −→ S be Z -invariant deformation. Assume the bound (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) L ∞ ( ◦ S ) + max ≤ s ≤ s max ( ◦ r ) − (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:46) ◦ δ. (9.2.14) Then, we have for any reduced scalar h defined on R(cid:107) h (cid:107) h s ( S ) (cid:46) r sup R | d ≤ s h | , for ≤ s ≤ s max . Also, if f ∈ h s ( S ) and f is its pull-back by ψ , we have (cid:107) f (cid:107) h s ( S ) = (cid:107) f (cid:107) h s ( ◦ S, g/ S , ) = (cid:107) f (cid:107) h s ( ◦ S, ◦ g/ ) (1 + O ( r − ◦ δ )) for ≤ s ≤ s max − . CHAPTER 9. GCM PROCEDURE
Proof.
See appendix C.1.
Corollary 9.2.7.
Under the same assumptions as Lemma 9.2.6, we have, for all ≤ k ≤ s max − , (cid:107) d ≤ Γ g (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) d ≤ Γ b (cid:107) h k ( S ) (cid:46) ◦ (cid:15), (9.2.15) (cid:107) d ≤ ( α, β, ˇ ρ, ˇ µ ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) d ≤ β (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) d ≤ α (cid:107) h k ( S ) (cid:46) ◦ (cid:15), (9.2.16) (cid:13)(cid:13)(cid:13)(cid:13) d ≤ (cid:18) γr − , b, e Φ r sin θ − (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15), (cid:13)(cid:13) d ≤ (Ω + Υ) (cid:13)(cid:13) h k ( S ) + (cid:13)(cid:13) d ≤ ( ς + 1) (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15)r. (9.2.17) Proof.
In view of Lemma 9.2.6 and assumptions
A1-A3 we have, for 0 ≤ s ≤ s max − (cid:13)(cid:13) d ≤ ˇΓ g (cid:13)(cid:13) h s ( S ) (cid:46) r sup R (cid:12)(cid:12) d ≤ s +2 ˇΓ g (cid:12)(cid:12) (cid:46) r − ◦ (cid:15). The other estimates are proved in the same manner.
We consider general null transformations introduced in Lemma 2.3.1, e (cid:48) = λ (cid:18) e + f e θ + 14 f e (cid:19) ,e (cid:48) θ = (cid:18) f f (cid:19) e θ + 12 f e + 12 f (cid:18) f f (cid:19) e ,e (cid:48) = λ − (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19) . (9.2.18) Definition 9.2.8.
Given a deformation
Ψ : ◦ S −→ S we say that a new frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) ,obtained from the standard frame ( e , e , e θ ) via the transformation (9.2.18) , is S -adaptedif we have, e (cid:48) θ = e S θ = 1( γ S ) / ψ ( ∂ θ ) . (9.2.19) .2. DEFORMATIONS OF S SURFACES
Proposition 9.2.9.
Consider a deformation
Ψ : ◦ S −→ S in R = R ( ◦ (cid:15), ◦ δ ) verifying theassumption A3 . The following statements hold true.1. A new frame e (cid:48) , e (cid:48) θ , e (cid:48) generated by ( f, f , λ = e a ) according to (9.2.18) is adapted to S = S ( ◦ u + U, ◦ s + S ) provided that, at all points θ ∈ [0 , π ] , (cid:112) γ (cid:16) − ς b U (cid:48) (cid:17) = (cid:0) ( γ S ) (cid:1) / (cid:18) f f ) (cid:19) ,ςU (cid:48) = (cid:0) ( γ S ) (cid:1) / f (cid:18) f f ) (cid:19) , (cid:16) S (cid:48) − ς U (cid:48) (cid:17) = (cid:0) ( γ S ) (cid:1) / f , (9.2.20) where, ( γ S ) = γ + ( ς ) (cid:18) Ω + 14 b γ (cid:19) ( ∂ θ U ) − ς ∂ θ U ∂ θ S − ( γςb ) ∂ θ U and denotes the pull back by ψ of the corresponding reduced scalars, i.e. forexample, f ( θ ) = f ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) .2. There exists a small enough constant δ such that for given f, f on R satisfying sup R (cid:16) | f | + | f | (cid:17) ≤ r − δ , we can uniquely solve the system (9.2.20) for U, S subject to the initial conditions, U (0) = 0 , S (0) = 0 . Thus, if ( ◦ u, ◦ s, corresponds to the south pole of ◦ S and f, f are given there exists aunique deformation S ⊂ R , given by U, S : [0 , π ] −→ R , adapted to frames generatedby ( f, f ) which passes through the same south pole. Moreover, sup [0 ,π ] | ( U (cid:48) , S (cid:48) ) | (cid:46) ◦ r sup S (cid:0) | f | + | f | (cid:1) (9.2.21) In later applications, we will have sup R ( | f | + | f | ) (cid:46) r − ◦ δ. Note that a is not restricted in this result. CHAPTER 9. GCM PROCEDURE and, for ≤ s ≤ s max , (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) L ∞ ( ◦ S ) + ( ◦ r ) − (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:46) (cid:107) f, f (cid:107) h s ( S ) (9.2.22) with (cid:107) f, f (cid:107) h s ( S ) = (cid:107) f (cid:107) h s ( S ) + (cid:107) f (cid:107) h s ( S ) .3. As a consequence of (9.2.22) the deformation thus obtained verifies the conclusionsof Lemmas 9.2.3-9.2.6 and Corollary 9.2.7. In particular,(a) We have, (cid:12)(cid:12)(cid:12) γ S , − ◦ γ (cid:12)(cid:12)(cid:12) (cid:46) δ ◦ r . (b) We have (cid:12)(cid:12)(cid:12)(cid:12) r S ◦ r − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ r − δ . Proof.
In view of Lemma 9.2.2, The Z -invariant vectorfield e S θ := γ S ) / Ψ ( ∂ θ ) can beexpressed by the formula, e S θ = 1( γ S ) / (cid:104) (cid:16) ∂ θ S − ς ∂ θ U (cid:17) e + ς ∂ θ U e + √ γ (cid:16) − ς b∂ θ U (cid:17) e θ (cid:105) . where ψ ( p ) = ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) and U (cid:48) = ∂ θ U ( θ ), S (cid:48) = ∂ θ S ( θ ). On the other hand,according to (9.2.18), at Ψ( p ) ∈ S , e (cid:48) θ = (cid:18) f f (cid:19) e θ + 12 f (cid:18) f f (cid:19) e + 12 f e . We deduce, at every θ ∈ [0 , π ], (cid:112) γ (cid:18) − ς b U (cid:48) (cid:19) = (cid:0) ( γ S ) (cid:1) / (cid:18) f f ) (cid:19) ,ς U (cid:48) = (cid:0) ( γ S ) (cid:1) / f (cid:18) f f ) (cid:19) , (cid:18) S (cid:48) − ς U (cid:48) (cid:19) = (cid:0) ( γ S ) (cid:1) / f , as desired. .2. DEFORMATIONS OF S SURFACES A = 1 + 12 ( f f ) , B = f (cid:18) f f ) (cid:19) , C = f , we have A − BC = 1 . Hence, squaring the first equation and subtracting the product ofthe other two we derive,( γ S ) = (cid:18)(cid:112) γ (cid:18) − ς b U (cid:48) (cid:19)(cid:19) − U (cid:48) ς (cid:18) S (cid:48) − ς U (cid:48) (cid:19) = γ (cid:18) − ( ςb ) U (cid:48) + 14 (cid:0) ( ςb ) U (cid:48) (cid:1) (cid:19) − ς U (cid:48) S (cid:48) + ( ς ) Ω ( U (cid:48) ) = γ + ( ς ) (cid:18) Ω + 14 (cid:0) b (cid:1) γ (cid:19) ( U (cid:48) ) − ς U (cid:48) S (cid:48) − γ ς b U (cid:48) (9.2.23)which coincides with the formula (9.2.5). It thus suffices to only consider the last twoequations in (9.2.20) which we write in the form, U (cid:48) = ( ς ) − (( γ S ) ) / f (cid:18) f f ) (cid:19) ,S (cid:48) = 12 (( γ S ) ) / f + 12 Ω (( γ S ) ) / f (cid:18) f + 14 ( f f ) (cid:19) , (9.2.24)i.e., U (cid:48) ( θ ) = (cid:20) ( ς ) − ( γ S ) / f (cid:18) f f ) (cid:19)(cid:21) ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) ,S (cid:48) ( θ ) = (cid:20)
12 ( γ S ) / f + 12 Ω( γ S ) / f (cid:18) f f (cid:19)(cid:21) ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) . Thus under the assumption sup R ( | f | + | f | ) ≤ ◦ r − δ , with δ sufficiently small, makingalso use of the expression (9.2.23) of γ S , and the estimates (9.1.15) for ( γ, b, Ω), for ◦ (cid:15) sufficiently small, we can uniquely solve for U, S subject to the initial conditions, U (0) = 0 , S (0) = 0 . Moreover the solution verifies,sup [0 ,π ] | ( U (cid:48) , S (cid:48) ) | (cid:46) ◦ r sup S (cid:0) | f | + | f | (cid:1) according to the Definition 9.2.1. Estimate (9.2.22) can be easily derived by taking higherderivatives and using A1-A3 . This concludes the proof of the lemma.48
CHAPTER 9. GCM PROCEDURE
For the convenience of the reader we start by recalling the transformation formulasrecorded in Proposition 2.3.4.
Proposition 9.3.1 (Transformation formulas-GCM) . Under a general transformation oftype (9.2.18) with λ = e a the Ricci coefficients and curvature components transform asfollows: ξ (cid:48) = λ (cid:18) ξ + 12 λ − e (cid:48) ( f ) + ωf + 14 f κ (cid:19) + λ Err ( ξ, ξ (cid:48) ) , Err ( ξ, ξ (cid:48) ) = 14 f ϑ + l.o.t. ,ξ (cid:48) = λ − (cid:18) ξ + 12 λe (cid:48) ( f ) + ω f + 14 f κ (cid:19) + λ − Err ( ξ, ξ (cid:48) ) , Err ( ξ, ξ (cid:48) ) = − λf e (cid:48) ( f ) + 14 f ϑ + l.o.t. , (9.3.1) ζ (cid:48) = ζ − e (cid:48) θ (log( λ )) + 14 ( − f κ + f κ ) + f ω − f ω + Err ( ζ, ζ (cid:48) ) , Err ( ζ, ζ (cid:48) ) = 12 f e (cid:48) θ ( f ) + 14 ( − f ϑ + f ϑ ) + l.o.t. ,η (cid:48) = η + 12 λe (cid:48) ( f ) + 14 κf − f ω + Err ( η, η (cid:48) ) , Err ( η, η (cid:48) ) = 14 f ϑ + l.o.t. ,η (cid:48) = η + 12 λ − e (cid:48) ( f ) + 14 κf − f ω + Err ( η, η (cid:48) ) , Err ( η, η (cid:48) ) = − f λ − e (cid:48) ( f ) + 14 f ϑ + l.o.t. , (9.3.2) κ (cid:48) = λ ( κ + d/ (cid:48) ( f )) + λ Err ( κ, κ (cid:48) ) , Err ( κ, κ (cid:48) ) = f ( ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t. ,κ (cid:48) = λ − (cid:0) κ + d/ (cid:48) ( f ) (cid:1) + λ − Err ( κ, κ (cid:48) ) , Err ( κ, κ (cid:48) ) = − f e (cid:48) θ ( f ) + f ( − ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t. , (9.3.3) .3. FRAME TRANSFORMATIONS ϑ (cid:48) = λ ( ϑ − d (cid:63) / (cid:48) ( f )) + λ Err ( ϑ, ϑ (cid:48) ) , Err ( ϑ, ϑ (cid:48) ) = f ( ζ + η ) + f ξ + 14 f f κ + f f ω − f ω + l.o.t. ϑ (cid:48) = λ − (cid:0) ϑ − d (cid:63) / (cid:48) ( f ) (cid:1) + λ − Err ( ϑ, ϑ (cid:48) ) , Err ( ϑ, ϑ (cid:48) ) = − f e (cid:48) θ ( f ) + f ( − ζ + η ) + f ξ + 14 f f κ + f f ω − f ω + l.o.t. , (9.3.4) ω (cid:48) = λ (cid:18) ω − λ − e (cid:48) (log( λ )) (cid:19) + λ Err ( ω, ω (cid:48) ) , Err ( ω, ω (cid:48) ) = 14 f e (cid:48) ( f ) + 12 ωf f − f η + 12 f ξ + 12 f ζ − κf + 18 f f κ − ωf + l.o.t. ,ω (cid:48) = λ − (cid:18) ω + 12 λe (cid:48) (log( λ )) (cid:19) + λ − Err ( ω, ω (cid:48) ) , Err ( ω, ω (cid:48) ) = − f e (cid:48) ( f ) + ωf f − f η + 12 f ξ − f ζ − κf + 18 f f κ − ωf + l.o.t. (9.3.5) The lower order terms we denote by l.o.t. are linear with respect to (cid:8) ξ, ξ, ϑ, κ, η, η, ζ, κ, ϑ (cid:9) and quadratic or higher order in f, f , and do not contain derivatives of these latter.Also, α (cid:48) = λ α + λ Err ( α, α (cid:48) ) , Err ( α, α (cid:48) ) = 2 f β + 32 f ρ + l.o.t. ,β (cid:48) = λ (cid:18) β + 32 ρf (cid:19) + λ Err ( β, β (cid:48) ) , Err ( β, β (cid:48) ) = 12 f α + l.o.t. ,ρ (cid:48) = ρ + Err ( ρ, ρ (cid:48) ) , Err ( ρ, ρ (cid:48) ) = 32 ρf f + f β + f β + l.o.t. ,β (cid:48) = λ − (cid:18) β + 32 ρf (cid:19) + λ − Err ( β, β (cid:48) ) , Err ( β, β (cid:48) ) = 12 f α + l.o.t. ,α (cid:48) = λ − α + λ − Err ( α, α (cid:48) ) , Err ( α, α (cid:48) ) = 2 f β + 32 f ρ + l.o.t. (9.3.6) The lower order terms we denote by l.o.t. are linear with respect to the curvature quantities α, β, ρ, β, α and quadratic or higher order in f, f , and do not contain derivatives of theselatter. CHAPTER 9. GCM PROCEDURE
In the following lemma we rewrite a subset of these transformations in a more useful form,
Lemma 9.3.2.
Under a general transformation of type (9.2.18) with λ = e a we have, inparticular, ζ (cid:48) = ζ − e (cid:48) θ ( a ) − f ω + f ω − f χ + 12 f χ + Err ( ζ, ζ (cid:48) ) , Err ( ζ, ζ (cid:48) ) = 12 f (cid:18) f f (cid:19) e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + 14 ( − f ϑ + f ϑ ) + l.o.t. (9.3.7) κ (cid:48) = e a ( κ + d/ (cid:48) f ) + e a Err ( κ, κ (cid:48) ) , Err ( κ, κ (cid:48) ) = 12 f f e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + f ( ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t. (9.3.8) κ (cid:48) = e − a (cid:0) κ + d/ (cid:48) f (cid:1) + e − a Err ( κ, κ (cid:48) ) , Err ( κ, κ (cid:48) ) = − f e (cid:48) θ (cid:18) f f + 18 f f (cid:19) + (cid:18) f f + 18 ( f f ) (cid:19) e (cid:48) θ ( f )+ 14 (cid:18) f f (cid:19) f e (cid:48) θ (cid:0) f f (cid:1) − f (cid:18) f f (cid:19) e (cid:48) θ (cid:0) f (cid:1) + f ( − ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t. (9.3.9) Also, ϑ (cid:48) = λ ( ϑ − d (cid:63) / (cid:48) ( f )) + λ Err ( ϑ, ϑ (cid:48) ) , Err ( ϑ, ϑ (cid:48) ) = 12 f f e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + f ( ζ + η ) + f ξ + 14 f f κ + f f ω − f ω + l.o.t. ϑ (cid:48) = λ − (cid:0) ϑ − d (cid:63) / (cid:48) ( f ) (cid:1) + λ − Err ( ϑ, ϑ (cid:48) ) , Err ( ϑ, ϑ (cid:48) ) = − f e (cid:48) θ (cid:18) f f + 18 f f (cid:19) + (cid:18) f f + 18 ( f f ) (cid:19) e (cid:48) θ ( f )+ 14 (cid:18) f f (cid:19) f e (cid:48) θ (cid:0) f f (cid:1) − f (cid:18) f f (cid:19) e (cid:48) θ (cid:0) f (cid:1) + f ( − ζ + η )+ f ξ + 14 f f κ + f f ω − f ω + l.o.t. , (9.3.10) The lower order terms we denote by l.o.t. are cubic or higher order in the small quantities ξ, ξ, ϑ, η, η, ζ, ϑ as well as f, f , and do not contain derivatives of these quantities. .3. FRAME TRANSFORMATIONS
We also have, β (cid:48) = λ (cid:18) β + 32 ρf (cid:19) + λ Err ( β, β (cid:48) ) , Err ( β, β (cid:48) ) = 12 f α + l.o.t. ,ρ (cid:48) = ρ + Err ( ρ, ρ (cid:48) ) , Err ( ρ, ρ (cid:48) ) = 32 ρf f + f β + f β + l.o.t. (9.3.11) The lower order terms above denoted by l.o.t. are cubic or higher order in the smallquantities ξ, ξ, ϑ, η, η, ζ, ϑ as well as a, f, f . Lemma 9.3.3.
The following transformation formula holds true µ (cid:48) = µ + ( d/ ) (cid:48) (cid:18) − ( d (cid:63) / ) (cid:48) a + f ω − f ω + 14 f κ − f κ (cid:19) + Err ( µ, µ (cid:48) ) , Err ( µ, µ (cid:48) ) = − d/ (cid:48) Err ( ζ, ζ (cid:48) ) − Err ( ρ, ρ (cid:48) ) + 14 (cid:0) ϑ (cid:48) ϑ (cid:48) − ϑϑ (cid:1) . The error term Err ( µ, µ (cid:48) ) is quadratic or higher order with respect to ( f, f , a, ˇΓ , ˇ R ) anddepends only on at most two angular derivatives e (cid:48) θ of f and one angular derivative e (cid:48) θ of a, f .Proof. Recall that µ = − d/ ζ − ρ + 14 ϑϑ. Therefore, µ (cid:48) = − d/ (cid:48) ζ (cid:48) − ρ (cid:48) + 14 ϑ (cid:48) ϑ (cid:48) = − d/ (cid:48) (cid:18) ζ − e (cid:48) θ ( a ) − f ω + f ω − f κ + 14 f κ + Err( ζ, ζ (cid:48) ) (cid:19) − ρ − Err( ρ, ρ (cid:48) ) + 14 ϑ (cid:48) ϑ (cid:48) = − d/ (cid:48) ζ − ρ + 14 ϑϑ − d/ (cid:48) (cid:18) ( d (cid:63) / ) (cid:48) a − f ω + f ω − f κ + 14 f κ (cid:19) − d/ (cid:48) Err( ζ, ζ (cid:48) ) − Err( ρ, ρ (cid:48) ) + 14 (cid:0) ϑ (cid:48) ϑ (cid:48) − ϑϑ (cid:1) . Note that, − d/ (cid:48) ζ − ρ + 14 ϑϑ = − d/ ζ − ρ + 14 ϑϑ + f e ζ + f e z e + l.o.t.= µ + f e ζ + f e ζ + l.o.t.52 CHAPTER 9. GCM PROCEDURE
Hence, µ (cid:48) = µ + d/ (cid:48) (cid:18) − ( d (cid:63) / ) (cid:48) a + f ω − f ω + 14 f κ − f κ (cid:19) + Err( µ, µ (cid:48) )where,Err( µ, µ (cid:48) ) = − d/ (cid:48) Err( ζ, ζ (cid:48) ) − Err( ρ, ρ (cid:48) ) + 14 (cid:0) ϑ (cid:48) ϑ (cid:48) − ϑϑ (cid:1) + f e ζ + f e ζ + l.o.t.In view of the transformation formulas for θ, θ and the structure of the error termsErr( ζ, ζ (cid:48) ) , Err( ρ, ρ (cid:48) ) , Err( ϑ, ϑ (cid:48) ) , Err( ϑ, ϑ (cid:48) ) in Lemma 9.3.2 we easily deduce that the errorterm Err( µ, µ (cid:48) ) depends only on at most two angular derivatives e (cid:48) θ of f and one angularderivative e (cid:48) θ of a, f .We shall also make use of the following, Lemma 9.3.4.
We have the transformation equations, e (cid:48) θ ( κ (cid:48) ) = e θ κ + e (cid:48) θ d/ (cid:48) f + κe (cid:48) θ a − κ ( f κ + f κ ) + κ ( f ω − ωf ) + f ρ + Err ( e (cid:48) θ κ (cid:48) , e θ κ ) ,e (cid:48) θ ( κ (cid:48) ) = e θ κ + e (cid:48) θ d/ (cid:48) f − κe (cid:48) θ a − κ ( f κ + f κ ) + κ ( f ω − ωf ) + f ρ + Err ( e (cid:48) θ κ (cid:48) , e θ κ ) ,e (cid:48) θ ( µ (cid:48) ) = e θ µ + e (cid:48) θ ( d/ ) (cid:48) (cid:18) − ( d (cid:63) / ) (cid:48) a + f ω − f ω + 14 f κ − f κ (cid:19) + 34 ρ ( f κ + f κ )+ Err ( e (cid:48) θ µ (cid:48) , e θ µ ) , (9.3.12) where,Err ( e (cid:48) θ κ (cid:48) , e θ κ ) = ( e a − (cid:16) e θ κ + e (cid:48) θ d/ (cid:48) f + 12 f e κ + 12 f e κ (cid:17) + e a (cid:20) e (cid:48) θ Err ( κ, κ (cid:48) ) + e (cid:48) θ ( a ) (cid:16) d/ (cid:48) f + Err ( κ, κ (cid:48) ) (cid:17) + 12 f f e θ κ + 18 f f e κ (cid:21) + 12 f (cid:18) d/ η − ϑ ϑ + 2( ξξ + η ) (cid:19) + 12 f f e θ κ + 18 f f e κ + 12 f (cid:18) d/ ξ − ϑ + 2( η + η + 2 ζ ) ξ (cid:19) , .3. FRAME TRANSFORMATIONS Err ( e (cid:48) θ κ (cid:48) , e θ κ ) = ( e − a − (cid:16) e θ κ + e (cid:48) θ d/ (cid:48) f + 12 f e κ + 12 f e κ (cid:17) + e − a (cid:20) e (cid:48) θ Err ( κ, κ (cid:48) ) + e (cid:48) θ ( a ) (cid:16) d/ (cid:48) f + Err ( κ, κ (cid:48) ) (cid:17) + 12 f f e θ κ + 18 f f e κ (cid:21) + 12 f (cid:18) d/ η − ϑ ϑ + 2( ξξ + η ) (cid:19) + 12 f f e θ κ + 18 f f e κ + 12 f (cid:18) d/ ξ − ϑ + 2( η + η − ζ ) ξ (cid:19) , and, Err ( e (cid:48) θ µ (cid:48) , e θ µ ) = e (cid:48) θ Err ( µ, µ (cid:48) ) + 12 f f e θ µ + 18 f f e µ − f (cid:16) d/ β − ϑ α − ζ β + 2( η β + ξ β ) (cid:17) − f (cid:16) d/ β − ϑ α + ζ β + 2( η β + ξ β ) (cid:17) + 12 f e (cid:18) − d/ ζ + 14 ϑϑ (cid:19) + 12 f e (cid:18) − d/ ζ + 14 ϑϑ (cid:19) . Proof.
Applying the vectorfield e (cid:48) θ to κ (cid:48) = e a ( κ + d/ (cid:48) f + Err( κ, κ (cid:48) ))we deduce, e (cid:48) θ ( κ (cid:48) ) = e a (cid:16) e (cid:48) θ κ + e (cid:48) θ d/ (cid:48) f + e (cid:48) θ (Err( κ, κ (cid:48) ) (cid:17) + e a e (cid:48) θ ( a ) (cid:16) κ + d/ (cid:48) f + Err( κ, κ (cid:48) ) (cid:17) . Hence, e − a e (cid:48) θ ( κ (cid:48) ) = e (cid:48) θ κ + e (cid:48) θ d/ (cid:48) f + e (cid:48) θ (Err( κ, κ (cid:48) ) + e (cid:48) θ ( a ) (cid:16) κ + d/ (cid:48) f + Err( κ, κ (cid:48) ) (cid:17) and thus e (cid:48) θ ( κ (cid:48) ) = e θ κ + e (cid:48) θ ( a ) κ + e (cid:48) θ d/ (cid:48) f + 12 f e κ + 12 f e κ + Err [ e θ ( κ ) , e (cid:48) θ ( κ (cid:48) )]with error term,Err [ e θ ( κ ) , e (cid:48) θ ( κ (cid:48) )] = ( e a − (cid:16) e θ κ + e (cid:48) θ d/ (cid:48) f + 12 f e κ + 12 f e κ (cid:17) + e a (cid:20) e (cid:48) θ (Err( κ, κ (cid:48) ) + e (cid:48) θ ( a ) (cid:16) d/ (cid:48) f + Err( κ, κ (cid:48) ) (cid:17) + 12 f f e θ κ + 18 f f e κ (cid:21) . CHAPTER 9. GCM PROCEDURE
Now, making use of e (cid:48) θ κ = (cid:18) f f (cid:19) e θ κ + 12 f e κ + 12 f (cid:18) f f (cid:19) e κ = e θ κ + 12 f e κ + 12 f e κ + 12 f f e θ κ + 18 f f e κ and the null structure equations, e ( κ ) + 12 κ κ − ωκ = 2 d/ η + 2 ρ − ϑ ϑ + 2( ξξ + η ) ,e κ + 12 κ + 2 ωκ = 2 d/ ξ − ϑ + 2( η + η + 2 ζ ) ξ, we deduce, e (cid:48) θ κ = e θ κ + 12 f (cid:18) − κ − ωκ (cid:19) + 12 f (cid:18) − κ κ + 2 ωκ + 2 ρ (cid:19) + 12 f f e θ κ + 18 f f e κ + 12 f (cid:18) d/ ξ − ϑ + 2( η + η + 2 ζ ) ξ (cid:19) + 12 f (cid:18) d/ η − ϑ ϑ + 2( ξξ + η ) (cid:19) . Hence, e (cid:48) θ ( κ (cid:48) ) = e θ κ + e (cid:48) θ ( a ) κ + e (cid:48) θ d/ (cid:48) f + κe (cid:48) θ a − κ ( f κ + f κ ) + κ ( f ω − ωf ) + f ρ + Err( e (cid:48) θ κ (cid:48) , e θ κ )where,Err( e (cid:48) θ κ (cid:48) , e θ κ ) = Err ( e (cid:48) θ κ (cid:48) , e θ κ ) + 12 f (cid:18) d/ η − ϑ ϑ + 2( ξξ + η ) (cid:19) + 12 f f e θ κ + 18 f f e κ + 12 f (cid:18) d/ ξ − ϑ + 2( η + η + 2 ζ ) ξ (cid:19) as desired. The formula for e (cid:48) θ ( κ (cid:48) ) is easily derived by symmetry from the one on e (cid:48) θ ( κ (cid:48) ).Note however that a becomes − a in the transformation.Applying the operator e (cid:48) θ = (cid:0) f f (cid:1) e θ + f e + f (cid:0) f f (cid:1) e to the transformationformula for µ , µ (cid:48) = µ + ( d/ ) (cid:48) (cid:18) − ( d (cid:63) / ) (cid:48) a + f ω − f ω + 14 f κ − f κ (cid:19) + Err( µ, µ (cid:48) ) .3. FRAME TRANSFORMATIONS e (cid:48) θ ( µ (cid:48) ) = e (cid:48) θ ( µ ) + e (cid:48) θ ( d/ ) (cid:48) (cid:18) − ( d (cid:63) / ) (cid:48) a + f ω − f ω + 14 f κ − f κ (cid:19) + e (cid:48) θ Err( µ, µ (cid:48) )= e θ ( µ ) + 12 f e µ + 12 f e µ + e (cid:48) θ ( d/ ) (cid:48) (cid:18) − ( d (cid:63) / ) (cid:48) a + f ω − f ω + 14 f κ − f κ (cid:19) + e (cid:48) θ Err( µ, µ (cid:48) ) + 12 f f e θ µ + 18 f f e µ. Recalling that µ = − d/ ζ − ρ + ϑϑ we find,12 f e µ + 12 f e µ = −
12 ( f e + f e ) ρ + 12 f e (cid:18) − d/ ζ + 14 ϑϑ (cid:19) + 12 f e (cid:18) − d/ ζ + 14 ϑϑ (cid:19) . Recalling the Bianchi equations for e ρ, e ρe ρ + 32 κρ = d/ β − ϑ α + ζ β + 2( η β + ξ β ) ,e ρ + 32 κρ = d/ β − ϑ α − ζ β + 2( η β + ξ β ) , we further deduce,12 f e µ + 12 f e µ = 34 ρ ( f κ + f κ ) − f (cid:16) d/ β − ϑ α − ζ β + 2( η β + ξ β ) (cid:17) − f (cid:16) d/ β − ϑ α + ζ β + 2( η β + ξ β ) (cid:17) + 12 f e (cid:18) − d/ ζ + 14 ϑϑ (cid:19) + 12 f e (cid:18) − d/ ζ + 14 ϑϑ (cid:19) . Therefore, e (cid:48) θ ( µ (cid:48) ) = e θ ( µ ) + 34 ρ ( f κ + f κ ) + e (cid:48) θ ( d/ ) (cid:48) (cid:18) − ( d (cid:63) / ) (cid:48) a + f ω − f ω + 14 f κ − f κ (cid:19) + Err( e θ µ, e θ µ )with, Err( e (cid:48) θ µ (cid:48) , e θ µ ) = e (cid:48) θ Err( µ, µ (cid:48) ) + 12 f f e θ µ + 18 f f e µ − f (cid:16) d/ β − ϑ α − ζ β + 2( η β + ξ β ) (cid:17) − f (cid:16) d/ β − ϑ α + ζ β + 2( η β + ξ β ) (cid:17) + 12 f e (cid:18) − d/ ζ + 14 ϑϑ (cid:19) + 12 f e (cid:18) − d/ ζ + 14 ϑϑ (cid:19) CHAPTER 9. GCM PROCEDURE as desired.Finally recalling the definition of the Hodge operators d/ , d (cid:63) / , ( d/ ) (cid:48) , ( d (cid:63) / ) (cid:48) and noticing that( d (cid:63) / ) (cid:48) ( κ (cid:48) ) = ( d (cid:63) / ) (cid:48) (ˇ κ (cid:48) ) , ( d (cid:63) / )( κ ) = ( d (cid:63) / )(ˇ κ ) , ( d (cid:63) / ) (cid:48) ( κ (cid:48) ) = ( d (cid:63) / ) (cid:48) (ˇ κ (cid:48) ) , ( d (cid:63) / )( κ ) = ( d (cid:63) / )(ˇ κ ) , ( d (cid:63) / ) (cid:48) ( µ (cid:48) ) = ( d (cid:63) / ) (cid:48) (ˇ µ (cid:48) ) , ( d (cid:63) / )( µ ) = ( d (cid:63) / )(ˇ µ ) , ( d (cid:63) / ) (cid:48) ( µ (cid:48) ) = ( d (cid:63) / ) (cid:48) (ˇ µ (cid:48) ) , ( d (cid:63) / )( µ ) = ( d (cid:63) / )(ˇ µ ) , we recast the results of Lemma 9.3.4 in the following form. Lemma 9.3.5.
We have the transformation equations, ( d (cid:63) / ) (cid:48) (ˇ κ (cid:48) ) = d (cid:63) / (ˇ κ ) + ( d (cid:63) / ) (cid:48) ( d/ ) (cid:48) f + κ ( d (cid:63) / ) (cid:48) a − ρf + 14 κ ( f κ + f κ ) − κ ( f ω − f ω ) − Err , ( d (cid:63) / ) (cid:48) (ˇ κ (cid:48) ) = d (cid:63) / (ˇ κ ) + ( d (cid:63) / ) (cid:48) ( d/ ) (cid:48) f − κ ( d (cid:63) / ) (cid:48) a − ρf + 14 κ ( f κ + f κ ) − κ ( f ω − f ω ) − Err , ( d (cid:63) / ) (cid:48) (ˇ µ (cid:48) ) = d (cid:63) / (ˇ µ ) + ( d (cid:63) / ) (cid:48) ( d/ ) (cid:48) (cid:18) − ( d (cid:63) / ) (cid:48) a + f ω − f ω + 14 f κ − f κ (cid:19) − ρ (cid:0) f κ + f κ (cid:1) − Err , (9.3.13) where,Err = Err ( e (cid:48) θ κ (cid:48) , e θ κ ) = e (cid:48) θ Err ( κ, κ (cid:48) ) + ae (cid:48) θ d/ (cid:48) f + e (cid:48) θ ( a ) d/ (cid:48) f + a (cid:18) e θ κ + 12 (cid:0) f e κ + f e κ (cid:1)(cid:19) + f d/ η + f d/ ξ + l.o.t. , Err = Err ( e (cid:48) θ κ (cid:48) , e θ κ ) = e (cid:48) θ Err ( κ, κ (cid:48) ) − ae (cid:48) θ d/ (cid:48) f − e (cid:48) θ ( a ) d/ (cid:48) f − a (cid:18) e θ κ + 12 (cid:0) f e κ + f e κ (cid:1)(cid:19) + f d/ η + f d/ ξ + l.o.t. , Err = Err ( e (cid:48) θ µ (cid:48) , e θ µ ) = e (cid:48) θ Err ( µ, µ (cid:48) ) − (cid:0) f d/ β + f d/ β (cid:1) −
12 ( f e + f e ) d/ ζ + l.o.t. , (9.3.14) where the terms denoted by l.o.t. are cubic or higher order in a, f, f , ˇΓ , ˇ R and contain noderivatives of ( a, f, f ) . .3. FRAME TRANSFORMATIONS Given a deformation Ψ : ◦ S −→ S and adapted frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) with e (cid:48) θ = e S θ we derivean elliptic system for the transition parameters ( a, f, f ). The system will later be used inthe construction of GCM surfaces.In what follows we denote by d/ S , d/ S , d/ S ,(cid:63) , d/ S ,(cid:63) the basic Hodge operators on S . Notingthat the transformation formulae in (9.3.13)–(9.3.14) contain only the operators ( d/ ) (cid:48) = d/ S , ( d (cid:63) / ) (cid:48) = d/ S ,(cid:63) applied to a, f, f we introduce the simplified notation, d/ S := ( d/ ) (cid:48) , d/ S ,(cid:63) := ( d (cid:63) / ) (cid:48) , A S := d/ S ,(cid:63) d/ S , d (cid:63) / := d (cid:63) / . (9.3.15)With these notation (9.3.13) takes the following form, d/ S ,(cid:63) ˇ κ S = d (cid:63) / ˇ κ + A S f + κ d/ S ,(cid:63) a − ρf + 14 κ ( f κ + f κ ) − κ ( f ω − f ω ) − Err ,d/ S ,(cid:63) ˇ κ S = d (cid:63) / ˇ κ + A S f − κ d/ S ,(cid:63) a − ρf + 14 κ ( f κ + f κ ) − κ ( f ω − f ω ) − Err ,d/ S ,(cid:63) ˇ µ S = d (cid:63) / ˇ µ + A S (cid:18) − d/ S ,(cid:63) a + f ω − f ω + 14 f κ − f κ (cid:19) − ρ (cid:0) f κ + f κ (cid:1) − Err , or, A S (cid:18) − d/ S ,(cid:63) a + f ω − f ω + 14 f κ − f κ (cid:19) − ρ ( κf + κf ) = d/ S ,(cid:63) ˇ µ S − d (cid:63) / ˇ µ + Err ,A S f + κ d/ S ,(cid:63) a − ρf + 14 κ ( f κ + f κ ) − κ ( f ω − f ω ) = d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ + Err ,A S f − κ d/ S ,(cid:63) a − ρf + 14 κ ( f κ + f κ ) − κ ( f ω − f ω ) = d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ + Err . (9.3.16)Since A S is invertible we can write, setting z := κf + κf , d/ S ,(cid:63) a = f ω − f ω + 14 f κ − f κ −
34 ( A S ) − (cid:0) ρz (cid:1) + ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ − Err (cid:1) . We have (cid:82) S f A S f = (cid:82) S ( d/ S f ) which in view of the identity (2.1.22) for d/ S and the definition of d/ S implies that A S is invertible. CHAPTER 9. GCM PROCEDURE
We can thus eliminate d/ S ,(cid:63) a from the last two equations, A S f + (cid:18) κκ − ρ (cid:19) f − κ ( A S ) − (cid:0) ρz (cid:1) = d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ − κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) + Err ,A S f + (cid:18) κκ − ρ (cid:19) f + 34 κ ( A S ) − (cid:0) ρz (cid:1) = d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ + κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) + Err , where, Err = Err + κ ( A S ) − Err , Err = Err − κ ( A S ) − Err . Therefore the system (9.3.16) is equivalent to the system, A S f + ( 12 κκ − ρ ) f − κ ( A S ) − (cid:0) ρz (cid:1) = d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ − κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) + Err ,A S f + ( 12 κκ − ρ ) f + 34 κ ( A S ) − (cid:0) ρz (cid:1) = d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ + κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) + Err ,d/ S ,(cid:63) a + 34 ( A S ) − (cid:0) ρz (cid:1) − f ω + f ω − f κ + 14 f κ = ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) − ( A S ) − Err . Furthermore, we have A S z = A S ( κf + κf ) = κA S f + κA S f + [ A S , κ ] f + [ A S , κ ] f . In view of the above equations for A S f and A S f , we infer A S z = κ (cid:26) − (cid:18) κκ − ρ (cid:19) f + 34 κ ( A S ) − (cid:0) ρz (cid:1) + d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ − κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1)(cid:27) + κ (cid:26) − (cid:18) κκ − ρ (cid:19) f − κ ( A S ) − (cid:0) ρz (cid:1) + d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ + κ ( A S ) − d (cid:63) / (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1)(cid:27) +[ A S , κ ] f + [ A S , κ ] f + κ Err + κ Err . Furthermore A S z = − (cid:18) κκ − ρ (cid:19) z + κ (cid:110) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ − κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1)(cid:111) + κ (cid:110) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ + κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1)(cid:111) +[ A S , κ ] f + [ A S , κ ] f + κ Err + κ Err = − (cid:18) κκ − ρ (cid:19) z + κ d/ S ,(cid:63) ˇ κ S + κ d/ S ,(cid:63) ˇ κ S − κ d (cid:63) / ˇ κ − κ d (cid:63) / ˇ κ + κ Err + κ Err + [ A S , κ ] f + [ A S , κ ] f . .3. FRAME TRANSFORMATIONS Lemma 9.3.6.
The original system (9.3.13) in ( a, f, f ) associated to a deformation sphere S is equivalent to the following (cid:0) A S + V (cid:1) z = κ d/ S ,(cid:63) ˇ κ S + κ d/ S ,(cid:63) ˇ κ S − κ d (cid:63) / ˇ κ − κ d (cid:63) / ˇ κ + κ Err + κ Err + [ A S , κ ] f + [ A S , κ ] f , (cid:0) A S + V (cid:1) f = 34 κ ( A S ) − (cid:0) ρz (cid:1) + d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ − κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) + Err , (cid:0) A S + V (cid:1) f = − κ ( A S ) − (cid:0) ρz (cid:1) + d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ + κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) + Err ,d/ S ,(cid:63) a = −
34 ( A S ) − (cid:0) ρz (cid:1) + f ω − f ω + 14 f κ − f κ + ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) − ( A S ) − Err , (9.3.17) where, z := κf + κf , V := − κκ + ρ. (9.3.18) The error terms are given by Err , Err , Err , defined in Lemma 9.3.5, andErr = Err + κ ( A S ) − Err , Err = Err − κ ( A S ) − Err . (9.3.19) Remark 9.3.7.
We note the following remarks concerning the system (9.3.17) .1. The right hand side of the equations is linear in the quantities, d/ S ,(cid:63) ˇ κ S , d/ S ,(cid:63) ˇ κ S , d/ S ,(cid:63) ˇ µ S , as well as d (cid:63) / ˇ κ, d (cid:63) / ˇ κ, d (cid:63) / ˇ µ. The first group is to be constrained by our GCM conditions in the next section whilethe second group depends on assumptions regarding the background foliation of R .2. The error terms contain only S -angular derivatives of ( a, f, f ) of order at mostequal to the order of the corresponding operators on the left hand sides, see Lemma9.3.8 below. Thus the system is in a standard quasilinear elliptic system form.3. In order to uniquely solve the equations for z , and then f and f , we need to thecoercivity of the operator A S + V . One can easily show that the potential V is positivefor small values of r , i.e. r near r H = 2 m (1 + δ H ) but negative for large r . In fact CHAPTER 9. GCM PROCEDURE A S + V has a nontrivial kernel for large r as one can easily see from the followingcalculation. Since, A S = d (cid:63) / d/ = d/ S d (cid:63) / S + 2 K, K = − ρ − κκ + 14 ϑϑ we deduce, A S + V = A S + 12 κκ − ρ = d/ S d (cid:63) / S − ρ + 12 ϑϑ. Thus for large enough r the operator A S + V behaves like d/ S d/ S ,(cid:63) which has anontrivial kernel.4. To be able to correct for the lack of coercivity of the system we need to control the (cid:96) = 1 modes of f, f , d/ S ,(cid:63) a . In subsection 9.3.4 we derive an equation for the last.The (cid:96) = 1 modes of ( f, f ) , on the other hand, have to be prescribed.5. The equations do not provide information on the average of a . For this we will needyet another equation derived in subsection 9.3.2. Lemma 9.3.8.
The error terms Err , . . . , Err can be written schematically as follows, r Err = ( d / S ) (cid:0) ( f, f , a ) (cid:1) + d / S (cid:0) ( f, f , a )( r ˇΓ g ) (cid:1) ,r Err = r − ( d / S ) (cid:0) ( f, f , a ) (cid:1) + d / S (cid:0) ( f, f , a )(ˇΓ) (cid:1) ,r Err = ( d / S ) (cid:0) ( f, f , a ) (cid:1) + ( d / S ) (cid:0) ( f, f , a )( r ˇΓ g ) (cid:1) , Err , Err = Err + ( A S ) − Err , (9.3.20) where the lower order terms denoted l.o.t. are cubic with respect to a, f, f , ˇΓ , ˇ R and mayinvolve fewer angular (along S ) derivatives of a, f, f . Remark 9.3.9.
Note that Err behaves worse in powers of r than Err . The reason isthe presence of the terms f e θ ξ, e θ ( f ξ ) in the formula for e S θ ( Err ( κ (cid:48) , κ )) .Proof. Note that in the spacetime region R of interest r and r S are comparable. Recall,see (9.3.14),Err = Err( e (cid:48) θ κ (cid:48) , e θ κ ) = e (cid:48) θ Err( κ, κ (cid:48) ) + ae (cid:48) θ d/ (cid:48) f + e (cid:48) θ ( a ) d/ (cid:48) f + a (cid:18) e θ κ + 12 (cid:0) f e κ + f e κ (cid:1)(cid:19) + f d/ η + f d/ ξ + l.o.t. , Err = Err( e (cid:48) θ κ (cid:48) , e θ κ ) = e (cid:48) θ Err( κ, κ (cid:48) ) − ae (cid:48) θ d/ (cid:48) f − e (cid:48) θ ( a ) d/ (cid:48) f − a (cid:18) e θ κ + 12 (cid:0) f e κ + f e κ (cid:1)(cid:19) + f d/ η + f d/ ξ + l.o.t. , Err = Err( e (cid:48) θ µ (cid:48) , e θ µ ) = e (cid:48) θ Err( µ, µ (cid:48) ) − (cid:0) f d/ β + f d/ β (cid:1) −
12 ( f e + f e ) d/ ζ + l.o.t. , .3. FRAME TRANSFORMATIONS ,Err( κ, κ (cid:48) ) = 12 f f e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + f ( ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t. , Err( κ, κ (cid:48) ) = − f e (cid:48) θ (cid:18) f f + 18 f f (cid:19) + (cid:18) f f + 18 ( f f ) (cid:19) e (cid:48) θ ( f )+ 14 (cid:18) f f (cid:19) f e (cid:48) θ (cid:0) f f (cid:1) − f (cid:18) f f (cid:19) e (cid:48) θ (cid:0) f (cid:1) + f ( − ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t.Also, Err( µ, µ (cid:48) ) = − e (cid:48) θ Err( ζ, ζ (cid:48) ) − Err( ρ, ρ (cid:48) ) + 14 (cid:0) ϑ (cid:48) ϑ (cid:48) − ϑϑ (cid:1) , Err( ζ, ζ (cid:48) ) = 12 f (cid:18) f f (cid:19) e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + 14 ( − f ϑ + f ϑ ) + l.o.t. , Err( ρ, ρ (cid:48) ) = 32 ρf f + f β + f β + l.o.t.We write schematically ,Err = ( f, f , a )( r − d / S ) ( f, f , a ) + ( r − d / S ( f, f , a )) + r − d / S (cid:0) ( f, f , a )ˇΓ g (cid:1) + r − d / S ( f ) + 12 a (cid:0) f e κ + f e κ (cid:1) + l.o.t.Making use of e ( κ ) + 12 κ κ − ωκ = 2 d/ η + 2 ρ − ϑ ϑ + 2( ξξ + η ) ,e κ + 12 κ + 2 ωκ = 2 d/ ξ − ϑ + 2( η + η + 2 ζ ) ξ, and treating the curvature terms that appear as ˇΓ g we easily derive, r Err = ( d / S ) (cid:0) ( f, f , a ) (cid:1) + d / S (cid:0) ( f, f , a )( r ˇΓ g ) (cid:1) . We obtain a worse estimate for Err because of the presence e (cid:48) θ ( f ξ ), since ξ ∈ ˇΓ b . In fact, r Err = r − ( d / S ) (cid:0) ( f, f , a ) (cid:1) + d / S (cid:0) ( f, f , a )ˇΓ (cid:1) . Recall also the outgoing geodesic conditions i.e. ξ = 0, ζ + η = 0, ζ − η = 0, ω = 0. The last term r − d / S ( f ) on the right of the identity below is due to the term e (cid:48) θ ( f ω ) in the expressionof e (cid:48) θ Err( κ, κ (cid:48) ). CHAPTER 9. GCM PROCEDURE
For Err we write similarly, treating the curvature terms that appear as ˇΓ g , e θ ( µ, µ (cid:48) ) = r − ( d / S ) (cid:0) ( f, f , a ) (cid:1) + r − ( d / S ) (cid:0) ( f, f , a )ˇΓ g (cid:1) + l.o.t.Using the null structure equations for ζ we infer that,Err = e θ ( µ, µ (cid:48) ) − (cid:0) f d/ β + f d/ β (cid:1) −
12 ( f e + f e ) d/ ζ + l.o.t.= r − ( d / S ) (cid:0) ( f, f , a ) (cid:1) + r − ( d / S ) (cid:0) ( f, f , a )ˇΓ g (cid:1) + l.o.t.as stated.Making use of the above lemma and the assumptions A1-A3 we can derive the following.
Lemma 9.3.10.
Assume given a deformation
Ψ : ◦ S −→ S in R and adapted frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) with e (cid:48) θ = e S θ with transition parameters a, f, f defined on S . Assume that thefollowing holds true (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) L ∞ ( ◦ S ) + max ≤ s ≤ s max ( ◦ r ) − (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:46) ◦ (cid:15). Then, for s = s max + 1 , with s max ≥ , (cid:107) Err , Err (cid:107) h s − ( S ) (cid:46) r − (cid:13)(cid:13)(cid:13) ( f, f , a ) (cid:13)(cid:13)(cid:13) h s ( S ) (cid:18) ◦ (cid:15) + (cid:13)(cid:13)(cid:13) f, f , a (cid:13)(cid:13)(cid:13) h s ( S ) (cid:19) , (cid:107) Err (cid:107) h s − ( S ) (cid:46) r − (cid:13)(cid:13)(cid:13) ( f, f , a ) (cid:13)(cid:13)(cid:13) h s ( S ) (cid:18) ◦ (cid:15) + (cid:13)(cid:13)(cid:13) f, f , a (cid:13)(cid:13)(cid:13) h s ( S ) (cid:19) , (cid:107) Err , Err (cid:107) h s − ( S ) (cid:46) r − (cid:13)(cid:13)(cid:13) ( f, f , a ) (cid:13)(cid:13)(cid:13) h s ( S ) (cid:18) ◦ (cid:15) + (cid:13)(cid:13)(cid:13) f, f , a (cid:13)(cid:13)(cid:13) h s ( S ) (cid:19) . (9.3.21) Proof.
The proof follows easily from Lemma 9.3.8, Corollary 9.2.7, coercivity of A S andobvious product estimates on S . Consider for example the termErr = r − ( d / S ) (cid:0) ( f, f , a ) (cid:1) + r − d / S (cid:0) ( f, f , a )(ˇΓ) (cid:1) . We write,( d / S ) k Err = r − ( d / S ) k (cid:0) ( f, f , a ) (cid:1) + r − ( d / S ) k (cid:0) ( f, f , a )(ˇΓ) (cid:1) + l.o.t.and ( d / S ) k (cid:0) ( f, f , a ) (cid:1) = (cid:88) i + j = k +2 d / i ( f, f , a ) · d / j ( f, f , a ) . This is assumption (9.2.14) of Lemma 9.2.6 with ◦ δ replaced to ◦ (cid:15) . .3. FRAME TRANSFORMATIONS i ≥ [ k +22 ] and i < [ k +22 ] and using Sobolevestimates for the terms involving fewer derivatives we derive, for [ k +22 ] + 2 ≤ k + 2, (cid:107) ( d / S ) k (cid:0) ( f, f , a ) (cid:1) (cid:107) L ( S ) (cid:46) r − (cid:107) ( a, f, f ) (cid:107) h k +2 ( S ) . Similarly, making use of our assumptions for ˇΓ,( d / S ) k (cid:0) ( f, f , a )(ˇΓ) (cid:1) (cid:46) r − (cid:107) ( a, f, f ) (cid:107) h k +1 ( S ) (cid:107) ˇΓ (cid:107) h k +1 ( S ) (cid:46) ◦ (cid:15)r − (cid:107) ( a, f, f ) (cid:107) h k +1 ( S ) . Thus, for all 2 ≤ k ≤ s max − (cid:107) ( d / S ) k Err (cid:107) L ( S ) (cid:46) r − (cid:107) ( f, f , a ) (cid:107) h k +2 ( S ) (cid:18) (cid:15) + (cid:13)(cid:13)(cid:13) ( f, f , a ) (cid:13)(cid:13)(cid:13) h k +2 ( S ) (cid:19) i.e., for 4 ≤ s ≤ s max + 1, (cid:107) Err (cid:107) h s − ( S ) (cid:46) r − (cid:13)(cid:13)(cid:13) ( f, f , a ) (cid:13)(cid:13)(cid:13) h s ( S ) (cid:18) ◦ (cid:15) + (cid:13)(cid:13)(cid:13) f, f , a (cid:13)(cid:13)(cid:13) h s ( S ) (cid:19) . All other terms can be treated similarly. a In the proof of existence and uniqueness of GCMS, see Theorem 9.4.2 we will need, inaddition of the equations derived so far, an equation for the average of a . To achieve thiswe make use of the transformation formula for κ of Lemma 9.3.2 κ (cid:48) = e a ( κ + d/ (cid:48) f ) + e a Err( κ, κ (cid:48) ) , Err( κ, κ (cid:48) ) = f ( ζ + η ) + f ξ + 12 f e f + 12 f e f + 14 f f κ + f f ω − ωf + l.o.t.which we rewrite in the form, κ S = e a (cid:18) r + ˇ κ + (cid:18) κ − r (cid:19) + d/ S f + Err( κ, κ (cid:48) ) (cid:19) . We deduce, ( e a −
1) 2 r = κ S − r − e a (cid:18) ˇ κ + κ − r + d/ S f (cid:19) − e a Err( κ, κ (cid:48) )= κ S − r S + (cid:18) r S − r (cid:19) − (cid:18) ˇ κ + κ − r + d/ S f (cid:19) − e a Err( κ, κ (cid:48) ) − ( e a − (cid:18) ˇ κ + κ − r + d/ S f (cid:19) CHAPTER 9. GCM PROCEDURE or, a r = κ S − r S + (cid:18) r S − r (cid:19) − (cid:18) ˇ κ + κ − r + d/ S f (cid:19) − e a Err( κ, κ (cid:48) ) − ( e a − (cid:18) ˇ κ + κ − r + d/ S f (cid:19) − ( e a − − a ) 2 r . We deduce, a = r S (cid:18) κ S − r S (cid:19) + (cid:18) − r S r (cid:19) − r S (cid:18) ˇ κ + κ − r + d/ S f (cid:19) + Err Err = − r S (cid:20) e a Err( κ, κ (cid:48) ) − ( e a − (cid:18) ˇ κ + κ − r + d/ S f (cid:19) − ( e a − − a ) 2 r (cid:21) − a (cid:18) r S r − (cid:19) . Taking the average on S we infer that, a S = r S (cid:18) κ SS − r S (cid:19) + (cid:18) − r S r (cid:19) S − r S (cid:18) ˇ κ + κ − r (cid:19) S + Err S (9.3.22)where h S denotes the average of h on S . Lemma 9.3.11.
Assume given a deformed sphere S ⊂ R with adapted null frame e S , e S , e S θ and transition functions ( a, f, f ) . We can extend a, f, f , and thus the frame e S , e S , e S θ ,in a small neighborhood of S such that the following hold true ξ S = 0 , ω S = 0 , η S + ζ S = 0 . (9.3.23) .3. FRAME TRANSFORMATIONS Proof.
According to Proposition 9.3.1 we have, ξ S = e a (cid:18) ξ + 12 e − a e S ( f ) + 14 f κ + f ω (cid:19) + e a Err( ξ, ξ S ) ,ξ S = e − a (cid:18) ξ + 12 e a e S ( f ) + 14 f κ + f ω (cid:19) + e − a Err( ξ, ξ S ) ,ζ S = ζ − e S θ ( a ) − f ω + f ω − f κ + 14 f κ + Err( ζ, ζ S ) ,η S = η + 12 e a e S f − f ω + 14 f κ + Err( η, η S ) ,η S = η + 12 e − a e S f − f ω + 14 f κ + Err( η, η S ) ,ω S = e a (cid:18) ω − e − a e a (cid:19) + e a Err( ω, ω S ) ,ω S = e − a (cid:18) ω + 12 e a e a (cid:19) + e − a Err( ω, ω S ) . Clearly the conditions ξ S = 0 , ω S = 0 allows us to determine e S f and e S a on S while thecondition η S + ζ S = 0 allows us to determine e S f on S . Remark 9.3.12.
Note that the equations above also allow us to impose, in addition,vanishing conditions on ξ and ω along S . Indeed these are determined by e S f and e S a . (cid:96) = 1 mode of d/ S ,(cid:63) a Recall the following change of frame formulas e (cid:48) θ ( κ (cid:48) ) = e θ κ + e (cid:48) θ d/ (cid:48) f + κe (cid:48) θ a − κ ( f κ + f κ ) + κ ( f ω − ωf ) + f ρ + Err( e (cid:48) θ κ (cid:48) , e θ κ ) . where e (cid:48) θ = − d/ S ,(cid:63) , d/ (cid:48) = d/ S , κ (cid:48) = κ S , β (cid:48) = β S . Making use of d (cid:63) / (cid:48) d/ (cid:48) = d/ (cid:48) d (cid:63) / (cid:48) + 2 K S , K = − ρ − κκ + 14 ϑϑ, we deduce, e (cid:48) θ ( κ (cid:48) ) = e θ κ − ( d/ ) (cid:48) ( d (cid:63) / ) (cid:48) f + κe (cid:48) θ ( a ) − κ (cid:18) κ + ω (cid:19) f + (cid:18) − κκ + κω + ρ − K (cid:19) f + Err( e (cid:48) θ κ (cid:48) , e θ κ ) − K S − K ) f = e θ κ − ( d/ ) (cid:48) ( d (cid:63) / ) (cid:48) f + κe (cid:48) θ ( a ) − κ (cid:18) κ + ω (cid:19) f + (cid:18) κκ + κω + 3 ρ (cid:19) f + Err( e (cid:48) θ κ (cid:48) , e θ κ ) − K S − K ) f − ϑϑf, CHAPTER 9. GCM PROCEDURE or, in view of the condition κ (cid:48) = κ S = r S we have e (cid:48) θ ( κ (cid:48) ) = − d/ S ,(cid:63) ( κ S ) = 0, e (cid:48) θ ( κ (cid:48) ) = e θ κ − ( d/ ) (cid:48) ( d (cid:63) / ) (cid:48) f + 2 r (cid:48) e (cid:48) θ ( a ) − κ (cid:18) κ + ω (cid:19) f + (cid:18) κκ + κω + 3 ρ (cid:19) f + Err( e (cid:48) θ κ (cid:48) , e θ κ ) + ( κ − r (cid:48) ) e (cid:48) θ ( a ) − K S − K ) f − ϑϑf. Multiplying by e Φ and integrating on S we derive, (cid:90) S e (cid:48) θ ( κ (cid:48) ) e Φ = (cid:90) S e θ ( κ ) e Φ + 2 r S (cid:90) S e S θ ( a ) e Φ − (cid:90) S κ (cid:18) κ + ω (cid:19) f e Φ + (cid:90) S (cid:18) κκ + κω + 3 ρ (cid:19) f e Φ + (cid:90) S (cid:18) Err( e S θ κ S , e θ κ ) − K S − K ) f − ( κ − r S ) d/ S ,(cid:63) a − ϑϑf (cid:19) e Φ . Therefore (recall that ω = 0),2 r S (cid:90) S e S θ ( a ) e Φ = (cid:90) S e S θ ( κ S ) e Φ − (cid:90) S e θ ( κ ) e Φ + 14 (cid:90) S κ f e Φ − (cid:90) S (cid:18) κκ + κω + 3 ρ (cid:19) f e Φ + Err , Err = (cid:90) S (cid:18) − Err( e S θ κ S , e θ κ ) + 2( K S − K ) f + ( κ − κ S ) d/ S ,(cid:63) a + 12 ϑϑf (cid:19) e Φ . We summarize the result in the following lemma.
Lemma 9.3.13.
We have, r S (cid:90) S e S θ ( a ) e Φ = (cid:90) S e S θ ( κ S ) e Φ − (cid:90) S e θ ( κ ) e Φ + 14 (cid:90) S κ f e Φ − (cid:90) S (cid:18) κκ + κω + 3 ρ (cid:19) f e Φ + Err , (9.3.24) where,Err = (cid:90) S (cid:18) − Err ( e S θ κ S , e θ κ ) + 2( K S − K ) f + (cid:18) κ − r S (cid:19) d/ S ,(cid:63) a + 12 ϑϑf (cid:19) e Φ . We now impose the GCM conditions on the deformed sphere S d/ S ,(cid:63) d/ S ,(cid:63) κ S = d/ S ,(cid:63) d/ S ,(cid:63) µ S = 0 , κ S = 2 r S , (cid:90) S f e Φ = Λ , (cid:90) S f e Φ = Λ , (9.4.1) .4. EXISTENCE OF GCM SPHERES f, f ) belong to the triplet ( f, f , λ = e a ) which denote the change of frame coef-ficients from the frame of ◦ S to the one of S . We show that under these conditions thedeformation parameters a, f, f verify a coercive elliptic system.We start with the following simple adaptation of Lemmas 9.3.17, 9.3.13 and the result ofsubsection 9.3.2. Proposition 9.4.1.
Assume that the deformed sphere S verifies the GCMS conditions (9.4.1) . Then the deformation parameters a, f, f verify the system (recall that z = κf + κf and V = κκ − ρ ) (cid:0) A S + V (cid:1) z = κ d/ S ,(cid:63) ˇ κ S − κ d (cid:63) / ˇ κ − κ d (cid:63) / ˇ κ + κ Err + κ Err + [ A S , κ ] f + [ A S , κ ] f , (cid:0) A S + V (cid:1) f = 34 κ ( A S ) − (cid:0) ρz (cid:1) − d (cid:63) / ˇ κ − κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) + Err , (cid:0) A S + V (cid:1) f = − κ ( A S ) − (cid:0) ρz (cid:1) + d/ S ,(cid:63) ˇ κ S + d (cid:63) / ˇ κ − κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) + Err ,d/ S ,(cid:63) a = −
34 ( A S ) − (cid:0) ρz (cid:1) + f ω + 14 f κ − f κ + ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) − ( A S ) − Err ,a S = (cid:18) − r S r (cid:19) S − r S (cid:18) ˇ κ + κ − r (cid:19) S + Err S , (9.4.2) and, r S (cid:90) S e S θ ( a ) e Φ = (cid:90) S e S θ ( κ S ) e Φ − (cid:90) S e θ ( κ ) e Φ + 14 (cid:90) S κ f e Φ − (cid:90) S (cid:18) κκ + κω + 3 ρ (cid:19) f e Φ + Err , (9.4.3) (cid:90) S e Φ f = Λ , (cid:90) S e Φ f = Λ . (9.4.4) The error terms are given by Err , Err , Err , defined in Lemma 9.3.5, andErr = Err + κ ( A S ) − Err , Err = Err − κ ( A S ) − Err , CHAPTER 9. GCM PROCEDURE
Err = − r S (cid:20) e a Err ( κ, κ (cid:48) ) − ( e a − (cid:18) ˇ κ + κ − r + d/ S f (cid:19) − ( e a − − a ) 2 r (cid:21) − a (cid:18) r S r − (cid:19) , Err = (cid:90) S (cid:18) − Err ( e S θ κ S , e θ κ ) + 2( K S − K ) f + (cid:18) r − r S (cid:19) d/ S ,(cid:63) a + 12 ϑϑf (cid:19) e Φ . We are ready to state the first main result of this chapter.
Theorem 9.4.2 (Existence of GCM spheres) . Let ◦ S = S ( ◦ u, ◦ s ) be a fixed sphere of the ( u, s ) outgoing geodesic foliation of a fixed spacetime region R . Assume in addition to A1 – A3 that the following estimates hold true on R , for all k ≤ s max , (cid:12)(cid:12) d k − d (cid:63) / d (cid:63) / κ (cid:12)(cid:12) (cid:46) ◦ δr − , (cid:12)(cid:12) d k − d (cid:63) / d (cid:63) / µ (cid:12)(cid:12) (cid:46) ◦ δr − , (cid:12)(cid:12) d k ˇ κ (cid:12)(cid:12) (cid:46) ◦ δr − , (cid:12)(cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ + 2 (cid:0) − mr (cid:1) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δr − . (9.4.5) For any fix Λ , Λ ∈ R verifying, | Λ | , | Λ | (cid:46) ◦ δr (9.4.6) there exists a unique GCM sphere S = S (Λ , Λ) , which is a deformation of ◦ S , such that theGCMS conditions (9.4.1) are verified. Moreover the following estimates hold true.1. We have (cid:12)(cid:12)(cid:12)(cid:12) r S ◦ r − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − ◦ δ. (9.4.7) In particular r, ◦ r and r S are all comparable in R .2. The unique functions ( λ, f, f ) on S , which relate the original frame e , e , e θ to thenew frame on e S , e S , e S θ according to (9.2.18) , verify the estimates (cid:13)(cid:13)(cid:13) f, f , log λ (cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) ◦ δ, k ≤ s max + 1 . (9.4.8) .4. EXISTENCE OF GCM SPHERES
3. The parameters
U, S of the deformation, see Definition 9.2.1, verify the estimate (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) L ∞ ( ◦ S ) + max ≤ s ≤ s max r − (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:46) ◦ δ. (9.4.9)
4. The Hawking mass m S verifies the estimate, (cid:12)(cid:12) m S − ◦ m (cid:12)(cid:12) (cid:46) ◦ δ. (9.4.10)
5. The curvature components ( α S , β S , ρ S , β S , α S ) , as well as µ S and the Ricci coeffi-cients ( κ S , ϑ S , ζ S , κ S , ϑ S ) on S , verify, for all k ≤ s max , (cid:107) ˇ κ S , ϑ S , ζ S , ˇ κ S (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) ϑ S (cid:107) h k ( S ) (cid:46) ◦ (cid:15), (cid:107) α S , β S , ˇ ρ S , µ S (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) β S (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) α S (cid:107) h k ( S ) (cid:46) ◦ (cid:15). (9.4.11)
6. The functions, ( λ, f, f ) uniquely defined above, can be smoothly extended to a smallneighborhood of S in such a way that the corresponding Ricci coefficients verify thefollowing transversality conditions ξ S = 0 , ω S = 0 , η S + ζ S = 0 . (9.4.12) In that case, the following estimates hold for all k ≤ s max − (cid:107) e ( f, f , log λ ) (cid:107) h k +1 ( S ) (cid:46) r − ◦ δ + r − ( | Λ | + | Λ | ) , (9.4.13) and, (cid:107) e S (ˇ κ S , ϑ S , ζ S , ˇ κ S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) e S ( ϑ S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) e S (cid:0) α S , β S , ˇ ρ S , µ S (cid:1) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) e S ( β S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) e S ( α S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − . (9.4.14) All other Ricci coefficients involve the transversal derivatives e S , e S of the frame. To be more precise one should replace r by ◦ r in the estimates below. Of course r and ◦ r are comparablein R , in particular on S . CHAPTER 9. GCM PROCEDURE
Remark 9.4.3.
In view of Propositions 9.2.9 and 9.4.1, to find a GCM sphere amountsto solve the following coupled system ς U (cid:48) = (cid:0) ( γ S ) (cid:1) / f (cid:18) f f ) (cid:19) ,S (cid:48) − ς U (cid:48) = 12 (cid:0) ( γ S ) (cid:1) / f , ( γ S ) = γ + ( ς ) (cid:18) Ω + 14 b γ (cid:19) ( ∂ θ U ) − ς ∂ θ U ∂ θ S − ( γςb ) ∂ θ U,U (0) = S (0) = 0 , (9.4.15) where the inputs ( a, f, f ) verifies (9.4.2) and (9.4.3) . Recall that for a reduced scalar h defined on S we write h ( ◦ u, ◦ s, θ ) = h ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) . We will solve the coupled system of equations (9.4.15) , (9.4.2) , (9.4.3) and (9.4.4) by aniteration argument which will be introduced below. Before doing this however it pays toobserve that the system (9.4.2) and (9.4.3) can be interpreted as an elliptic system on afixed surface S for ( a, f, f ) . In the next subsection we state a result which establishes thecoercivity of the system. The full proof of the theorem is detailed in the remaining part ofthis section, see subsection 9.4.1 –9.6.3. The following result plays the main role in the proof of Theorem 9.4.2.
Proposition 9.4.4 (Apriori Estimates for GCMS) . Let a fixed spacetime region R veri-fying assumptions A1 − A3 and (9.4.5) . Assume S is a given surface in R such that thearea radius of S verifies (cid:12)(cid:12)(cid:12)(cid:12) r S ◦ r − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − ◦ δ. (9.4.16) Then, for every Λ , Λ , | Λ | + | Λ | (cid:46) ◦ δ , there exists a unique solution ( λ, f, f ) of the system (9.4.2) - (9.4.3) verifying the estimate, (cid:107) (log λ, f, f ) (cid:107) h smax +1 ( S ) (cid:46) ◦ δ. (9.4.17) Recall also from (9.1.16) that r − ◦ r = O ( ◦ δ ◦ (cid:15) − / ). Thus r, ◦ r and r S are all comparable. .4. EXISTENCE OF GCM SPHERES More precisely, (cid:13)(cid:13)(cid:13) f, f , log λ (cid:13)(cid:13)(cid:13) h smax +1 ( S ) (cid:46) ◦ δ + | Λ | + | Λ | . (9.4.18)As an immediate corollary we derive the following rigidity result for GCM spheres. Corollary 9.4.5 (Rigidity I) . Let a fixed spacetime region R verifying assumptions A1 − A3 and (9.4.5) . Assume S is a sphere in R which verifies the the GCM condi-tions κ S = 2 r S , d (cid:63) / S d (cid:63) / S κ S = d (cid:63) / S d (cid:63) / S µ S = 0 . (9.4.19) Then the transition functions ( f, f , log λ ) from the background frame of R to to that of S verifies the estimates (cid:107) ( f, f , log( λ )) (cid:107) h smax +1 ( S ) (cid:46) ◦ δ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) . The proof of Proposition 9.4.4 will be given in the next section. The proof of Corollary9.4.5 is an immediate consequence of the proposition.We assume in addition that S is a deformation of ◦ S and prove the following corollary ofProposition 9.4.4. Corollary 9.4.6.
Let a fixed spacetime region R verifying assumptions A1 − A3 and (9.4.5) , and let Ψ : ◦ S −→ S be a fixed deformation in R given by the scalars U, S . Weassume that the deformation verifies, see (9.2.14) , (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) L ∞ ( ◦ S ) (cid:46) ◦ δ. (9.4.20) Then ( a, f, f ) verify the estimate (cid:107) ( a, f, f ) (cid:107) h smax +1 ( S ) (cid:46) ◦ δ. (9.4.21) Proof.
In view of Proposition 9.4.4, it suffices to prove that (9.4.16) holds. (9.4.16) actu-ally follows from the comparison Lemma 9.2.3.72
CHAPTER 9. GCM PROCEDURE
We establish the estimate (9.4.10) concerning the Hawking mass m S . Recall that, m S = r S (cid:18) π (cid:90) S κ S κ S (cid:19) , ◦ m = ◦ r (cid:18) π (cid:90) ◦ S ◦ κ ◦ κ (cid:19) . We write2 (cid:16) m S r S − ◦ m ◦ r (cid:17) = 116 π (cid:20)(cid:90) S (cid:0) κ S κ S − κκ (cid:1) + (cid:18)(cid:90) S κκ − (cid:90) ◦ S κκ (cid:19) − (cid:90) ◦ S (cid:16) κκ − ◦ κ ◦ κ (cid:17)(cid:21) = I + I + I . In view of Proposition 9.2.9 we have | r S − ◦ r | (cid:46) ◦ δ and (cid:12)(cid:12) γ S , − ◦ γ (cid:12)(cid:12) (cid:46) ◦ δ ◦ r . Making use ofCorollary 9.2.5 and the assumptions A1-A3 for κ, κ we deduce, (cid:12)(cid:12) I (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S κκ − (cid:90) ◦ S κκ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δr − . Similarly, taking into account the definition of R := (cid:26) | u − ◦ u | ≤ ◦ δ, | s − ◦ s | ≤ ◦ δ (cid:27) , (cid:12)(cid:12) I (cid:12)(cid:12) (cid:46) ◦ δr − . Finally, making also use of the transformation formula from the original frame ( e , e , e θ )to the frame ( e S , e S , e S θ ) of S κ S κ S = (cid:0) κ + d/ S f + Err( κ, κ S ) (cid:1) (cid:0) κ + d/ S f + Err( κ, κ S ) (cid:1) and the estimates for ( f, f , a = log λ ) we deduce, (cid:12)(cid:12) κ S κ S − κκ (cid:12)(cid:12) (cid:46) r − ◦ δ. Hence, (cid:12)(cid:12)(cid:12) I (cid:12)(cid:12)(cid:12) (cid:46) ◦ δr − . We infer that, (cid:12)(cid:12)(cid:12) m S r S − ◦ m ◦ r (cid:12)(cid:12)(cid:12) (cid:46) ◦ δr − from which the desired estimate (9.4.10) easily follows. .4. EXISTENCE OF GCM SPHERES We solve the coupled system of equations (9.4.15), (9.4.2) and (9.4.3) by an iterationargument as follows.Starting with the trivial quintet Q (0) := ( U (0) , S (0) , a (0) , f (0) , f (0) ) = (0 , , , , , corresponding to the undeformed sphere ◦ S , we define iteratively the quintet Q ( n +1) = ( U ( n +1) , S ( n +1) , a ( n +1) , f ( n +1) , f ( n +1) )from Q ( n ) = ( U ( n ) , S ( n ) , a ( n ) , f ( n ) , f ( n ) )as follows.1. The pair ( U ( n ) , S ( n ) ) defines the deformation sphere S ( n ) and the corresponding pullback map n given by the map Ψ ( n ) : ◦ S −→ S ( n ),( ◦ u, ◦ s, θ, ϕ ) −→ ( ◦ u + U ( n ) ( θ ) , ◦ s + S ( n ) ( θ ) , θ, ϕ ) . (9.4.22)By induction we assume that the following estimates hold true: (cid:13)(cid:13)(cid:13) ( a ( n ) , f ( n ) , f ( n ) ) (cid:13)(cid:13)(cid:13) h smax − ( S ( n )) (cid:46) ◦ δ, (9.4.23)and (cid:107) ∂ θ (cid:0) U ( n ) , S ( n ) (cid:1) (cid:107) L ∞ ( ◦ S ) + max ≤ s ≤ s max − r − (cid:107) ∂ θ (cid:0) U ( n ) , S ( n ) (cid:1) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:46) ◦ δ. (9.4.24)2. We then define the triplet ( a ( n +1) , f ( n +1) , f ( n +1) ) by solving the system on S ( n )74 CHAPTER 9. GCM PROCEDURE consisting of the equations (9.4.25), (9.4.28) and (9.4.29) below. (cid:0) A S ( n ) + V (cid:1) z ( n +1) = E ( n +1) ,E ( n +1) : = κ d/ S ( n ) ,(cid:63) ˇ κ S ( n ) − κ d (cid:63) / ˇ κ − κ d (cid:63) / ˇ κ + κ Err ( n +1)4 + κ Err ( n +1)5 + [ A S ( n ) , κ ] f ( n +1) + [ A S ( n ) , κ ] f ( n +1) , (cid:0) A S ( n ) + V (cid:1) f ( n +1) = 34 κ ( A S ( n ) ) − ( ρz ( n +1) ) + E ( n +1) ,E ( n +1) : = − d (cid:63) / ˇ κ − κ ( A S ( n ) ) − (cid:0) − d/ S ( n ) ,(cid:63) ˇ µ S ( n ) + d (cid:63) / ˇ µ (cid:1) + Err ( n +1)4 , (cid:0) A S ( n ) + V (cid:1) f ( n +1) = − κ ( A S ( n )) ) − ( ρz ( n +1) ) + E ( n +1) ,E ( n +1) : = d/ S ( n ) ,(cid:63) ˇ κ S ( n ) − d (cid:63) / ˇ κ + κ ( A S ( n ) ) − (cid:0) − d/ S ( n ) ,(cid:63) ˇ µ ( S ( n ) + d (cid:63) / ˇ µ (cid:1) + Err ( n +1)5 ,d (cid:63) / S ( n ) a ( n +1) = −
34 ( A S ( n ) ) − (cid:0) ρz ( n +1) (cid:1) + f ( n +1) ω + 14 f ( n +1) κ − f ( n +1) κ (cid:101) E ( n +1) , (cid:101) E ( n +1) : = ( A S ( n ) ) − (cid:0) − d/ S ( n ) ,(cid:63) ˇ µ S ( n ) + d (cid:63) / ˇ µ (cid:1) − ( A S ( n ) ) − (cid:0) Err ( n +1)3 (cid:1) , (9.4.25)where, z ( n +1) = κf ( n +1) + κf ( n +1) , (9.4.26)and the error terms,Err ( n +1)1 , Err ( n +1)2 , Err ( n +1)3 , Err ( n +1)4 , Err ( n +1)5 , (9.4.27)are obtained from the error terms Err , Err , Err , Err , Err by setting ( a, f, f ) =( a ( n +1) , f ( n +1) , f ( n +1) ) and their derivatives by the corresponding ones on the sphere S ( n ).We also set,2 r S ( n ) (cid:90) S ( n ) e S ( n ) θ ( a ( n +1) ) e Φ = 14 (cid:90) S ( n ) κ f ( n +1) e Φ − (cid:90) S ( n ) (cid:18) κκ + κω + 3 ρ (cid:19) f ( n +1) e Φ − (cid:90) S ( n ) e θ ( κ ) e Φ + Err ( n +1)7 , (cid:90) S ( n ) e Φ f ( n +1) = Λ , (cid:90) S ( n ) e Φ f ( n +1) = Λ , (9.4.28) .4. EXISTENCE OF GCM SPHERES a ( n +1) S ( n ) = (cid:18) − r S ( n ) r (cid:19) S ( n ) − r S ( n ) (cid:18) ˇ κ + κ − r (cid:19) S ( n ) + Err ( n +1)6 S ( n ) , (9.4.29)where Err ( n +1)6 , Err ( n +1)7 are obtained from the error terms Err , Err as above in(9.4.27), by setting ( a, f, f ) = ( a ( n +1) , f ( n +1) , f ( n +1) ) and their derivatives by thecorresponding ones on the sphere S ( n ).3. In view of the induction hypothesis (9.4.24) we are thus in a position to applyProposition 9.4.4, more precisely its Corollary 9.4.6, to construct a unique solutionverifying the estimate, uniformly for n ∈ N , (cid:13)(cid:13)(cid:13) ( a ( n +1) , f ( n +1) , f ( n +1) ) (cid:13)(cid:13)(cid:13) h smax − ( S ( n )) (cid:46) ◦ δ. (9.4.30)4. We use the new pair ( f ( n +1) , f ( n +1) ) to solve the equations on ◦ S , ς n ∂ θ U (1+ n ) = ( γ ( n ) ) / ( f (1+ n ) ) n (cid:18) (cid:16) f (1+ n ) f (1+ n ) (cid:17) n (cid:19) ,∂ θ S (1+ n ) − ς n Ω n ∂ θ U (1+ n ) = 12 ( γ ( n ) ) / ( f (1+ n ) ) n ,γ ( n ) = γ n + (cid:0) ς n (cid:1) (cid:18) Ω + 14 b γ (cid:19) n ( ∂ θ U ( n ) ) − ς n ∂ θ U ( n ) ∂ θ S ( n ) − (cid:16) γςb (cid:17) n ∂ θ U ( n ) ,U (1+ n ) (0) = S (1+ n ) (0) = 0 , (9.4.31)where, we repeat, the pull back n is defined with respect to the mapΨ ( n ) ( ◦ u, ◦ s, θ ) = ( ◦ u + U ( n ) ( θ ) , ◦ s + S ( n ) ( θ ) , θ ) , and γ ( n ) := γ S ( n ) , n . The equation (9.4.31) admits a unique solution U (1+ n ) , S (1+ n ) , according to theproposition below. The new pair ( U ( n +1) , S ( n +1) ) defines the new polarized sphere S ( n + 1) and we can proceed with the next step of the iteration.76 CHAPTER 9. GCM PROCEDURE
Proposition 9.4.7.
The equation (9.4.31) admits a unique solution U (1+ n ) , S (1+ n ) veri-fying the estimate, (cid:107) ∂ θ (cid:0) U ( n +1) , S ( n +1) (cid:1) (cid:107) L ∞ ( ◦ S ) + r − (cid:107) ∂ θ (cid:0) U ( n +1) , S ( n +1) (cid:1) (cid:107) h smax − ( ◦ S, ◦ g/ ) (cid:46) ◦ δ uniformly for all n ∈ N .Proof. The existence and uniqueness part of the proposition is an immediate consequenceof the standard results for ODE’s.To prove the desired estimate, we use the equations for ( U (1+ n ) , S (1+ n ) ) and infer, for s = s max − (cid:107) ∂ θ U ( n +1) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:46) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( γ ( n ) ) / (cid:0) ς n (cid:1) − ( f (1+ n ) ) n (cid:18) (cid:16) f (1+ n ) f (1+ n ) (cid:17) n (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) . Together with the non sharp product estimate on ( ◦ S, ◦ g/ ), see Lemma 9.1.5, we infer that,for s = s max − (cid:107) ∂ θ U ( n +1) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:46) r − (cid:13)(cid:13)(cid:13) ( f (1+ n ) ) n , ( f (1+ n ) ) n (cid:13)(cid:13)(cid:13) h smax − ( ◦ S, ◦ g/ ) (cid:13)(cid:13)(cid:13)(cid:0) ς n (cid:1) − ( γ ( n ) ) / (cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) × (cid:18) (cid:13)(cid:13)(cid:13) ( f (1+ n ) ) n , ( f (1+ n ) ) n (cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) (cid:19) . In view of Lemma 9.2.6, Corollary 9.2.7, and (9.4.30), we deduce (cid:107) ∂ θ U ( n +1) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:46) ◦ δr − (cid:13)(cid:13)(cid:13)(cid:0) ς n (cid:1) − ( γ ( n ) ) / (cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) . We recall that, γ ( n ) = γ n + (cid:0) ς n (cid:1) (cid:0) Ω + b γ (cid:1) n ( ∂ θ U ( n ) ) − ς n ∂ θ U ( n ) − (cid:16) γςb (cid:17) n ∂ θ U ( n ) . In view of our assumptions on the Ricci coefficients and the non-sharp product estimatesof Lemma 9.1.5 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ς (cid:0) Ω + 14 b γ (cid:1)(cid:19) n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h smax − ( ◦ S, ◦ g/ ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) γb (cid:17) n (cid:13)(cid:13)(cid:13)(cid:13) h smax − ( ◦ S, ◦ g/ ) (cid:46) ◦ (cid:15)r .4. EXISTENCE OF GCM SPHERES (cid:13)(cid:13)(cid:13)(cid:0) ς n (cid:1) − γ ( n ) (cid:13)(cid:13)(cid:13) h smax − ( ◦ S, ◦ g/ ) (cid:46) (cid:13)(cid:13) γ n (cid:13)(cid:13) h smax − ( ◦ S, ◦ g/ ) + ◦ δr . Together with Lemma 9.2.6 and Corollary 9.2.7, we deduce (cid:13)(cid:13)(cid:13)(cid:0) ς n (cid:1) − γ ( n ) (cid:13)(cid:13)(cid:13) h smax − ( ◦ S, ◦ g/ ) (cid:46) r and therefore, (cid:107) ∂ θ U ( n +1) (cid:107) h smax − ( ◦ S, ◦ g/ ) (cid:46) ◦ δr. Proceeding in the same manner with equation ∂ θ S (1+ n ) − ς n Ω n ∂ θ U (1+ n ) = 12 ( γ ( n ) ) / ( f (1+ n ) ) n we infer that, r − (cid:107) ∂ θ U ( n +1) , ∂ θ S ( n +1) (cid:107) h smax − ( ◦ S, ◦ g/ ) (cid:46) ◦ δ. This, together with the Sobolev inequality, concludes the proof of Proposition 9.4.7.
To finish the proof of Theorem 9.4.2 it remains to prove convergence of the iterates.
Step 1.
In order to prove the convergence of the iterative scheme, we introduce thefollowing quintet P ( n ) P (0) = (0 , , , , , P ( n ) = (cid:16) U ( n ) , S ( n ) , ( a ( n ) ) n − , ( f ( n ) ) n − , ( f ( n ) ) n − (cid:17) , n ≥ . Since ( a ( n ) , f ( n ) , f ( n ) ) are defined on S ( n − ( n − is definedon ◦ S so that P ( n ) is a quintet of functions on ◦ S for any n and we may introduce thefollowing norms to compare the elements of the sequence (cid:107) P ( n ) (cid:107) k : = (cid:107) ∂ θ (cid:0) U ( n ) , S ( n ) (cid:1) (cid:107) L ∞ ( ◦ S ) + r − (cid:107) ∂ θ (cid:0) U ( n ) , S ( n ) (cid:1) (cid:107) h k − ( ◦ S ) + (cid:13)(cid:13)(cid:13)(cid:16) ( a ( n ) ) n − , ( f ( n ) ) n − , ( f ( n ) ) n − (cid:17)(cid:13)(cid:13)(cid:13) h k − ( ◦ S ) . (9.4.32)Here are the substeps needed to implement a convergence argument.78 CHAPTER 9. GCM PROCEDURE
1. The quintets P ( n ) are bounded with respect to the norm (9.4.32) for the choice k = s max .2. The quintets P ( n ) are contractive with respect to the norm (9.4.32) for the choice k = 3.The precise statements are given in the following propositions. Proposition 9.4.8.
We have, uniformly for all n ∈ N , (cid:107) P ( n ) (cid:107) s max (cid:46) ◦ δ. Proof.
The proof is an immediate consequence of Propositions 9.4.4, 9.4.7 and the esti-mate, (cid:13)(cid:13)(cid:13) (Ψ ( n − ) (cid:16) f ( n ) , f ( n ) , a ( n ) (cid:17)(cid:13)(cid:13)(cid:13) h smax − ( ◦ S ) (cid:46) (cid:13)(cid:13)(cid:13)(cid:16) f ( n ) , f ( n ) , a ( n ) (cid:17)(cid:13)(cid:13)(cid:13) h smax − ( S ( n − (9.4.33)which is a consequence of Lemma 9.2.6. Proposition 9.4.9.
We have, uniformly for all n ∈ N , the contraction estimate, (cid:107) P ( n +1) − P ( n ) (cid:107) (cid:46) ◦ δ (cid:107) P ( n ) − P ( n − (cid:107) . The proof of Proposition 9.4.9 is postponed to section 9.6.
Step 2.
In view of Proposition 9.4.9, we have (cid:107) P ( n +1) − P ( n ) (cid:107) (cid:46) ( ◦ δ ) n (cid:107) P (1) − P (0) (cid:107) which in view of Proposition 9.4.8 yields (cid:107) P ( n +1) − P ( n ) (cid:107) (cid:46) ( ◦ δ ) n +1 . Together with a simple interpolation argument on ◦ S and Proposition 9.4.8, we infer (cid:107) P ( n +1) − P ( n ) (cid:107) k (cid:46) ( ◦ δ ) ( smax − ksmax − ) n , ≤ k ≤ s max . We infer the existence of a quintet of functions P ( ∞ ) on ◦ S such that (cid:107) P ( ∞ ) (cid:107) s max (cid:46) ◦ δ (9.4.34) .4. EXISTENCE OF GCM SPHERES n → + ∞ (cid:107) P ( n ) − P ( ∞ ) (cid:107) s max − = 0 . (9.4.35)Also, we have P ( ∞ ) = (cid:16) U ( ∞ ) , S ( ∞ ) , a ( ∞ )0 , f ( ∞ )0 , f ( ∞ )0 (cid:17) , where all function are defined on ◦ S . The functions ( U ( ∞ ) , S ( ∞ ) ) defines a sphere S ( ∞ )and we introduce the mapΨ ( ∞ ) ( ◦ u, ◦ s, θ, ϕ ) = (cid:16) ◦ u + U ( ∞ ) ( θ ) , ◦ s + S ( ∞ ) ( θ ) , θ, ϕ (cid:17) so that Ψ ( ∞ ) is a map from ◦ S to S ( ∞ ). Then, let a ( ∞ ) = a ( ∞ )0 ◦ (Ψ ( ∞ ) ) − , f ( ∞ ) = f ( ∞ )0 ◦ (Ψ ( ∞ ) ) − , f ( ∞ ) = f ( ∞ )0 ◦ (Ψ ( ∞ ) ) − so that a ( ∞ ) , f ( ∞ ) , f ( ∞ ) are defined on S ( ∞ ) and a ( ∞ )0 = ( a ( ∞ ) ) ∞ , f ( ∞ )0 = ( f ( ∞ ) ) ∞ , f ( ∞ )0 = ( f ( ∞ ) ) ∞ . From these definitions, the above control of P ( ∞ ) and Lemma 9.2.6, we infer r − (cid:107) ( U ( ∞ ) , S ( ∞ ) ) (cid:107) h smax − ( ◦ S ) + (cid:107) ( a ( ∞ ) , f ( ∞ ) , f ( ∞ ) ) (cid:107) h smax − ( S ( ∞ )) (cid:46) ◦ δ. In particular, using the Sobolev embedding on ◦ S , we have (cid:107) ( U ( ∞ ) (cid:48) , S ( ∞ ) (cid:48) ) (cid:107) L ∞ ( ◦ S ) (cid:46) ◦ δ , andhence, in view of Corollary 9.4.6, we deduce (cid:107) ( a ( ∞ ) , f ( ∞ ) , f ( ∞ ) ) (cid:107) h smax +1 ( S ( ∞ )) (cid:46) ◦ δ. This estimate allows to argue as in Proposition 9.4.7 with ( U ( n +1) , S ( n +1) ) replaced by( U ( ∞ ) , S ( ∞ ) ) and s max − s max . We finally obtain r − (cid:107) ( U ( ∞ ) (cid:48) , S ( ∞ ) (cid:48) ) (cid:107) h smax ( ◦ S ) + (cid:107) ( a ( ∞ ) , f ( ∞ ) , f ( ∞ ) ) (cid:107) h smax +1 ( S ( ∞ )) (cid:46) ◦ δ. (9.4.36) Step 3.
We proceed to control the area radius r S ( ∞ ) and the Hawking mass m S ( ∞ ) of thesphere S ( ∞ ). First, note from (9.4.36) and the Sobolev embedding on ◦ S that we have (cid:107) ( U ( ∞ ) , S ( ∞ ) ) (cid:107) L ∞ ( ◦ S ) (cid:46) ◦ δ. (9.4.37)80 CHAPTER 9. GCM PROCEDURE
Together with Lemma 9.2.3, we infer that (cid:12)(cid:12)(cid:12)(cid:12) r S ( ∞ ) r − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. (9.4.38)Next, we denote by Γ S ( ∞ ) the connection coefficients of S ( ∞ ). We have in view of thetransformation formula from the original frame ( e , e , e θ ) to the frame ( e S ( ∞ )4 , e S ( ∞ )3 , e S ( ∞ ) θ )of S ( ∞ ) κ S ( ∞ ) κ S ( ∞ ) = (cid:0) κ + d/ S ( ∞ ) f ( ∞ ) + Err( κ, κ S ( ∞ ) ) (cid:1) (cid:16) κ + d/ S ( ∞ ) f ( ∞ ) + Err( κ, κ S ( ∞ ) ) (cid:17) . Together with the estimate (9.4.36) for f ( ∞ ) and f ( ∞ ) and the assumptions A1-A3 for ˇΓcorresponding to the original frame ( e , e , e θ ), we infer (cid:12)(cid:12) κ S ( ∞ ) κ S ( ∞ ) − κκ (cid:12)(cid:12) (cid:46) ◦ δr − . Recall that (see (9.4.5)) (cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12) (cid:46) ◦ δr − , (cid:12)(cid:12)(cid:12) κ + 2 (cid:0) − mr (cid:1) r (cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. Thus, since κ = κ + ˇ κ , κκ = − (cid:0) − mr (cid:1) r + 2 r ˇ κ + O ( ◦ δ ) r − . We deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ S ( ∞ ) κ S ( ∞ ) + 4 (cid:0) − mr (cid:1) r − r ˇ κ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δr − . Thus, in view of (9.4.38), (cid:90) S ( ∞ ) κ S ( ∞ ) κ S ( ∞ ) = − (cid:90) S ( ∞ ) (cid:0) − mr (cid:1) r + O ( ◦ δ ) r − . Making use of the definition of the Hawking mass m S ( ∞ ) = r S ( ∞ ) (cid:16) π (cid:82) S ( ∞ ) κ S ( ∞ ) κ S ( ∞ ) (cid:17) we easily deduce (cid:12)(cid:12)(cid:12) m S ( ∞ ) − m (cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. (9.4.39) Here, we also use the fact that, on S ( ∞ ) , we have | r − ◦ r | (cid:46) (cid:107) ( U ( ∞ ) , S ( ∞ ) ) (cid:107) L ∞ ( ◦ S ) (cid:46) ◦ δ. See also subsection 9.4.2. .5. PROOF OF PROPOSITION 9.4.4
Step 4.
We make use of Lemma 9.3.11 to extend ( a ∞ , f ∞ , f ∞ ) as well as the frame (cid:0) e S ( ∞ )3 , e S ( ∞ )4 , e S ( ∞ ) θ (cid:1) in a small neighborhood of S ( ∞ ) such that we have, ξ S ( ∞ ) = 0 , ω S ( ∞ ) = 0 , η S ( ∞ ) + ζ S ( ∞ ) = 0 , (9.4.40)and then provide estimates for the corresponding Ricci coefficients and curvature compo-nents ˇΓ S ( ∞ ) , ˇ R S ( ∞ ) . More precisely we make use of the assumption A1 , the estimates in(9.4.36) for ( a ∞ , f ∞ , f ∞ ), and the transformation formulae to derive the desired estimates(9.4.11) for s max derivative of the Ricci coefficients and curvature components of S ( ∞ ). Step 5.
Thanks to (9.4.35), we can pass to the limit in (9.4.25) (9.4.29). We deduce thatall equations recorded in Proposition 9.4.1 hold true and thus that all the desired GCMShold true.
Since all estimates below take place on S , we simplify our notation and denote the norms (cid:107) · (cid:107) h k ( S ) simply by (cid:107) · (cid:107) k in what follows . In the particular case k = 0 we also write (cid:107) · (cid:107) = (cid:107) · (cid:107) S . The sup norms (cid:107) · (cid:107) h ∞ k ( S ) , though rarely needed, will be denoted by | · | k .Since r and r S are comparable we will freely choose one or the other throughout the proof.We introduce the notation (recall that a = log λ ) (cid:107) F (cid:107) k = (cid:107) a (cid:107) k + (cid:107) f (cid:107) k + (cid:107) f (cid:107) k ,I ( f, f ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) ,I ( a, f, f ) = I ( f, f ) + r S (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( e θ a ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) . (9.5.1)We assume that a solution exists verifying the auxiliary bootstrap assumption (cid:107) ( a, f, f ) (cid:107) s max +1 (cid:46) (cid:113) ◦ (cid:15) (9.5.2)based on which we will prove the stronger estimate, (cid:107) ( a, f, f ) (cid:107) s max +1 (cid:46) ◦ δ. (9.5.3) To remind the reader about our convention we shall in fact alternately use both notations. This is not really needed, we only use it to simplify the exposition. CHAPTER 9. GCM PROCEDURE
Step 1.
We write the first equation in (9.4.2) in the form (cid:0) A S + V (cid:1) z = 2 r (cid:0) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:1) − r d (cid:63) / ˇ κ + Err( z ) , Err( z ) := κ Err + κ Err + [ A S , κ ] f + [ A S , κ ] f + (cid:18) κ − r (cid:19) (cid:0) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:1) + (cid:18) κ + 2Υ r (cid:19) d (cid:63) / ˇ κ. Thus, (cid:13)(cid:13)(cid:0) A S + V (cid:1) z (cid:13)(cid:13) k (cid:46) r − (cid:13)(cid:13) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:13)(cid:13) k + r − (cid:107) d (cid:63) / ˇ κ (cid:107) k + (cid:107) Err( z ) (cid:107) k . In view of Lemma 9.3.10 and the auxiliary bootstrap assumption (9.5.2) we have , for k ≤ s max − (cid:107) Err , Err (cid:107) h k ( S ) (cid:46) r − (cid:13)(cid:13)(cid:13) f, f , a (cid:13)(cid:13)(cid:13) h k +2 ( S ) (cid:18) ◦ (cid:15) + (cid:13)(cid:13)(cid:13) f, f , a (cid:13)(cid:13)(cid:13) h k +2 ( S ) (cid:19) (cid:46) r − (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +2 . Also, (cid:13)(cid:13) [ A S , κ ] f + [ A S , κ ] f (cid:13)(cid:13) h k ( S ) (cid:46) r − (cid:113) ◦ (cid:15) (cid:13)(cid:13) ( f, f ) (cid:13)(cid:13) h k +1 ( S ) . In view of (9.4.5) we have (cid:13)(cid:13) ˇ κ, κ − r (cid:13)(cid:13) h k ( S ) (cid:46) ◦ δr − and (cid:107) d (cid:63) / ˇ κ (cid:107) h k − ( S ) (cid:46) ◦ δr − for all k ≤ s max ,and thus (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) κ − r (cid:19) (cid:0) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:1)(cid:13)(cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) ◦ δr − (cid:13)(cid:13) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:13)(cid:13) h k ( S ) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) κ + 2Υ r (cid:19) d (cid:63) / ˇ κ (cid:13)(cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) ◦ δ ◦ (cid:15)r − , i.e., (cid:107) Err( z ) (cid:107) k (cid:46) r − (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +2 + ◦ δr − (cid:13)(cid:13) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:13)(cid:13) h k ( S ) + ◦ δ ◦ (cid:15)r − . Hence, (cid:13)(cid:13)(cid:0) A S + V (cid:1) z (cid:13)(cid:13) k (cid:46) r − (cid:13)(cid:13) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:13)(cid:13) k + r − ◦ δ + r − (cid:113) ◦ (cid:15) (cid:13)(cid:13) F (cid:13)(cid:13) k +2 . (9.5.4) We also assume tacitly, throughout the estimates below, that k ≥ s max / (cid:107) · (cid:107) k . .5. PROOF OF PROPOSITION 9.4.4 (cid:13)(cid:13) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:13)(cid:13) k we proceed as follows. Note, using the GCMS conditions,that d/ S ,(cid:63) (cid:16) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:17) = − d/ S ,(cid:63) d (cid:63) / ˇ κ = − d (cid:63) / d (cid:63) / ˇ κ − ( d/ S ,(cid:63) − d (cid:63) / ) d (cid:63) / ˇ κ. According to the first inequality in (9.4.5) (cid:107) d (cid:63) / d (cid:63) / κ (cid:107) h k ( S ) (cid:46) ◦ δr − . Hence, (cid:13)(cid:13)(cid:13) d/ S ,(cid:63) (cid:16) d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ (cid:17)(cid:13)(cid:13)(cid:13) k (cid:46) (cid:107) d (cid:63) / d (cid:63) / ˇ κ (cid:107) k + (cid:107) ( d/ S ,(cid:63) − d (cid:63) / ) d (cid:63) / ˇ κ (cid:107) k (cid:46) ◦ δr − + (cid:107) ( d/ S ,(cid:63) − d (cid:63) / ) d (cid:63) / ˇ κ (cid:107) k . Recalling the transformation formulas (9.2.18) we have schematically, d/ S ,(cid:63) − (cid:18) f f (cid:19) d (cid:63) / = 12 f e + 12 f (cid:18) f f (cid:19) e + l.o.t.Therefore, (cid:107) ( d/ S ,(cid:63) − d (cid:63) / ) d (cid:63) / ˇ κ (cid:107) k (cid:46) r − (cid:113) ◦ (cid:15) (cid:107) ( f, f ) (cid:107) k and thus setting B := d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ, (9.5.5)we have (cid:13)(cid:13)(cid:13) d/ S ,(cid:63) B (cid:13)(cid:13)(cid:13) k (cid:46) r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) ( f, f ) (cid:107) k (cid:19) . (9.5.6)According to the elliptic estimate given by Lemma 9.1.4 we have, (cid:107) B (cid:107) k (cid:46) r (cid:107) d (cid:63) / B (cid:107) k + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Be Φ (cid:12)(cid:12)(cid:12)(cid:12) i.e., (cid:107) B (cid:107) k (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Be Φ (cid:12)(cid:12)(cid:12)(cid:12) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) ( f, f ) (cid:107) k (cid:19) . (9.5.7)To estimate the (cid:96) = 1 mode of B , i.e. its projection on the kernel of d (cid:63) / , we recall thetransformation formula, see Lemma 9.3.4, e (cid:48) θ ( κ (cid:48) ) = e θ κ + e (cid:48) θ d/ (cid:48) f − κe (cid:48) θ a − κ ( f κ + f κ ) − κ ωf + f ρ + Err( e (cid:48) θ κ (cid:48) , e θ κ )84 CHAPTER 9. GCM PROCEDURE i.e., B = d/ S ,(cid:63) d/ S f + κ d/ S f + (cid:18) κ − κ ω (cid:19) f + (cid:18) κκ − ρ (cid:19) f − Err( e (cid:48) θ κ (cid:48) , e θ κ )= A S f + κ d/ S f + c S f + c S f + Err( B )where, c S = (cid:18) Υ S r S (cid:19) − m S ( r S ) , c S = − Υ S ( r S ) + 2 m S ( r S ) are constants on S verifying (cid:12)(cid:12)(cid:12) ( c S , c S ) (cid:12)(cid:12)(cid:12) (cid:46) r − . We also denote, c = (cid:18) κ − κ ω (cid:19) = (cid:18) Υ r (cid:19) − mr c = (cid:18) κκ − ρ (cid:19) = − Υ r + 2 mr . The error term Err( B ) is given byErr( B ) = − Err( e (cid:48) θ κ (cid:48) , e θ κ ) + f (cid:20)(cid:18) κ − κ ω (cid:19) − c (cid:21) + f (cid:20)(cid:18) κκ − ρ (cid:19) − c (cid:21) . We deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Be Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S A S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + r − (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Err( B ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) . (9.5.8)We can easily check, using the assumptions of the theorem, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κ − κ ω − (cid:18) κ − κ ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15)r − , (cid:12)(cid:12)(cid:12)(cid:12) κκ − ρ − κκ − ρ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15)r − , (cid:12)(cid:12)(cid:12) c S − c (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c S − c (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15)r − . For the last inequality we note that (cid:12)(cid:12)(cid:12) r S ) − r (cid:12)(cid:12)(cid:12) (cid:46) ( r S ) − (cid:12)(cid:12)(cid:12) r S r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r S r + 1 (cid:12)(cid:12)(cid:12) (cid:46) r − ◦ (cid:15). .5. PROOF OF PROPOSITION 9.4.4 for Err in Lemma 9.3.10, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Err( B ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r (cid:107) Err( B ) (cid:107) S (cid:46) r (cid:107) Err (cid:107) S + (cid:107) ( f, f ) (cid:107) S (cid:46) (cid:113) ◦ (cid:15) (cid:107) F (cid:107) . Consequently, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Be Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S A S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + r − (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) . We can evaluate the integral (cid:82) S A S f e Φ by using the identity A S = d/ S d/ S ,(cid:63) + K . Thus, (cid:90) S A S f e Φ = (cid:90) S d/ S d/ S ,(cid:63) f e Φ + (cid:90) S Kf e Φ = (cid:90) S d/ S ,(cid:63) f d/ S ,(cid:63) ( e Φ ) + (cid:90) S Kf e Φ = (cid:90) S Kf e Φ = ( r S ) − (cid:90) S f e Φ + (cid:90) S (cid:18)(cid:18) K − r (cid:19) + 1 r − r S ) (cid:19) f e Φ . Hence, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S A S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:113) ◦ (cid:15) (cid:107) f (cid:107) . Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Be Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) . (9.5.9)Recalling (9.5.7), (cid:107) B (cid:107) k (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Be Φ (cid:12)(cid:12)(cid:12)(cid:12) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k (cid:19) (cid:46) r − (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k (cid:19) . Returning to (9.5.4) and recalling the definition of B = d/ S ,(cid:63) ˇ κ S − d (cid:63) / ˇ κ , (cid:13)(cid:13)(cid:0) A S + V (cid:1) z (cid:13)(cid:13) k (cid:46) r − (cid:107) B (cid:107) k + r − ◦ δ + r − (cid:113) ◦ (cid:15) (cid:13)(cid:13)(cid:13) F (cid:13)(cid:13)(cid:13) k +2 (cid:46) r − (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +2 (cid:19) Note that Err( e (cid:48) θ κ (cid:48) , e θ κ ) = Err + l.o.t. from the definition of Err . CHAPTER 9. GCM PROCEDURE or, recalling the definition of I ( f, f ), (cid:13)(cid:13)(cid:0) A S + V (cid:1) z (cid:13)(cid:13) k (cid:46) r − I ( f, f ) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +2 (cid:19) . (9.5.10)To estimate z we recall that, A S + V = d/ S d/ S ,(cid:63) − ρ + 12 ϑϑ. Thus, (cid:16) ( A S + V ) z, z (cid:17) S = (cid:107) d/ S ,(cid:63)s z (cid:107) S + (cid:18) mr + O ( r − (cid:113) ◦ (cid:15) ) (cid:19) (cid:107) z (cid:107) S . We deduce, (cid:107) d/ S ,(cid:63)s z (cid:107) S (cid:46) (cid:16) ( A S + V ) z, z (cid:17) S (cid:46) (cid:107) ( A S + V ) z (cid:107) S (cid:107) z (cid:107) S . We make use of the elliptic estimate, see Lemma 9.1.4, (cid:107) z (cid:107) S (cid:46) r (cid:107) d/ S ,(cid:63)s z (cid:107) S + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ze Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r (cid:107) ( A S + V ) z (cid:107) S (cid:107) z (cid:107) S + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ze Φ (cid:12)(cid:12)(cid:12)(cid:12) from which we infer that (cid:107) z (cid:107) S (cid:46) r (cid:107) ( A S + V ) z (cid:107) S + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ze Φ (cid:12)(cid:12)(cid:12)(cid:12) . (9.5.11)Recalling the definition of z , z = κf + κf = 2 r f − r f + (cid:18) κ − r (cid:19) f + (cid:18) κ + 2Υ r (cid:19) f = 2 r S f − S r S f − (cid:18) r S − r (cid:19) f + (cid:18) S r S − r (cid:19) f + (cid:18) κ − r (cid:19) f + (cid:18) κ + 2Υ r (cid:19) f, and making use of our assumption (in particular (cid:12)(cid:12)(cid:12) r S r − (cid:12)(cid:12)(cid:12) (cid:46) r − ◦ (cid:15) ) we deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ze Φ − r S (cid:90) S f e Φ + 2Υ S r S (cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:113) ◦ (cid:15) r (cid:0) (cid:107) f (cid:107) + (cid:107) f (cid:107) (cid:1) . .5. PROOF OF PROPOSITION 9.4.4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ze Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − I ( f, f ) + (cid:113) ◦ (cid:15) r (cid:107) F (cid:107) and therefore in view of (9.5.11) and (9.5.10) (for k = 0) (cid:107) z (cid:107) (cid:46) r (cid:107) ( A S + V ) z (cid:107) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ze Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r (cid:18) r − I ( f, f ) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) (cid:19)(cid:19) + r − (cid:18) r − I ( f, f ) + (cid:113) ◦ (cid:15) r (cid:107) F (cid:107) (cid:19) (cid:46) r − I ( f, f ) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) (cid:19) . By standard elliptic theory, (cid:107) z (cid:107) k +2 (cid:46) (cid:107) z (cid:107) + r (cid:107) ( A S + V ) z (cid:107) k . We infer that, for k ≤ s max − (cid:107) z (cid:107) k +2 (cid:46) r − I ( f, f ) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k (cid:19) . (9.5.12) Step 2.
We now proceed in the same manner to derive an estimate for f . Using thesecond equation in (9.4.2) (cid:0) A S + V (cid:1) f = 34 κ ( A S ) − (cid:0) ρz (cid:1) − d (cid:63) / ˇ κ − κ ( A S ) − (cid:0) − d/ S ,(cid:63) ˇ µ S + d (cid:63) / ˇ µ (cid:1) + Err , we derive (note that (cid:107) ( A S ) − h (cid:107) k (cid:46) r (cid:107) h (cid:107) k − ), (cid:107) (cid:0) A S + V (cid:1) f (cid:107) k (cid:46) r (cid:107) ρz (cid:107) k − + (cid:107) d (cid:63) / ˇ κ (cid:107) k + r (cid:107) d/ S ,(cid:63) ˇ µ S − d (cid:63) / ˇ µ (cid:107) k − + (cid:107) Err (cid:107) k + (cid:113) ◦ (cid:15) r − (cid:107) F (cid:107) k (cid:46) r − (cid:107) z (cid:107) k − + ◦ δr − + r (cid:107) C (cid:107) k − + (cid:113) ◦ (cid:15) r − (cid:107) F (cid:107) k +2 (9.5.13)where, C : = d/ S ,(cid:63) ˇ µ S − d (cid:63) / ˇ µ. (9.5.14)To estimate C we write, proceeding as in the estimate for B is Step 1. Making use of theGCM condition d/ S ,(cid:63) d/ S ,(cid:63) µ S = 0 we derive, as in (9.5.6), (cid:13)(cid:13)(cid:13) d/ S ,(cid:63) C (cid:13)(cid:13)(cid:13) k (cid:46) r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k (cid:19) (9.5.15) Since most estimates are similar as in Step 1 above we skip some of the intermediary steps. CHAPTER 9. GCM PROCEDURE which implies, as in (9.5.7), (cid:107) C (cid:107) k (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Ce Φ (cid:12)(cid:12)(cid:12)(cid:12) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k (cid:19) . (9.5.16)To estimate the (cid:96) = 1 mode of C we recall the transformation formula, see Lemma 9.3.4, e (cid:48) θ ( µ (cid:48) ) = e θ µ + e (cid:48) θ ( d/ ) (cid:48) (cid:18) − ( d (cid:63) / ) (cid:48) a + f ω − f ω + 14 f κ − f κ (cid:19) + 34 ρ ( f κ + f κ )+ Err( e (cid:48) θ µ (cid:48) , e θ µ ) , i.e., C = A S (cid:18) − d/ S ,(cid:63) a + f ω + 14 f κ − f κ (cid:19) − ρz + Err( e (cid:48) θ µ (cid:48) , e θ µ ) . Recall that, A S = d/ S d/ S ,(cid:63) = d/ S d/ S ,(cid:63) + K. Therefore, for any h ∈ s , (cid:90) S A S he Φ = (cid:90) S hKe Φ . We deduce, (cid:90) S Ce Φ = (cid:90) S K (cid:18) − d/ S ,(cid:63) a + f ω + 14 f κ − f κ (cid:19) e Φ − (cid:90) S ( ρz ) e Φ + (cid:90) S Err( e (cid:48) θ µ (cid:48) , e θ µ ) e Φ = ( r S ) − (cid:90) S (cid:18) − d/ S ,(cid:63) a + f ω + 14 f κ − f κ (cid:19) e Φ − (cid:90) S ( ρz ) e Φ + (cid:90) S Err( e (cid:48) θ µ (cid:48) , e θ µ ) e Φ + (cid:90) S (cid:0) ( K − r − ) + ( r − − ( r S ) − ) (cid:1) (cid:18) − d/ S ,(cid:63) a + f ω + 14 f κ − f κ (cid:19) e Φ . Also, (cid:90) S (cid:18) f ω + 14 f κ − f κ (cid:19) e Φ = m S ( r S ) (cid:90) S f e Φ − Υ S r S (cid:90) S f e Φ − r S (cid:90) S f e Φ + l.o.t. (cid:90) S ρze Φ = − m S ( r S ) (cid:90) S ze Φ + l.o.t.Note that, since Err( e (cid:48) θ µ (cid:48) , e θ µ ) = Err , we have in view of Lemma 9.3.10, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Err( e (cid:48) θ µ (cid:48) , e θ µ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r (cid:107) Err (cid:107) (cid:46) r − (cid:107) F (cid:107) . .5. PROOF OF PROPOSITION 9.4.4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Ce Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − I ( a, f, f ) + r − (cid:107) F (cid:107) . (9.5.17)Back to (9.5.16) we infer that, (cid:107) C (cid:107) k (cid:46) r − I ( a, f, f ) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k (cid:19) . (9.5.18)Back to (9.5.13), (cid:107) (cid:0) A S + V (cid:1) f (cid:107) k (cid:46) r − (cid:107) z (cid:107) k − + ◦ δr − + r (cid:107) C (cid:107) k − + (cid:113) ◦ (cid:15) r − (cid:107) F (cid:107) k +2 (cid:46) r − (cid:107) z (cid:107) k − + ◦ δr − + (cid:113) ◦ (cid:15) r − (cid:107) F (cid:107) k +2 + r − I ( a, f, f )i.e., recalling the estimate (9.5.12) for (cid:107) z (cid:107) k in Step 1, (cid:107) (cid:0) A S + V (cid:1) f (cid:107) k (cid:46) ◦ δr − + (cid:113) ◦ (cid:15) r − (cid:107) F (cid:107) k +2 + r − I ( a, f, f ) . (9.5.19)To estimate (cid:107) f (cid:107) = (cid:107) f (cid:107) S we proceed as in the proof of the estimate for (cid:107) z (cid:107) in Step 1and deduce, as in (9.5.11), (cid:107) f (cid:107) S (cid:46) r (cid:107) ( A S + V ) f (cid:107) S + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) . (9.5.20)Hence, (cid:107) f (cid:107) (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) + r − I ( a, f, f ) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) i.e., (cid:107) f (cid:107) (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) + r − I ( a, f, f ) . Finally, (cid:107) f (cid:107) k +2 (cid:46) (cid:107) f (cid:107) + r (cid:107) ( A S + V ) f (cid:107) k (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k + r − I ( a, f, f ) . Hence, (cid:107) f (cid:107) k +2 (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +2 + r − I ( a, f, f ) . (9.5.21)90 CHAPTER 9. GCM PROCEDURE
Since by the definition of z we have, f = κ − ( z − κf )we also derive, (cid:107) f (cid:107) k +2 (cid:46) r − (cid:107) z (cid:107) k +2 + (cid:107) f (cid:107) k +2 . Thus, making use of the estimate (9.5.12) for z , we also deduce the estimate for f , (cid:107) f (cid:107) k +2 (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +2 + r − I ( a, f, f )i.e., (cid:107) (cid:0) f, f (cid:1) (cid:107) k +2 (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +2 + r − I ( a, f, f ) . (9.5.22) Step 3.
It remains to estimate a by making use of the equation (see Proposition 9.4.2and definition of C = d/ S ,(cid:63) ˇ µ S − d (cid:63) / ˇ µ in (9.5.14) ) d/ S ,(cid:63) a = −
34 ( A S ) − (cid:0) ρz (cid:1) + f ω + 14 f κ − f κ − ( A S ) − C − ( A S ) − Err . Proceeding as before we deduce, (cid:107) d/ S ,(cid:63) a (cid:107) k (cid:46) r − (cid:107) z (cid:107) k − + r − (cid:107) ( f, f ) (cid:107) k + r (cid:107) C (cid:107) k − + r (cid:107) Err (cid:107) k − . Making use of (9.5.12) and,(9.5.18) and the estimate of Lemma 9.3.10 for Err , (cid:107) z (cid:107) k − (cid:46) r − I ( f, f ) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k − (cid:19) , (cid:107) C (cid:107) k − (cid:46) r − I ( a, f, f ) + r − (cid:18) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k − (cid:19) , (cid:107) Err (cid:107) k − (cid:46) r − (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +1 , we further deduce, (cid:107) d/ S ,(cid:63) a (cid:107) k (cid:46) r − ( ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +1 ) + r − I ( a, f, f ) . (9.5.23)Combining this with (9.5.22), r S (cid:107) d/ S ,(cid:63) a (cid:107) k +1 + (cid:107) (cid:0) f, f (cid:1) (cid:107) k +2 (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +2 + r − I ( a, f, f ) . (9.5.24) .5. PROOF OF PROPOSITION 9.4.4 Step 4.
We make use of the equations (9.4.3) to estimate I ( a, f, f ). Recall that, I ( a, f, f ) = I ( f, f ) + r S (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( e S θ a ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) . Step 4a.
We estimate the integral (cid:82) S ( e S θ a ) e Φ making use of the the first equation in(9.4.3).2 r S (cid:90) S e S θ ( a ) e Φ = − (cid:90) S e θ ( κ ) e Φ + 14 (cid:90) S κ f e Φ − (cid:90) S (cid:18) κκ + κω + 3 ρ (cid:19) f e Φ + Err = − J + J − J + Err where,Err = (cid:90) S (cid:18) − Err( e S θ κ S , e θ κ ) + 2( K S − K ) f + (cid:18) κ − r S (cid:19) d/ S ,(cid:63) a + 12 ϑϑf (cid:19) e Φ . We deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e S θ ( a ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ( | J | + | J | + | J | ) + r | Err | . We make use of e S θ = (cid:18) f f (cid:19) e θ + 12 f e + 12 f (cid:18) f f (cid:19) e and the assumptions of Theorem 9.4.4, in particular, (cid:107) e S θ κ (cid:107) S (cid:46) ◦ δr − , to deduce, | J | (cid:46) r (cid:90) S (cid:0) | e S θ ( κ ) | + | e S θ ( κ ) − e θ ( κ ) | (cid:1) (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) ( f, f ) (cid:107) S . Also, | J | (cid:46) r S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + r (cid:90) S | f | (cid:20)(cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) r S ) − r (cid:12)(cid:12)(cid:12)(cid:21) (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:113) ◦ (cid:15) (cid:107) f (cid:107) S . Similarly, | J | (cid:46) r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:113) ◦ (cid:15) (cid:107) f (cid:107) S . recall that ω = 0. CHAPTER 9. GCM PROCEDURE
Also, | Err | (cid:46) r (cid:107) Err( e (cid:48) θ κ (cid:48) , e θ κ ) (cid:107) S + r (cid:18) (cid:107) K S − K (cid:107) S (cid:107) f (cid:107) S + (cid:107) κ − r S (cid:107) S (cid:107) d/ S ,(cid:63) a (cid:107) S + (cid:107) ϑϑ (cid:107) S (cid:107) f (cid:107) S (cid:19) . Thus, proceeding as before, | Err | (cid:46) (cid:113) ◦ (cid:15) (cid:107) F (cid:107) . We deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e S θ ( a ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ( | J | + | J | + | J | ) + r | Err | (cid:46) r ◦ δ + (cid:113) ◦ (cid:15) r (cid:107) ( f, f ) (cid:107) S + r − (cid:20)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S f e Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) + r (cid:113) ◦ (cid:15) (cid:107) F (cid:107) i.e., (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e S θ ( a ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ◦ δ + r − I ( f, f ) + r (cid:113) ◦ (cid:15) (cid:107) F (cid:107) . (9.5.25)As a consequence we deduce, I ( a, f, f ) (cid:46) I ( f, f ) + r ◦ δ. (9.5.26) Step 5.
We combine estimate (9.5.24) with (9.5.26) with r S (cid:107) d/ S ,(cid:63) a (cid:107) k +1 + (cid:107) (cid:0) f, f (cid:1) (cid:107) k +2 (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) F (cid:107) k +2 + r − I ( f, f ) . Also, taking into account that (cid:107) F (cid:107) k +2 = (cid:107) a (cid:107) + r S (cid:107) d/ S ,(cid:63) a (cid:107) k +1 + (cid:107) (cid:0) f, f (cid:1) (cid:107) k +2 and the smallness of ◦ (cid:15) , we deduce, r S (cid:107) d/ S ,(cid:63) a (cid:107) k +1 + (cid:107) (cid:0) f, f (cid:1) (cid:107) k +2 (cid:46) ◦ δ + (cid:113) ◦ (cid:15) (cid:107) a (cid:107) + r − I ( f, f ) . (9.5.27) Step 6.
It remains to get an estimate for (cid:107) a (cid:107) = (cid:107) a (cid:107) L ( S ) . For this we need to makeuse of the last equation in for the average of a in (9.4.2) as well as the GCM condition κ S = r S , a S = (cid:16) − r S r (cid:17) S − r S (cid:0) ˇ κ + κ − r (cid:1) S + Err S .5. PROOF OF PROPOSITION 9.4.4 = − r S (cid:20) e a Err( κ, κ (cid:48) ) − ( e a − (cid:18) ˇ κ + κ − r + d/ S f (cid:19) − ( e a − − a ) 2 r (cid:21) − a (cid:18) r S r − (cid:19) . Hence, | a S | (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − r S r (cid:19) S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + r (cid:18) | ˇ κ | + (cid:12)(cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + r (cid:12)(cid:12)(cid:12) Err (cid:12)(cid:12)(cid:12) . Making use of the assumption | a S | (cid:46) r − ◦ δ. (9.5.28) Step 7.
By the standard Poincare inequality on S we have, (cid:13)(cid:13) a − a S (cid:13)(cid:13) S (cid:46) r S (cid:107) d/ S ,(cid:63) a (cid:107) S . Combining this with (9.5.27) and (9.5.28) we deduce, (cid:107) (cid:0) a, f, f (cid:1) (cid:107) k +2 (cid:46) ◦ δ + r − I ( f, f ) . (9.5.29)Finally, making use of the condition (cid:90) S f e Φ = Λ , (cid:90) S f e Φ = Λ , Λ , Λ = O ( ◦ δr )we derive the desired estimate, (cid:107) (cid:0) a, f, f (cid:1) (cid:107) k +2 (cid:46) ◦ δ. (9.5.30)The uniqueness part is obvious and left to the reader. This ends the proof of Theorem9.4.4. Note that full strength of the assumption is only used here, otherwise the weaker assumption, with ◦ δ replaced by ◦ (cid:15) , suffices in the rest of the argument. CHAPTER 9. GCM PROCEDURE
According to Proposition 9.4.8 we may assume valid the uniform bounds for the quintets P ( n ) . To establish a contraction estimate we need to compare the quantities, h ( n ) : = (Ψ ( n − ) f ( n ) , h ( n ) := (Ψ ( n − ) f ( n ) , w ( n ) := (Ψ ( n − ) z ( n ) ,e ( n ) : = (Ψ ( n − ) a ( n ) , and, h ( n +1) : = (Ψ ( n ) ) f ( n +1) , h ( n +1) := (Ψ ( n ) ) f ( n +1) , w ( n +1) := (Ψ ( n ) ) z ( n +1) ,e ( n +1) : = (Ψ ( n ) ) a ( n +1) . According to Lemma 9.2.2 we have,(Ψ ( n ) ) (cid:0) d/ S ( n ) f ( n +1) (cid:1) = d/ ( n ) h ( n +1) , (Ψ ( n ) ) (cid:0) A S ( n ) f ( n +1) (cid:1) = A ( n ) h ( n +1) , where d/ ( n ) , d (cid:63) / ( n ) , A ( n ) are the corresponding Hodge operators on ◦ S defined with respectto the metric g/ ( n ) := (Ψ ( n ) ) ( g/ S ( n ) ) given by, g/ ( n ) = γ ( n ) dθ + e n dϕ . Consequently the system (9.4.25) takes the form, A ( n ) w ( n +1) + V ( n ) w ( n +1) = (Ψ ( n ) ) E ( n +1) ,A ( n ) h ( n +1) + V ( n ) h ( n +1) = 34 κ ( n ) ( A ( n ) ) − ( ρ ( n ) w ( n +1) ) + (Ψ ( n ) ) E ( n +1) ,A ( n ) h ( n +1) + V ( n ) h ( n +1) = − κ ( n ) ( A ( n ) ) − ( ρ ( n ) w ( n +1) ) + (Ψ ( n ) ) E ( n +1) ,d (cid:63) / ( n ) e ( n +1) = 34 ( A ( n ) ) − ( ρ ( n ) w ( n +1) ) − h ( n +1) ω ( n ) − h ( n +1) κ ( n ) + 14 h ( n +1) κ ( n ) + (Ψ ( n ) ) (cid:101) E ( n +1) , (9.6.1)where, κ ( n ) , κ ( n ) , ρ ( n ) , ω ( n ) , ω ( n ) are the pull backs by Ψ ( n ) of κ, κ, ρ, ω, ω . Also V ( n ) is thepull back by Ψ ( n ) of the potential V = κκ − ρ . .6. PROOF OF PROPOSITION 9.4.9 r S ( n ) (cid:90) ◦ S,g/ ( n ) d (cid:63) / ( n ) e ( n +1) e Φ n = − (cid:90) ◦ S,g/ ( n ) ( κ ( n ) ) h ( n +1) e Φ n + (cid:90) ◦ S,g/ ( n ) (cid:18) κκ + κω + 3 ρ (cid:19) ( n ) h ( n +1) e Φ n − (cid:90) ◦ S,g/ ( n ) d (cid:63) / ( n ) ( κ ( n ) ) e Φ n da g/ ( n ) + (Ψ ( n ) ) Err ( n +1)6 , (cid:90) ◦ S,g/ ( n ) h ( n +1) e Φ n = Λ , (cid:90) ◦ S,g/ ( n ) h ( n +1) e Φ n = Λ , (9.6.2)Equation (9.4.29) takes the form, e ( n +1) ◦ S,g/ ( n ) = (cid:18) − r S ( n ) r (cid:19) ( n ) ◦ S,g/ ( n ) − r S ( n ) (cid:18) ˇ κ + κ − r (cid:19) ( n ) ◦ S,g/ ( n ) + Err ( n +1)8 ◦ S,g/ ( n ) . (9.6.3)Finally the system (9.4.31) takes the form, ς n ∂ θ U (1+ n ) = ( γ ( n ) ) / h (1+ n ) (cid:18) h (1+ n ) h (1+ n ) (cid:19) ,∂ θ S (1+ n ) − ς n Ω n ∂ θ U (1+ n ) = 12 ( γ ( n ) ) / h (1+ n ) ,γ ( n ) = γ n + (cid:0) ς n (cid:1) (cid:18) Ω n + 14 ( b n ) γ n (cid:19) ( ∂ θ U ( n ) ) − ς n ∂ θ U ( n ) ∂ θ S ( n ) − γ n ς n b n ∂ θ U ( n ) ,U (1+ n ) (0) = S (1+ n ) (0) = 0 . (9.6.4)We recall, see (9.4.32), the definition of the norm for the quintets P ( n ) in the particularcase k = 3 (cid:107) P ( n ) (cid:107) : = (cid:107) ∂ θ (cid:0) U ( n ) , S ( n ) (cid:1) (cid:107) L ∞ ( ◦ S ) + r − (cid:107) ∂ θ (cid:0) U ( n ) , S ( n ) (cid:1) (cid:107) h ( ◦ S ) + (cid:13)(cid:13)(cid:13)(cid:16) ( a ( n ) ) n − , ( f ( n ) ) n − , ( f ( n ) ) n − (cid:17)(cid:13)(cid:13)(cid:13) h ( ◦ S ) . To prove the estimate (cid:107) P ( n +1) − P ( n ) (cid:107) (cid:46) ◦ δ (cid:107) P ( n ) − P ( n − (cid:107) we set, δw ( n +1) = w ( n +1) − w ( n ) , δh ( n +1) = h ( n +1) − h ( n ) , δh ( n +1) = h ( n +1) − h ( n ) ,δe ( n +1) = e ( n +1) − e ( n ) , δU ( n +1) = U ( n +1) − U ( n ) , δS ( n +1) = S ( n +1) − S ( n ) , CHAPTER 9. GCM PROCEDURE and, δw ( n ) = w ( n ) − w ( n − , δh ( n ) = h ( n ) − h ( n − , δh ( n ) = h ( n ) − h ( n − ,δe ( n ) = e ( n ) − e ( n − , δU ( n ) = U ( n ) − U ( n − , δS ( n ) = S ( n ) − S ( n − . We will derive below the following estimates (cid:107) δh ( n +1) , δh ( n +1) , δe ( n +1) ) (cid:107) h ( ◦ S ) (cid:46) ◦ δr − (cid:107) ∂ θ (cid:0) δU ( n ) , δS ( n ) (cid:1) (cid:107) h ( ◦ S ) (9.6.5)and r − (cid:107) ∂ θ (cid:0) δU ( n +1) , δS ( n +1) (cid:1) (cid:107) h ( ◦ S ) (cid:46) (cid:107) δh ( n +1) , δh ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15)r − (cid:107) ∂ θ (cid:0) δU ( n ) , δS ( n ) (cid:1) (cid:107) h ( ◦ S ) . (9.6.6)Proposition 9.4.9 is then an immediate consequence of (9.6.5) (9.6.6). Thus, from nowon, we focus on the proof of (9.6.5) (9.6.6). To this end, we will rely on the followinglemmas. Lemma 9.6.1.
Let F be a reduced scalar function defined in a neighborhood of ◦ S in R and define its pull back F ( n ) = (Ψ ( n ) ) F to ◦ S , i.e., F ( n ) ( θ ) = F ( ◦ u + U ( n ) ( θ ) , ◦ s + S ( n ) ( θ ) , θ ) ,F ( n − ( θ ) = F ( ◦ u + U ( n − ( θ ) , ◦ s + S ( n − ( θ ) , θ ) . Then , for all ≤ p ≤ ∞ , with δ n U = U ( n +1) − U ( n ) , δ n S = S ( n +1) − S ( n ) (cid:107) δ n F (cid:107) L p ( ◦ S ) (cid:46) (cid:16) (cid:107) δ n U (cid:107) L p ( ◦ S ) + (cid:107) δ n S (cid:107) L p ( ◦ S ) (cid:17) sup R (cid:0)(cid:12)(cid:12) ∂ s F (cid:12)(cid:12) + (cid:12)(cid:12) ∂ u F (cid:12)(cid:12)(cid:1) . (9.6.7) Also, (cid:107) δ n F (cid:107) h ( ◦ S ) (cid:46) (cid:16) (cid:107) δ n U (cid:107) h ( ◦ S ) + (cid:107) δ n S (cid:107) h ( ◦ S ) (cid:17) sup R | d ≤ d F | (9.6.8) where δ n U = U ( n +1) − U ( n ) , δ n S = S ( n +1) − S ( n ) . Recall ∂ s = e , ∂ u = ς (cid:0) e − Ω e − bγ / e θ (cid:1) . .6. PROOF OF PROPOSITION 9.4.9 Proof.
We write, δ n F := F ( u + U ( n ) ( θ ) , s + S ( n ) ( θ ) , θ ) − F ( u + U ( n − ( θ ) , s + S ( n − ( θ ) , θ )= (cid:90) ddt F (cid:0) u + tU ( n ) ( θ ) + (1 − t ) U ( n − ( θ ) , s + tS ( n ) ( θ ) + (1 − t ) S ( n − ( θ ) , θ (cid:1) , i.e., denoting δ n U = U ( n ) − U ( n − , δ n S = S ( n ) − S ( n − , | δ n F | (cid:46) (cid:12)(cid:12) δ n U (cid:12)(cid:12) (cid:90) (cid:12)(cid:12) ∂ u F (cid:0) u + tU ( n ) ( θ ) + (1 − t ) U ( n − ( θ ) , s + tS ( n ) ( θ ) + (1 − t ) S ( n − ( θ ) , θ (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) δ n S (cid:12)(cid:12) (cid:90) (cid:12)(cid:12) ∂ s F (cid:0) u + tU ( n ) ( θ ) + (1 − t ) U ( n − ( θ ) , s + tS ( n ) ( θ ) + (1 − t ) S ( n − ( θ ) , θ (cid:1)(cid:12)(cid:12) i.e., | δ n F | (cid:46) (cid:12)(cid:12) U ( n ) ( θ ) − U ( n − ( θ ) (cid:12)(cid:12) sup ◦ S + ◦ δ ◦ S | ∂ u F | + (cid:12)(cid:12) S ( n ) ( θ ) − S ( n − ( θ ) (cid:12)(cid:12) sup ◦ S + (cid:15) ◦ S | ∂ s F | from which (9.6.7) easily follows.Also, in view of ∂ s = e , ∂ u = (cid:0) e − Ω e − bγ / e θ (cid:1) and our assumptions for Ω and b , | δ n F | (cid:46) (cid:0) | U ( n ) − U ( n − | + | S ( n ) − S ( n − | (cid:1) | sup R d F | and by integration, (cid:107) δ n F (cid:107) L p ( ◦ S ) (cid:46) (cid:16) (cid:107) δ n U (cid:107) L p ( ◦ S ) + (cid:107) δ n S (cid:107) L p ( ◦ S ) (cid:17) sup R | d F | . Similarly, (cid:107) d /δ n F (cid:107) L ( ◦ S ) (cid:46) (cid:16) (cid:107) δ n U (cid:107) h ( ◦ S ) + (cid:107) δ n S (cid:107) h ( ◦ S ) (cid:17) sup R | d ≤ d F | . Hence, (cid:107) δ n F (cid:107) h ( ◦ S ) (cid:46) (cid:16) (cid:107) δ n U (cid:107) h ( ◦ S ) + (cid:107) δ n S (cid:107) h ( ◦ S ) (cid:17) sup R | d ≤ d F | as desired. Lemma 9.6.2.
Let ψ, h ∈ s ( ◦ S ) , and δA ( n ) = A ( n ) − A ( n − . The following formula holdstrue. (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( ◦ S,g/ ( n ) ) ψδA ( n ) h (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) (cid:16) (cid:107) ψ (cid:107) h ( ◦ S ) (cid:107) h (cid:107) h ( ◦ S ) + (cid:107) ψ (cid:107) h ( ◦ S ) (cid:107) h (cid:107) h ( ◦ S ) (cid:17) . CHAPTER 9. GCM PROCEDURE
Proof.
Recall that the metric g/ ( n ) is given by g/ ( n ) = γ ( n ) dθ + e n dϕ so that the operator A ( n ) = − d (cid:63) / ( n ) d/ ( n ) , applied to s tensors h on ◦ S is given by A ( n ) h = − (cid:112) γ ( n ) ∂ θ (cid:32) (cid:112) γ ( n ) (cid:0) ∂ θ h + ∂ θ (Φ n ) h (cid:1)(cid:33) . This yields δA ( n ) h = − (cid:32) (cid:112) γ ( n ) − (cid:112) γ ( n − (cid:33) ∂ θ (cid:32) (cid:112) γ ( n ) (cid:0) ∂ θ h + ∂ θ (Φ n ) h (cid:1)(cid:33) − (cid:112) γ ( n − ∂ θ (cid:32)(cid:32) (cid:112) γ ( n ) − (cid:112) γ ( n − (cid:33) (cid:0) ∂ θ h + ∂ θ (Φ n ) h (cid:1)(cid:33) − (cid:112) γ ( n − ∂ θ (cid:32) (cid:112) γ ( n − ∂ θ (Φ n − Φ n − ) h (cid:33) . Using the previous formula to integrate ψδA ( n ) h on ◦ S with the volume of g/ ( n ) , and afterintegration by parts, we infer (cid:90) ( ◦ S,g/ ( n ) ) ψδA ( n ) h = (cid:90) ( ◦ S,g/ ( n ) ) (cid:112) γ ( n ) ∂ θ (cid:32)(cid:32) − (cid:112) γ ( n ) (cid:112) γ ( n − (cid:33) ψ (cid:33) (cid:112) γ ( n ) (cid:0) ∂ θ h + ∂ θ (Φ n ) h (cid:1) + (cid:90) ( ◦ S,g/ ( n ) ) (cid:112) γ ( n ) ∂ θ (cid:32) (cid:112) γ ( n ) (cid:112) γ ( n − ψ (cid:33) (cid:32) (cid:112) γ ( n ) − (cid:112) γ ( n − (cid:33) (cid:0) ∂ θ h + ∂ θ (Φ n ) h (cid:1) + (cid:90) ( ◦ S,g/ ( n ) ) (cid:112) γ ( n ) ∂ θ (cid:32) (cid:112) γ ( n ) (cid:112) γ ( n − ψ (cid:33) (cid:112) γ ( n − ∂ θ (cid:0) Φ n − Φ n − (cid:1) h. We now make us of the bounds (9.1.15) for (Ω , b, γ ) involved in the definition of γ ( n − and γ ( n ) , the uniform bound of P ( n ) provided by Proposition 9.4.8 and the Sobolev inequalityto deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( ◦ S,g/ ( n ) ) ψδA ( n ) h (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:107) γ ( n ) − γ ( n − (cid:107) h ( ◦ S ) (cid:16) (cid:107) ψ (cid:107) h ( ◦ S ) (cid:107) h (cid:107) h ( ◦ S ) + ( (cid:107) ψ (cid:107) h ( ◦ S ) (cid:107) h (cid:107) h ( ◦ S ) (cid:17) + r − (cid:13)(cid:13) ∂ θ (cid:0) Φ n − Φ n − (cid:1) h (cid:13)(cid:13) L ( ◦ S ) (cid:107) ψ (cid:107) h ( ◦ S ) . (9.6.9) .6. PROOF OF PROPOSITION 9.4.9 γ ( n ) − γ ( n − we recall that, γ ( n ) = γ n + (cid:0) ς n (cid:1) (cid:0) Ω n + ( b n ) γ n (cid:1) ( ∂ θ U ( n ) ) − ς n ∂ θ U ( n ) ∂ θ S ( n ) − γ n ς n b n ∂ θ U ( n ) γ ( n − = γ n − + (cid:0) ς n − (cid:1) (cid:0) Ω n − + ( b n − ) γ n − (cid:1) ( ∂ θ U ( n − ) − ς n − ∂ θ U ( n − ∂ θ S ( n − − γ n − ς n − b n − ∂ θ U ( n − . The principal term γ n − γ n − can be estimated with the help of Lemma 9.6.1, theuniform bound of P ( n ) provided by Proposition 9.4.8, and the bounds provided by A3 .All other terms can be estimated in a similar fashion. We derive, (cid:107) γ ( n ) − γ ( n − (cid:107) h ( ◦ S ) (cid:46) r (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) (9.6.10)where, (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) := (cid:107) ∂ θ ( U ( n ) − U ( n − ) (cid:107) h ( ◦ S ) + (cid:107) ∂ θ ( S ( n ) − S ( n − ) (cid:107) h ( ◦ S ) . We deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( ◦ S,g/ ( n ) ) ψδA ( n ) h (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) (cid:16) (cid:107) ψ (cid:107) h ( ◦ S ) (cid:107) h (cid:107) h ( ◦ S ) + ( (cid:107) ψ (cid:107) h ( ◦ S ) (cid:107) h (cid:107) h ( ◦ S ) (cid:17) + r − (cid:13)(cid:13) ∂ θ (cid:0) Φ n − Φ n − (cid:1) h (cid:13)(cid:13) L ( ◦ S ) (cid:107) ψ (cid:107) h ( ◦ S ) . (9.6.11)The proof of 9.6.2. is now an immediate consequence of the following. Lemma 9.6.3.
The following estimate holds true for a reduced scalar h ∈ s ( ◦ S ) (cid:13)(cid:13) ∂ θ (cid:0) Φ n − Φ n − (cid:1) h (cid:13)(cid:13) L ( ◦ S ) (cid:46) r (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) (cid:107) h (cid:107) h ( ◦ S ) . (9.6.12) Proof.
We write, ∂ θ (cid:0) Φ n − Φ n − (cid:1) = (cid:26)(cid:18) ∂ θ S ( n ) −
12 Ω ∂ θ U ( n ) (cid:19) e Φ + 12 ∂ θ U ( n ) e Φ + √ γ (cid:18) − b∂ θ U ( n ) (cid:19) e θ Φ (cid:27) n − (cid:26)(cid:18) ∂ θ S ( n − −
12 Ω ∂ θ U ( n − (cid:19) e Φ + 12 ∂ θ U ( n − e Φ + √ γ (cid:18) − b∂ θ U ( n − (cid:19) e θ Φ (cid:27) n − = (cid:18) ∂ θ S ( n ) −
12 Ω n ∂ θ U ( n ) (cid:19) ( e Φ) n + 12 ∂ θ U ( n ) ( e Φ) n + √ γ n (cid:18) − b n ∂ θ U ( n ) (cid:19) ( e θ Φ) n − (cid:18) ∂ θ S ( n − −
12 Ω n − ∂ θ U ( n − (cid:19) ( e Φ) n − − ∂ θ U ( n − ( e Φ) n − −√ γ n − (cid:18) − b n − ∂ θ U ( n − (cid:19) ( e θ Φ) n − CHAPTER 9. GCM PROCEDURE i.e., grouping the terms appropriately, ∂ θ (cid:0) Φ n − Φ n − (cid:1) = J + J + J ,J = (cid:18) ∂ θ S ( n ) −
12 Ω n ∂ θ U ( n ) (cid:19) ( e Φ) n − (cid:18) ∂ θ S ( n − −
12 Ω n − ∂ θ U ( n − (cid:19) ( e Φ) n − ,J = 12 ∂ θ U ( n ) ( e Φ) n − ∂ θ U ( n − ( e Φ) n − ,J = √ γ n (cid:18) − b n ∂ θ U ( n ) (cid:19) ( e θ Φ) n − √ γ n − (cid:18) − b n − ∂ θ U ( n − (cid:19) ( e θ Φ) n − , and, J = J + J ,J = ( e θ Φ) n − (cid:18) √ γ n (cid:18) − b n ∂ θ U ( n ) (cid:19) − √ γ n − (cid:18) − b n − ∂ θ U ( n − (cid:19)(cid:19) ,J = √ γ n (cid:18) − b n ∂ θ U ( n ) (cid:19) (cid:0) ( e θ Φ) n − ( e θ Φ) n − (cid:1) . The contribution to the estimate of of Lemma 9.6.3 given by J , J , J can be easilyestimated by making use of the uniform bound of P ( n ) provided by Proposition 9.4.8, thebound (9.1.15) for (Ω , b, γ ), Lemma 9.2.6 as well as Lemma 9.6.1. We thus derive, (cid:107) ( J , J , J ) h (cid:107) L ( ◦ S ) (cid:46) r (cid:107) ∂ θ Ψ ( n ) − ∂ θ Ψ ( n − (cid:107) h ( ◦ S ) (cid:107) h (cid:107) h ( ◦ S ) . It remains to estimate the term (cid:107) J h (cid:107) L ( ◦ S ) which presents a difficulty at the axis ofsymmetry where sin θ = 0. We can write, J = J + J ,J = ( e θ Φ) n − (cid:18) √ γ n (cid:18) − b n ∂ θ U ( n ) (cid:19) − √ γ n − (cid:18) − b n − ∂ θ U ( n − (cid:19)(cid:19) ,J = √ γ n (cid:18) − b n ∂ θ U ( n ) (cid:19) (cid:0) ( e θ Φ) n − ( e θ Φ) n − (cid:1) . Clearly, (cid:107) J h (cid:107) L ( ◦ S ) (cid:46) (cid:13)(cid:13)(cid:0) ( e θ Φ) n − ( e θ Φ) n − (cid:1) h (cid:13)(cid:13) L ( ◦ S ) . We are thus left to estimate theterm (cid:13)(cid:13)(cid:0) ( e θ Φ) n − ( e θ Φ) n − (cid:1) h (cid:13)(cid:13) L ( ◦ S ) . Proceeding as in the proof of Lemma 9.6.1 we write,for F = e θ Φ, | δ n F | (cid:46) (cid:12)(cid:12) δ n U (cid:12)(cid:12) (cid:90) (cid:12)(cid:12) ∂ u F (cid:0) u + tU ( n ) ( θ ) + (1 − t ) U ( n − ( θ ) , s + tS ( n ) ( θ ) + (1 − t ) S ( n − ( θ ) , θ (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) δ n S (cid:12)(cid:12) (cid:90) (cid:12)(cid:12) ∂ s F (cid:0) u + tU ( n ) ( θ ) + (1 − t ) U ( n − ( θ ) , s + tS ( n ) ( θ ) + (1 − t ) S ( n − ( θ ) , θ (cid:1)(cid:12)(cid:12) . .6. PROOF OF PROPOSITION 9.4.9 , where sin θ = 0, to the integral terminvolving ∂ u ( e θ Φ) = 12 (cid:0) e − Ω e − bγ / e θ (cid:1) e θ Φ . This leads us to consider the integral, (cid:90) [ be θ ( e θ (Φ))] ( ◦ u + tU ( n ) ( θ ) + (1 − t ) U ( n − ( θ ) , ◦ s, θ ) dt and the L norm of its product with h on ◦ S . We recall (see Lemma 2.1.13) that (cid:52) / Φ = − K .and Therefore, (cid:12)(cid:12) e θ ( e θ Φ) (cid:12)(cid:12) (cid:46) r − + | e θ Φ) | The contribution due to K does not present anydifficulties on the axis therefore we are led to consider the integral I ( θ ) := (cid:90) (cid:2) b ( e θ (Φ)) (cid:3) ( ◦ u + tU ( n ) ( θ ) + (1 − t ) U ( n − ( θ ) , ◦ s, θ ) dt and the L norm of its product with h on ◦ S . Making use of (9.1.17) and then the firstestimate of (9.1.18) of Lemma 9.1.3 together with our assumption A3 we derive thebound, (cid:12)(cid:12) I ( θ ) h ( θ ) (cid:12)(cid:12) (cid:46) θ (cid:18)(cid:90) (cid:12)(cid:12)(cid:12) b ( ◦ u + tU ( n ) ( θ ) + (1 − t ) U ( n − ( θ ) , ◦ s, θ ) (cid:12)(cid:12)(cid:12) dt (cid:19) | h ( θ ) | (cid:46) (cid:12)(cid:12)(cid:12) h ( θ )sin θ (cid:12)(cid:12)(cid:12) sup R (cid:12)(cid:12)(cid:12)(cid:12) b sin θ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r (cid:12)(cid:12)(cid:12) h ( θ ) e Φ (cid:12)(cid:12)(cid:12) sup R (cid:12)(cid:12)(cid:12)(cid:12) be Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15)r (cid:12)(cid:12)(cid:12) h ( θ ) e Φ (cid:12)(cid:12)(cid:12) . Making use of the second estimate in (9.1.17) we then derive, (cid:107) I · h (cid:107) L ( ◦ S ) (cid:46) ◦ (cid:15)r (cid:13)(cid:13)(cid:13)(cid:13) he Φ (cid:13)(cid:13)(cid:13)(cid:13) L ( S ) (cid:46) ◦ (cid:15) (cid:107) h (cid:107) h ( S ) . This shows that the behavior along the axis in (9.6.12) is not an issue. This ends theproof of both Lemma 9.6.3 and Lemma 9.6.2.
Lemma 9.6.4.
Consider equations on ◦ S , of the form, A ( n ) h ( n +1) + V ( n ) h ( n +1) = H ( n ) ,A ( n − h ( n ) + V ( n − h ( n ) = H ( n − . (9.6.13) Indeed the term e θ ( e θ Φ) is quite singular on the axis. CHAPTER 9. GCM PROCEDURE
Also, (cid:90) ◦ S,g/ ( n ) h ( n +1) e Φ n = B ( n ) , (cid:90) ◦ S,g/ ( n − h ( n ) e Φ n − = B ( n − . Then we have, (cid:107) h ( n +1) − h ( n ) (cid:107) h ( ◦ S ) (cid:46) ◦ δr − (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) + r (cid:107) H ( n +1) − H ( n ) (cid:107) L ( ◦ S ) + r − (cid:12)(cid:12)(cid:12) B ( n ) − B ( n − (cid:12)(cid:12)(cid:12) . (9.6.14) Proof.
Setting δh ( n +1) : = h ( n +1) − h ( n ) , δA ( n ) := A ( n ) − A ( n − , δV ( n ) := V ( n ) − V ( n − ,δH ( n +1) : = H ( n +1) − H ( n ) , δB ( n ) := B ( n ) − B ( n − , and subtracting the equations (9.6.13) we derive,( A ( n ) + V ( n ) ) δh ( n +1) = − ( δA ( n ) ) h ( n ) − δV ( n ) h ( n ) + δH ( n +1) . (9.6.15)Also writing (cid:82) ◦ S,g/ ( n ) f = (cid:82) ◦ S f da g/ ( n ) (cid:90) ◦ S,g/ ( n ) δh ( n +1) e Φ n = δB n − (cid:90) ◦ S h ( n ) (cid:16) e Φ n da g/ ( n ) − e Φ n − da g/ ( n − (cid:17) . (9.6.16)Throughout the proof below we make systematic use of the boundedness result of Propo-sition 9.4.8 and comparison Lemma 9.2.3, according to which the spaces h k ( ◦ S, g/ ( n ) ) areall uniformly equivalent to h k ( ◦ S ). Step 1.
Estimates for (cid:107) d (cid:63) / ( n )2 (cid:0) δh ( n +1) (cid:1) (cid:107) h ( ◦ S ) .As in Remark 9.3.7, see also the proof of Proposition 9.4.4 in the previous section, wewrite, A ( n ) = d (cid:63) / n ) d/ n ) = d/ n ) d (cid:63) / n ) − (Ψ ( n ) ) (cid:18) − ρ + 12 ϑϑ (cid:19) . Hence, (cid:90) ( ◦ S,g/ ( n ) ) δh ( n +1) ( A ( n ) + V ( n ) ) δh ( n +1) = (cid:90) ( ◦ S,g/ ( n ) ) | d (cid:63) / ( n )2 ( δh ( n +1) ) | + (cid:18) mr + O ( r − ◦ δ ) (cid:19) (cid:90) ( ◦ S,g/ ( n ) ) | δh ( n +1) | . .6. PROOF OF PROPOSITION 9.4.9 ◦ δ is small, (cid:90) ( ◦ S,g/ ( n ) ) | d (cid:63) / n ) ( δh ( n +1) ) | (cid:46) (cid:90) ( ◦ S,g/ ( n ) ) δh ( n +1) ( A ( n ) + V ( n ) ) δh ( n +1) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( ◦ S,g/ ( n ) ) δh ( n +1) ( δA ( n ) h ( n ) ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:90) ( ◦ S,g/ ( n ) ) (cid:12)(cid:12)(cid:12) δh ( n +1) δV ( n ) h ( n ) (cid:12)(cid:12)(cid:12) + (cid:90) ( ◦ S,g/ ( n ) ) (cid:12)(cid:12)(cid:12) δh ( n +1) δH ( n +1) (cid:12)(cid:12)(cid:12) . We deduce, making use of the comparison Lemma and the boundedness result of Propo-sition 9.4.8, (cid:107) d (cid:63) / ( n )2 δh ( n +1) (cid:107) h ( ◦ S ) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( ◦ S,g/ ( n ) ) δh ( n +1) ( δA ( n ) h ( n ) ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) ◦ δr − (cid:107) δV ( n ) (cid:107) L ( ◦ S ) + (cid:107) δH ( n +1) (cid:107) L ( ◦ S ) (cid:19) (cid:107) δh ( n +1) (cid:107) L ( ◦ S ) . (9.6.17) Step 1a.
We estimate the term (cid:12)(cid:12)(cid:12)(cid:82) ( ◦ S,g/ ( n ) ) δh ( n +1) ( δA ( n ) h ( n ) ) (cid:12)(cid:12)(cid:12) using Lemma 9.6.2 with ψ = δh ( n +1) and h = h ( n ) . We deduce, making use of the boundedness (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( ◦ S,g/ ( n ) ) δh ( n +1) ( δA ( n ) h ( n ) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) (cid:107) h ( n ) (cid:107) h ( ◦ S ) (cid:46) r − ◦ δ (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) . Back to (9.6.17) we deduce, (cid:107) d (cid:63) / ( n )2 δh ( n +1) (cid:107) h ( ◦ S ) (cid:46) (cid:18) ◦ δr − (cid:107) δV ( n ) (cid:107) L ( ◦ S ) + r − ◦ δ (cid:107) ∂ θ ( δ Ψ ( n ) ) (cid:107) h ( ◦ S ) + (cid:107) δH ( n +1) (cid:107) L ( ◦ S ) (cid:19) (cid:107) δh ( n +1) (cid:107) L ( ◦ S ) . (9.6.18) Step 1b.
We estimate (cid:107) δV ( n ) (cid:107) L ( ◦ S ) . Recall that δV ( n ) = (cid:18) κκ − ρ (cid:19) n − (cid:18) κκ − ρ (cid:19) n − . In view of Lemma 9.6.1 and our assumptions
A1-A3 we derive, (cid:107) δV ( n ) (cid:107) L ( ◦ S ) (cid:46) r − (cid:107) ( δU ( n ) , δS ( n ) ) (cid:107) L ( ◦ S ) . CHAPTER 9. GCM PROCEDURE
Therefore, back to (9.6.18), we deduce (cid:107) d (cid:63) / ( n )2 δh ( n +1) (cid:107) h ( ◦ S ) (cid:46) (cid:18) r − ◦ δ (cid:107) ∂ θ ( δ Ψ ( n ) ) (cid:107) h ( ◦ S ) + (cid:107) δH ( n +1) (cid:107) L ( ◦ S ) (cid:19) (cid:107) δh ( n +1) (cid:107) L ( ◦ S ) . (9.6.19) Step 2.
We make use of the equation (9.6.16) (cid:90) ◦ S,g/ ( n ) δh ( n +1) e Φ n = δB n − (cid:90) ◦ S h ( n ) (cid:16) e Φ n da g/ ( n ) − e Φ n − da g/ ( n − (cid:17) from which we deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ◦ S,g/ ( n ) δh ( n +1) e Φ n (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ◦ S h ( n ) (cid:16) e Φ n da g/ ( n ) − e Φ n − da g/ ( n − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) δB n (cid:12)(cid:12)(cid:12) (cid:46) (cid:107) h ( n ) (cid:107) L ( ◦ S ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:32) e Φ n (cid:112) γ ( n ) (cid:112) γ (0) − e Φ n − (cid:112) γ ( n − (cid:112) γ (0) (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( ◦ S ) + (cid:12)(cid:12)(cid:12) δB n (cid:12)(cid:12)(cid:12) . On the other hand, (cid:32) e Φ n (cid:112) γ ( n ) (cid:112) γ (0) − e Φ n − (cid:112) γ ( n − (cid:112) γ (0) (cid:33) = (cid:16) e Φ n − e Φ n − (cid:17) (cid:112) γ ( n ) (cid:112) γ (0) + e Φ n − (cid:32) (cid:112) γ ( n ) (cid:112) γ (0) − (cid:112) γ ( n − (cid:112) γ (0) (cid:33) . Hence, proceeding as before, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:32) e Φ n (cid:112) γ ( n ) (cid:112) γ (0) − e Φ n − (cid:112) γ ( n − (cid:112) γ (0) (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( ◦ S ) (cid:46) (cid:13)(cid:13)(cid:13) e Φ n − e Φ n − (cid:13)(cid:13)(cid:13) L ( ◦ S ) + r − (cid:13)(cid:13) γ n − γ n − (cid:13)(cid:13) L ( ◦ S ) (cid:46) r (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) . Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ◦ S,g/ ( n ) δh ( n +1) e Φ n (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δr (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) + (cid:12)(cid:12)(cid:12) δB n (cid:12)(cid:12)(cid:12) . (9.6.20) Step 3.
Making use of Lemma 9.1.4 together with (9.6.19), (9.6.20) and the comparisonLemma and we deduce, (cid:107) δh ( n +1) (cid:107) L ( ◦ S,g/ ( n ) ) (cid:46) r (cid:107) d (cid:63) / ( n )2 δh ( n +1) (cid:107) h ( ◦ S,g/ ( n ) ) + r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ◦ S,g/ ( n ) δh ( n +1) e Φ n (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r (cid:107) d (cid:63) / ( n )2 δh ( n +1) (cid:107) h ( ◦ S ) + ◦ δr − (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) + r − (cid:12)(cid:12)(cid:12) δB n (cid:12)(cid:12)(cid:12) (cid:46) r (cid:18) r − ◦ δ (cid:107) ∂ θ ( δ Ψ ( n ) ) (cid:107) h ( ◦ S ) + (cid:107) δH ( n +1) (cid:107) L ( ◦ S ) (cid:19) / (cid:107) δh ( n +1) (cid:107) / L ( ◦ S ) + ◦ δr − (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) + r − (cid:12)(cid:12)(cid:12) δB n (cid:12)(cid:12)(cid:12) . .6. PROOF OF PROPOSITION 9.4.9 (cid:107) δh ( n +1) (cid:107) L ( ◦ S,g/ ( n ) ) (cid:46) ◦ δr − (cid:107) ∂ θ (cid:0) Ψ ( n ) − Ψ ( n − (cid:1) (cid:107) h ( ◦ S ) + r (cid:107) δH ( n +1) (cid:107) L ( ◦ S ) + r − (cid:12)(cid:12)(cid:12) δB n (cid:12)(cid:12)(cid:12) which, together with (9.6.19), ends the proof of Lemma 9.6.4. (9.6.5) , (9.6.6) We are now in position to prove (9.6.5) (9.6.6).
Step 1.
With start by estimating δh ( n +1) , δh ( n +1) . To this end, we need to apply Lemma9.6.4 to the equations for δw ( n +1) , δh ( n +1) , δh ( n +1) , derived from the first three equationsin (9.6.1), and estimate the corresponding δH ( n +1) on the right-hand side. This is tediousbut straightforward and one derives, in the case of estimates of δw ( n +1) (cid:107) δH ( n +1) (cid:107) L ( ◦ S ) (cid:46) ◦ (cid:15) (cid:40) (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δe ( n +1) (cid:107) h ( ◦ S ) (cid:41) and in the case of δh ( n +1) , δh ( n +1) (cid:107) δH ( n +1) (cid:107) L ( ◦ S ) (cid:46) (cid:107) δw ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:40) (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δe ( n +1) (cid:107) h ( ◦ S ) (cid:41) . Remark 9.6.5.
Note that the presence of the inverse operators ( A ( n ) ) − in the right-hand side of the equations for δh ( n +1) , δh ( n +1) do not create any difficulties when takingdifferences. Indeed we can write, ( A ( n ) ) − − ( A ( n − ) − = ( A ( n ) ) − (cid:0) A ( n − − A ( n ) (cid:1) ( A ( n − ) − and estimate the difference δA ( n ) = A ( n ) − A ( n − as in the proof of Lemma 9.6.4. We infer from Lemma 9.6.4 and the above estimates (cid:107) δw ( n +1) (cid:107) h ( ◦ S ) (cid:46) ◦ (cid:15) (cid:40) (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δe ( n +1) (cid:107) h ( ◦ S ) (cid:41) CHAPTER 9. GCM PROCEDURE and (cid:107) δh ( n +1) , δh ( n +1) (cid:107) h ( ◦ S ) (cid:46) (cid:107) δw ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:40) (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δe ( n +1) (cid:107) h ( ◦ S ) (cid:41) . This thus yields (cid:107) δh ( n +1) , δh ( n +1) (cid:107) h ( ◦ S ) (cid:46) ◦ (cid:15) (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) δe ( n +1) (cid:107) h ( ◦ S ) . (9.6.21) Step 2.
Next, we estimate d (cid:63) / δe ( n +1) . We make use of the last equation in (9.6.1) whichwe write int the form d (cid:63) / ( n ) e ( n +1) = H ( n +1) with, H ( n +1) = 34 ( A ( n ) ) − ( ρ ( n ) w ( n +1) ) − h ( n +1) ω ( n ) + h ( n +1) ω ( n ) − h ( n +1) κ ( n ) + 14 f ( n +1) κ ( n ) + ( A ( n ) ) − (cid:0) − d (cid:63) / ( n ) ˇ µ ( n ) + (Ψ ( n ) ) Err ( n +1)3 (cid:1) . Hence, d (cid:63) / ( n ) ( δe ( n +1) ) + ( d (cid:63) / ( n ) − d (cid:63) / ( n − ) e ( n ) = δH ( n +1) which can be written in the form, (cid:112) γ (0) (cid:112) γ ( n ) d (cid:63) / ( δe ( n +1) ) = − ( d (cid:63) / ( n ) − d (cid:63) / ( n − ) e ( n ) + δH ( n +1) = − (cid:32) (cid:112) γ (0) (cid:112) γ ( n ) − (cid:112) γ (0) (cid:112) γ ( n − (cid:33) d (cid:63) / e ( n ) + δH ( n +1) or, d (cid:63) / ( δe ( n +1) ) = (cid:32) − (cid:112) γ ( n ) (cid:112) γ ( n − (cid:33) d (cid:63) / e ( n ) + (cid:112) γ ( n ) (cid:112) γ (0) δH ( n +1) . This yields (cid:107) d (cid:63) / ( δe ( n +1) ) (cid:107) L ( ◦ S ) (cid:46) ◦ (cid:15) (cid:107) γ ( n ) − γ ( n − (cid:107) L ( ◦ S ) + (cid:107) δH ( n +1) (cid:107) L ( ◦ S ) .6. PROOF OF PROPOSITION 9.4.9 P ( n ) provided by Proposition 9.4.8 and the bound(9.1.15) for (Ω , b, γ ). γ ( n ) − γ ( n − can be estimated as in the proof of Lemma 9.6.4, andhence (cid:107) d (cid:63) / ( δe ( n +1) ) (cid:107) L ( ◦ S ) (cid:46) ◦ (cid:15) (cid:16) (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) (cid:17) + (cid:107) δH ( n +1) (cid:107) L ( ◦ S ) . The differences H ( n +1) − H ( n ) can also be easily estimated and deduce, (cid:107) H ( n ) − H ( n − (cid:107) L ( ◦ S ) (cid:46) (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:16) (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) (cid:17) . We obtain (cid:107) d (cid:63) / ( δe ( n +1) ) (cid:107) L ( ◦ S ) (cid:46) (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) (9.6.22)+ ◦ (cid:15) (cid:16) (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) (cid:17) . Step 3.
Next, we estimate the average of δe ( n +1) . Recall from (9.6.3) e ( n +1) ◦ S,g/ ( n ) = (cid:18) − r S ( n ) r (cid:19) ( n ) ◦ S,g/ ( n ) − r S ( n ) (cid:18) ˇ κ + κ − r (cid:19) ( n ) ◦ S,g/ ( n ) + Err ( n +1)8 ◦ S,g/ ( n ) . and e ( n ) ◦ S,g/ ( n ) = (cid:18) − r S ( n − r (cid:19) ( n − ◦ S,g/ ( n − − r S ( n − (cid:18) ˇ κ + κ − r (cid:19) ( n ) ◦ S,g/ ( n − + Err ( n )8 ◦ S,g/ ( n − . Taking the difference, recalling that we have in the ( θ, ϕ ) coordinates system dvolg/ ( n ) = (cid:112) γ ( n ) e Φ n dθdϕ, r S ( n ) = (cid:90) ◦ S (cid:112) γ ( n ) e Φ n dθdϕ and dvolg/ ( n − = (cid:112) γ ( n − e Φ n − dθdϕ, r S ( n − = (cid:90) ◦ S (cid:112) γ ( n − e Φ n − dθdϕ and using the uniform bound of P ( n ) provided by Proposition 9.4.8 and the bounds A1 for ˇΓ, we infer (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δe ( n +1) ◦ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) (cid:110) (cid:107) γ ( n ) − γ ( n − (cid:107) L ( ◦ S ) + (cid:107) Φ n − Φ n − (cid:107) L ( ◦ S ) + (cid:107) δ Err ( n +1)6 (cid:107) L ( ◦ S ) (cid:111) . CHAPTER 9. GCM PROCEDURE
Arguing as above, we deduce (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δe ( n +1) ◦ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) (cid:110) (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) (cid:107) h ( ◦ S ) + (cid:107) δe ( n +1) (cid:107) h ( ◦ S ) + (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) (cid:111) . Together with (9.6.21) (9.6.22), and an elliptic estimate for d (cid:63) / , we infer (cid:107) δh ( n +1) , δh ( n +1) , δe ( n +1) (cid:107) h ( ◦ S ) (cid:46) ◦ (cid:15) (cid:107) U ( n ) − U ( n − (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) S ( n ) − S ( n − (cid:107) h ( ◦ S ) (9.6.23)which concludes the proof of (9.6.5). Step 4.
Finally, we focus on (9.6.6). Recall (9.6.4) ς n ∂ θ U (1+ n ) = ( γ ( n ) ) / h (1+ n ) (cid:18) h (1+ n ) h (1+ n ) (cid:19) ,∂ θ S (1+ n ) − ς n Ω n ∂ θ U (1+ n ) = 12 ( γ ( n ) ) / h (1+ n ) ,γ ( n ) = γ n + (cid:0) ς n (cid:1) (cid:18) Ω n + 14 ( b n ) γ n (cid:19) ( ∂ θ U ( n ) ) − ς n ∂ θ U ( n ) ∂ θ S ( n ) − γ n ς n b n ∂ θ U ( n ) ,U (1+ n ) (0) = S (1+ n ) (0) = 0 . (9.6.24)Taking the difference and arguing as above, we derive (cid:107) ∂ θ δU (1+ n ) (cid:107) h ( ◦ S ) (cid:46) (cid:107) δh ( n +1) , δh ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) γ ( n ) − γ ( n − (cid:107) h ( ◦ S ) , (cid:107) ∂ θ δS (1+ n ) (cid:107) h ( ◦ S ) (cid:46) (cid:107) δU (1+ n ) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) δS (1+ n ) (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) , δh ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) γ ( n ) − γ ( n − (cid:107) h ( ◦ S ) . Since δU (1+ n ) = δS (1+ n ) = 0, we deduce (cid:107) δU (1+ n ) (cid:107) h ( ◦ S ) (cid:46) (cid:107) δh ( n +1) , δh ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) γ ( n ) − γ ( n − (cid:107) h ( ◦ S ) , (cid:107) δS (1+ n ) (cid:107) h ( ◦ S ) (cid:46) (cid:107) δU (1+ n ) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) δS (1+ n ) (cid:107) h ( ◦ S ) + (cid:107) δh ( n +1) , δh ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) γ ( n ) − γ ( n − (cid:107) h ( ◦ S ) and hence (cid:107) δU ( n +1) , δS ( n +1) (cid:107) h ( ◦ S ) (cid:46) (cid:107) δh ( n +1) , δh ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) δU ( n ) , δS ( n ) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) γ ( n ) − γ ( n − (cid:107) h ( ◦ S ) . .7. A COROLLARY TO THEOREM 9.4.2 γ ( n ) − γ ( n − as above, we infer (cid:107) δU ( n +1) , δS ( n +1) (cid:107) h ( ◦ S ) (cid:46) (cid:107) δh ( n +1) , δh ( n +1) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) δU ( n ) , δS ( n ) (cid:107) h ( ◦ S ) + ◦ (cid:15) (cid:107) δU ( n ) , δS ( n ) (cid:107) h ( ◦ S ) . This is the desired estimate (9.6.6) and hence concludes the proof of Proposition 9.4.9.
The following result is a simple Corollary of Theorem 9.4.2.
Theorem 9.7.1 (Existence of GCM spheres) . In addition to the assumptions of Theorem9.4.2 we assume that, for any background sphere S in R , r (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. (9.7.1) Then there exists a unique GCM sphere S , which is a deformation of ◦ S , such that theGCMS conditions hold true d/ S ,(cid:63) d/ S ,(cid:63) κ S = d/ S ,(cid:63) d/ S ,(cid:63) µ S = 0 , κ S = 2 r S , (cid:90) S β S e Φ = 0 , (cid:90) S e S θ ( κ S ) e Φ = 0 . (9.7.2) Moreover all other estimates of Theorem 9.4.2 hold true.Proof.
The proof of the theorem follows easily in view of Theorem 9.4.2 and the followinglemma.
Lemma 9.7.2.
Let S be a deformation of ◦ S as in Theorem 9.4.2 with Λ = (cid:82) S f e Φ , Λ = (cid:82) S f e Φ . The following identities hold true. Λ = r m (cid:18)(cid:90) ◦ S βe Φ − (cid:90) S β S e Φ (cid:19) + F (Λ , Λ) , Λ = r mr ) (cid:18)(cid:90) ◦ S ( e θ κ ) e Φ − (cid:90) S ( e S θ κ S ) e Φ + Υ (cid:90) ◦ S e θ ( κ ) e Φ (cid:19) + Υ2(1 + mr ) Λ + F (Λ , Λ) , (9.7.3) where F , F are continuous in Λ , Λ , with F (0 ,
0) = F (0 ,
0) = 0 , verifying the estimates, (cid:12)(cid:12) F | + (cid:12)(cid:12) F (cid:12)(cid:12) (cid:46) ◦ (cid:15) ◦ δr . In fact smooth. CHAPTER 9. GCM PROCEDURE
Proof.
To prove (9.7.3) we start with the change of frame formula, β S = e a (cid:18) β + 32 ρf (cid:19) + e a Err( β, β S ) , Err( β, β S ) = 12 f α + l.o.t.We write β S = β + 32 ρf + ( e a − (cid:18) β + 32 ρf (cid:19) + e a Err( β, β S )= β + 32 (cid:18) − mr (cid:19) f + 32 (cid:18) ρ + 2 mr (cid:19) f + ( e a − (cid:18) β + 32 ρf (cid:19) + e a Err( β, β S )and deduce, β S + 3 m S ( r S ) f = β + Err (cid:48) ( β, β S )with error term Err (cid:48) ( β, β S ),Err (cid:48) ( β, β S ) = (cid:18) m S ( r S ) − mr f (cid:19) + 32 (cid:18) ρ + 2 mr (cid:19) f + ( e a − (cid:18) β + 32 ρf (cid:19) + e a Err( β, β S ) . Making use of the assumptions
A1-A3 , the estimates of Theorem 9.4.2 for ( f, f , a ) aswell as the bounds for ◦ r − r S , ◦ m − m S we deduce (cid:12)(cid:12)(cid:12) Err (cid:48) ( β, β S ) (cid:12)(cid:12)(cid:12) (cid:46) r − ◦ δ ◦ (cid:15). Thus, 3 m S ( r S ) (cid:90) S f e Φ = (cid:90) S βe Φ − (cid:90) S β S e Φ + (cid:90) S Err (cid:48) ( β, β S ) e Φ = (cid:90) ◦ S βe Φ − (cid:90) S β S e Φ + (cid:90) S Err (cid:48) ( β, β S ) e Φ + (cid:16) (cid:90) S βe Φ − (cid:90) ◦ S βe Φ (cid:17) or, 3 mr (cid:90) S f e Φ = (cid:90) S βe Φ − (cid:90) S β S e Φ + (cid:90) S Err (cid:48) ( β, β S ) e Φ + (cid:16) (cid:90) S βe Φ − (cid:90) S βe Φ (cid:17) + (cid:16) mr − m S ( r S ) (cid:17) (cid:90) S f e Φ . Here ( r, m ) represents the area radius and Hawking of ◦ S while ( r S , m S ) represent the area radius andHawking of S . Since | r S r − | (cid:46) ◦ δ , | m S − m | (cid:46) ◦ δ we can interchange freely r S with r and m S with m . .7. A COROLLARY TO THEOREM 9.4.2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S Err (cid:48) ( β, β S ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ◦ δ ◦ (cid:15). Also, proceeding exactly as in Corollary 9.2.5 we deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ − (cid:90) ◦ S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ (cid:18) sup R ◦ r (cid:12)(cid:12) d ≤ (cid:37) ( βe Φ ) (cid:12)(cid:12) + sup R ◦ r (cid:12)(cid:12) e ( βe Φ (cid:1) | (cid:19) . Thus, in view of the assumptions
A1-A3 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S βe Φ − (cid:90) ◦ S βe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ ◦ (cid:15)r . (9.7.4)We deduce, Λ = r m (cid:16) (cid:90) ◦ S βe Φ − (cid:90) S β S e Φ (cid:17) + F (Λ , Λ)where the error term F (Λ , Λ) is a continuous function of Λ , Λ verifying the estimate, (cid:12)(cid:12)(cid:12) F (Λ , Λ) (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) ◦ δr . We also recall, see Lemma 9.3.4 e S θ ( κ S ) = e θ κ − d/ S ,(cid:63) d/ S f − κe S θ a − κ ( f κ + f κ ) + κ ( f ω − ωf ) + f ρ + Err( e S θ κ S , e θ κ )where,Err( e S θ κ S , e θ κ ) = ( e − a − (cid:16) e θ κ − d/ S ,(cid:63) d/ S f + 12 f e κ + 12 f e κ (cid:17) + e − a (cid:20) e S θ Err( κ, κ S ) + e S θ ( a ) (cid:16) d/ S f + Err( κ, κ S ) (cid:17) + 12 f f e θ κ + 18 f f e κ (cid:21) + 12 f (cid:18) d/ η − ϑ ϑ + 2( ξξ + η ) (cid:19) + 12 f f e θ κ + 18 f f e κ + 12 f (cid:18) d/ ξ − ϑ + 2( η + η − ζ ) ξ (cid:19) . Making use of the identity d/ S ,(cid:63) d/ S = d/ S d/ S ,(cid:63) + 2 K we deduce, e S θ ( κ S ) + (cid:18) κκ − ρ + 2 K (cid:19) f = e θ κ − d/ S d/ S ,(cid:63) f + κe S θ a − κ f − κ ωf + Err( e S θ κ S , e θ κ ) . CHAPTER 9. GCM PROCEDURE
Writing, k = r + ( κ − r ), κ = − r + ( κ + r , ρ = − mr + ( ρ + mr ), K = r + ( K − r )14 κκ − ρ + 2 K = 1 r + 4 mr + 12 r (cid:18) κ + 2Υ r (cid:19) − Υ2 r (cid:18) κ − r (cid:19) + (cid:18) ρ + 2 mr (cid:19) + 2 (cid:18) K − r (cid:19) = 1 r + 4 mr + O ( ◦ (cid:15)r − ) . Also, κ = − r + (cid:18) κ + 2Υ r (cid:19) = − r + O ( r − ◦ (cid:15) ) ,κ ω = − m Υ r + O ( r − ◦ (cid:15) ) , and, since a, f, f = O ( r − ◦ δ ), Err( e S θ κ S , e θ κ ) = O ( r − ◦ δ ◦ (cid:15) ) . We deduce, e S θ ( κ S ) + (cid:18) r + 4 mr (cid:19) f = e θ κ − d/ d (cid:63) / f − r e S θ a + 2 m Υ r f + Err with error term (cid:12)(cid:12)(cid:12) Err (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) ◦ δr − . Projecting on e Φ and proceeding as before, (cid:18) r + 4 mr (cid:19) (cid:90) S f e Φ = (cid:90) ◦ S ( e θ κ ) e Φ − (cid:90) S ( e S θ κ S ) e Φ − r (cid:90) S ( e S θ a ) e Φ + 2 m Υ r (cid:90) S f e Φ + I (Λ , Λ) (9.7.5)where the error term I is continuous in Λ , Λ and verifies the estimate (cid:12)(cid:12)(cid:12) I (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) ◦ δ. We now calculate (cid:82) S ( e S θ a ) e Φ . In view of formula (9.4.3) we have,2 r S (cid:90) S e S θ ( a ) e Φ = − (cid:90) S e θ ( κ ) e Φ + 14 (cid:90) S κ f e Φ − (cid:90) S (cid:18) κκ + κω + 3 ρ (cid:19) f e Φ + Err = − (cid:90) ◦ S e θ ( κ ) e Φ + 1 r Λ − r (1 + mr )Λ + I (Λ , Λ) .7. A COROLLARY TO THEOREM 9.4.2 (cid:12)(cid:12)(cid:12) I (Λ , Λ) (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) ◦ δ. Indeed, using once more Corollary 9.2.5, we note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e θ ( κ ) e Φ − (cid:90) ◦ S e θ ( κ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) ◦ δ. All other error terms are easily estimated.Back to (9.7.5) we deduce, (cid:18) r + 4 mr (cid:19) Λ = (cid:90) ◦ S ( e θ κ ) e Φ − (cid:90) S ( e S θ κ S ) e Φ − r (cid:90) S ( e S θ a ) e Φ + 2 m Υ r Λ + I (Λ , Λ)= (cid:90) ◦ S ( e θ κ ) e Φ − (cid:90) S ( e S θ κ S ) e Φ + 2 m Υ r Λ + I (Λ , Λ) − Υ (cid:18) − (cid:90) ◦ S e θ ( κ ) e Φ + 1 r Λ − r (1 + mr )Λ + I (Λ , Λ) (cid:19) = (cid:90) ◦ S ( e θ κ ) e Φ − (cid:90) S ( e S θ κ S ) e Φ + Υ (cid:90) ◦ S e θ ( κ ) e Φ − Υ r Λ + Υ r (1 + 3 mr )Λ+ I (Λ , Λ) − Υ I (Λ , Λ) . Hence, 2 r (cid:16) mr (cid:17) Λ = (cid:90) ◦ S ( e θ κ ) e Φ − (cid:90) S ( e S θ κ S ) e Φ + Υ (cid:90) ◦ S e θ ( κ ) e Φ + Υ r (1 + 3 mr )Λ+ I (Λ , Λ) − Υ I (Λ , Λ) . Thus,Λ = r mr ) (cid:18)(cid:90) ◦ S ( e θ κ ) e Φ − (cid:90) S ( e S θ κ S ) e Φ + Υ (cid:90) ◦ S e θ ( κ ) e Φ (cid:19) + Υ2(1 + mr ) Λ + F (Λ , Λ)with error term F (Λ , Λ) continuous in Λ , Λ and verifying the estimate, (cid:12)(cid:12)(cid:12) F (Λ , Λ) (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) ◦ δr . This ends the proof of the lemma.Under the assumptions of the theorem the systemΛ = r m (cid:90) ◦ S βe Φ + F (Λ , Λ) , Λ = r mr ) (cid:18)(cid:90) ◦ S ( e S θ κ ) e Φ + Υ (cid:90) ◦ S e θ ( κ ) e Φ (cid:19) + Υ2(1 + mr ) Λ + F (Λ , Λ) , CHAPTER 9. GCM PROCEDURE has a unique solution Λ , Λ verifying the estimate | Λ | + | Λ | (cid:46) ◦ δr . Taking Λ = Λ , Λ = Λ in (9.7.3) we deduce, (cid:90) S β S e Φ = 0 , (cid:90) S e S θ ( κ S ) e Φ = 0 , as stated. Corollary 9.7.3 (Rigidity II) . Assume that the spacetime region R satisfies assumptions A1 − A3 and (9.4.5) , as well as (9.7.1) . Assume given a sphere S ⊂ R endowed with acompatible frame e S , e S θ , e S which verifies the GCM conditions κ S = 2 r S , d (cid:63) / S d (cid:63) / S κ S = d (cid:63) / S d (cid:63) / S µ S = 0 . (9.7.6) In addition, we have r (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S β S e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e S θ ( κ S ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. (9.7.7) Then the transition functions ( f, f , log λ ) from the background frame of R to that of S verifies the estimates (cid:107) ( f, f , log( λ )) (cid:107) h smax − ( S ) (cid:46) ◦ δ. The proof is an immediate consequence of Lemma 9.7.2 and the rigidity result of Corollary9.4.5. Note that in the corollary, the sphere S is reinterpreted as a deformation spherefrom the unique background sphere sharing the same south pole. We are ready to state our main result concerning the construction of GCM hypersurfaces.
Theorem 9.8.1.
Let a fixed spacetime region R verifying assumptions A1 − A3 and (9.4.5) . In addition we assume that, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( u,s ) ηe φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ◦ δ, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( u,s ) ξe φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ◦ δ, (9.8.1) .8. CONSTRUCTION OF GCM HYPERSURFACES and, everywhere on Σ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ς + Ω (cid:19) (cid:12)(cid:12)(cid:12) SP − − mr (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ (9.8.2) where SP denotes the South pole, i.e. θ = 0 relative to the adapted geodesic coordinates u, s, θ .Let S = S [ ◦ u, ◦ s, Λ , Λ ] be a fixed GCMS provided by Theorem 9.4.2. Then, there existsthen a unique, local , smooth, Z -invariant spacelike hypersurface Σ passing through S ,a scalar function u S defined on Σ , whose level surfaces are topological spheres denoted by S , and a smooth collection of constants Λ S , Λ S verifying, Λ S = Λ , Λ S = Λ , such that the following conditions are verified:1. The surfaces S of constant u S verifies all the properties stated in Theorem 9.4.2 forthe prescribed constants Λ S , Λ S . In particular they come endowed with null frames ( e S , e S θ , e S ) such thati. For each S the GCM conditions (9.4.1) with Λ = Λ S , Λ = Λ S , are verified.ii. The transition functions ( f, f , a = log λ ) verify the estimates (9.4.8) .iii. The transversality conditions (9.4.12) are verified.iv. The corresponding Ricci and curvature coefficients verify the estimates (9.4.11) and (9.4.14) .2. Denoting r S to be the area radius of the spheres S we have, for some constant c ∗ , u S + r S = c ∗ , along Σ . (9.8.3)
3. Let ν S be the unique vectorfield tangent to the hypersurface Σ , normal to S , andnormalized by g ( ν S , e S ) = − . There exists a unique scalar function a S on Σ suchthat ν S is given by ν S = e S + a S e S . The following normalization condition holds true at the South Pole SP of everysphere S , i.e. at θ = 0 , a S (cid:12)(cid:12)(cid:12) SP = − − m S r S . (9.8.4) i.e. in a neighborhood of S . CHAPTER 9. GCM PROCEDURE
4. We extend u S and r S in a small neighborhood of Σ such that the following transver-sality conditions are verified on Σ , e S ( u S ) = 0 , e ( r s ) = r S κ S = 1 . (9.8.5)
5. In view of (9.8.5) the Ricci coefficients η S , ξ S are well defined for every S ⊂ Σ andverify (cid:90) S η S e Φ = (cid:90) S ξ S e Φ = 0 . (9.8.6)
6. The following estimates hold true for all k ≤ s max , (cid:107) η S (cid:107) h k ( S ) (cid:46) ◦ (cid:15), (9.8.7) (cid:107) ξ S (cid:107) h k ( S ) (cid:46) ◦ (cid:15), (9.8.8) (cid:13)(cid:13)(cid:13)(cid:13) a S + 1 + 2 m S r S (cid:13)(cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15). (9.8.9) The e S derivatives of ˇ κ S , ϑ S , ζ S , ˇ κ S , ϑ S , α S , β S , ˇ ρ S , µ S , β S are well defined on Σ andwe have, for all k ≤ s max − (cid:107) e S (ˇ κ S , ϑ S , ζ S , ˇ κ S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) e S ( ϑ S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15), (cid:107) e S (cid:0) α S , β S , ˇ ρ S , µ S (cid:1) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) e S ( β S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) e S ( α S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15). (9.8.10)
7. The transition functions from the background foliation to that of Σ verify (cid:107) d ≤ s max +1 ( f, f , log λ ) (cid:107) L ( S ) (cid:46) ◦ δ. (9.8.11) Corollary 9.8.2 (Rigidity III) . Let a fixed spacetime region R verifying assumptions A1 − A3 and the small GCM conditions (9.4.5) . Assume given a GCM hypersurface Σ ⊂ R foliated by surfaces S such that κ S = 2 r S , d (cid:63) / S d (cid:63) / S κ S = d (cid:63) / S d (cid:63) / S µ S = 0 , (cid:90) S η S e Φ = 0 , (cid:90) S ξ S e Φ = 0 . Here the average of κ S is taken on S . In view of the GCM conditions (9.8.13) we deduce e S ( r S ) = 1. .8. CONSTRUCTION OF GCM HYPERSURFACES Assume in addition that for a specific sphere S on Σ , the transition functions f, f fromthe background foliation to S verify (cid:90) S f e Φ = O ( ◦ δ ) , (cid:90) S f e Φ = O ( ◦ δ ) . (9.8.12) Then, for all derivatives of the transition functions along S , (cid:107) d ≤ s max +1 ( f, f , log λ ) (cid:107) L ( S ) (cid:46) ◦ δ. We give below the proof of Theorem 9.8.1 and a short discussion of the proof of theCorollary. Σ As stated in the theorem we assume given a spacetime region R = {| u − ◦ u | ≤ δ R , | s − ◦ s | ≤ δ R } (see definition(9.1.6)) endowed with a background foliation such that the condition A1-A3 hold true. We also assume given a deformation sphere S := S [ ◦ u, ◦ s, Λ , Λ ]of a given sphere ◦ S = S ( ◦ u, ◦ s ) of the background foliation which verify the conclusions ofTheorem 9.4.2. We then proceed to construct, in a small neighborhood of S , a spacelikehypersurface Σ initiating at S verifying all the desired properties mentioned above. Inwhat follows we outline the main steps in the construction. Step 1.
According to Theorem 9.4.2, for every value of the parameters ( u, s ) in R (i.e.such that the background spheres S ( u, s ) ⊂ R ) and every real numbers (Λ , Λ), there existsa unique GCM sphere S [ u, s, Λ , Λ], as a Z -polarized deformation of S ( u, s ). In particularthe following are verified: • S coincides with S ( u, s ) at their south poles (i.e. for θ = 0 in the adapted coordi-nates). • On S , the following GCMS conditions hold κ S = 2 r S , d/ S ,(cid:63) d/ S ,(cid:63) κ S = 0 , d/ S ,(cid:63) d/ S ,(cid:63) µ S = 0 , (9.8.13)18 CHAPTER 9. GCM PROCEDURE (cid:90) S f e Φ = Λ S , (cid:90) S f e Φ = Λ S , (9.8.14)where ( f, f , λ ) are the transition parameters of the frame transformation from thebackground frame ( e , e θ , e ) to the adapted frame ( e S , e S θ , e S ). The constants Λ S , Λ S depend smoothly on the surfaces S andΛ S = Λ , Λ S = Λ . • There is a map Ξ : S ( u, s ) −→ S given byΞ : ( u, s, θ ) = (cid:16) u + U ( θ, u, s, Λ , Λ) , s + S ( θ, u, s, Λ , Λ) , θ (cid:17) (9.8.15)with U, S vanishing at θ = 0. • The transversality conditions (9.4.12) hold, i.e. ξ S = ω S = ζ S + η S = 0. Note thatthese specify the e S derivatives of ( f, f , λ ) on S . • The Ricci coefficients κ S , κ S , ϑ S , ϑ S , ζ S are well defined on each sphere S of Σ , andhence on Σ . The same holds true for all curvature coefficients α S , β S , ρ S , β S , α S .Taking into account our transversality condition we remark that the only ill definedRicci coefficients are η S , ξ S , ω S . • Let ν S be the unique vectorfield tangent to the hypersurface Σ , normal to S , andnormalized by g ( ν S , e S ) = −
2. There exists a unique scalar function a S on Σ suchthat ν S is given by ν S = e S + a S e S . We deduce that the quantities g ( D ν S e S , e S θ ) = 2 η S + 2 a S ξ S = 2 η S ,g ( D ν S e S , e S θ ) = 2 ξ S + 2 a S η S = 2( ξ S − a S ζ S ) ,g ( D ν S e S , e S ) = 4 ω S − a S ω S = 4 ω S , are well defined on Σ . Thus the scalar a S allows us to specify the remaining Riccicoefficients, η S , ξ S , ω S along Σ , which we do below. Consequently the Hawking mass m S is also well defined. .8. CONSTRUCTION OF GCM HYPERSURFACES Σ We analyze the extrinsic properties of the hypersurfaces Σ defined in Step 1. Step 2.
We define the scalar function u S on Σ as u S := c − r S , (9.8.16)where r S is the area radius of S and the contant c is such that u S (cid:12)(cid:12) S = ◦ u , i.e. c = ◦ u + r S (cid:12)(cid:12) S . Step 3.
We extend u S and r S in a small neighborhood of Σ such that the followingtransversality conditions are verified. e S ( u S ) = 0 , e S ( r S ) = r S κ S , (9.8.17)where the average of κ S is taken on S . In view of the GCM conditions (9.8.13) we deduce e S ( r S ) = 1. Step 4.
Note that e S ( u S , r S ) remain undetermined. On the other hand, since e S θ ( u S ) = e S θ ( r S ) = 0, we deduce in view of (9.8.17) e S θ ( e S ( u S )) = [ e S θ , e S ] u S = (cid:20)
12 ( κ S + ϑ S ) e S θ + ( ζ S − η S ) e S + ξ S e S (cid:21) u S = ( ζ S − η S ) e S ( u S ) ,e S θ ( e S ( r S )) = [ e S θ , e S ] r S = (cid:20)
12 ( κ S + ϑ S ) e S θ + ( ζ S − η S ) e S + ξ S e S (cid:21) r S = ( ζ S − η S ) e S ( r S ) + ξ S . Thus introducing the scalars ς S := 2 e S ( u S ) , (9.8.18)and, A S := 2 r S ( e S ( r S ) + Υ S ) , (9.8.19)we deduce, e S θ (log ς S ) = ( η S − ζ S ) , (9.8.20) e S θ ( A S ) = − S r S ( ζ S − η S ) − r S ξ S + ( ζ S − η S ) A S . (9.8.21)20 CHAPTER 9. GCM PROCEDURE
We infer that e S θ (log ς S ) and e S θ ( A S ) are determined in terms of η, ξ . Step 5.
In view of the definition of ν S and ς S we make use of (9.8.17) to deduce ν S ( u S ) = e S ( u S ) + a S e S ( u S ) = 2 ς S . On the other hand, since u S := c − r S along Σ , ν S ( u S ) = − ν S ( r S ) = − e S ( r S ) − a S e S ( r S ) = Υ S − r S A S − a S and therefore, a S = − ς S + Υ S − r S A S = − ς S − Ω S (9.8.22)where, Ω S := e S ( r S ) = − Υ S − r S A S . (9.8.23) Step 6.
The following lemma will be used, in particular , to determine the A S . Lemma 9.8.3.
For every scalar function h we have the formula ν S (cid:18)(cid:90) S h (cid:19) = ( ς S ) − (cid:90) S ς S (cid:0) ν S ( h ) + ( κ S + a S κ S ) h (cid:1) . (9.8.24) In particular ν S ( r S ) = r S ς S ) − ς S ( κ S + a S κ S ) where the average is with respect to S .Proof. We consider the coordinates u S , θ S along Σ with ν S ( θ S ) = 0. In these coordinateswe have, ν S = 2 ς S ∂ u S . The lemma follows easily by expressing the volume element of the surfaces S ⊂ Σ withrespect to the coordinates u S , θ S (see also the proof of Proposition 2.2.9). It will also be used below to derive equations for Λ , Λ. .8. CONSTRUCTION OF GCM HYPERSURFACES Step 7.
Note that the GCM condition κ S = r S together with the definition of theHawking mass implies that, κ S = − S r S , Υ S = 1 − m S r S , where the average is taken with respect to S . Thus in view of Lemma 9.8.3 we deduce e S ( r S ) + a S = ν S ( r S ) = r S ς S ) − ς S ( κ S + a S κ S ) = r S ς S ) − (cid:0) ς S κ S + ˇ ς S ˇ κ S (cid:1) + ( ς S ) − ς S a S = − Υ S ( ς S ) − ς S + r S ς S ) − ˇ ς S ˇ κ S + ( ς S ) − ς S a S . Since according to (9.8.19) e S ( r S ) = − Υ S + r S A S , we deduce A S = 2 r S (cid:18) Υ S − a S − Υ S ( ς S ) − ς S + r S ς S ) − ˇ ς S ˇ κ S + ( ς S ) − ς S a S (cid:19) . In particular, multiplying by ς S and taking the average, we infer ς S A S = ˇ ς S ˇ κ S , and hence A S = 1 ς S (cid:16) ˇ ς S ˇ κ S − ˇ ς S ˇ A S (cid:17) . (9.8.25) Step 8.
We summarize the results in Steps 1-7 in the following.
Proposition 9.8.4.
Let Σ be a smooth spacelike hypersurface foliated by framed spheres ( S , e S , e S θ .e S ) whose Ricci coefficients verify the GCM condition κ S = r S and transversalitycondition (9.4.12) . Define u S as in (9.8.16) such that u S + r S is constant on Σ with r S the area radius of the spheres S . Extend u S and r S in a neighborhood of Σ such thatthe transversality conditions (9.8.17) are verified. Then, defining the scalars ς S , A S as in (9.8.18) , (9.8.19) we establish the following relations between η S , ξ S and ς S , A S and a S ,where the latter scalar is defined in Step 1, e S θ (log ς S ) = ( η S − ζ S ) ,e S θ ( A S ) = − S r S ( ζ S − η S ) − r S ξ S + ( ζ S − η S ) A S ,A S = 1 ς S (cid:16) ˇ ς S ˇ κ S − ˇ ς S ˇ A S (cid:17) ,a S = − ς S + Υ S − r S A S . (9.8.26) i.e. differentiable spheres S endowed with adapted null frames ( e S , e S θ , e S ). CHAPTER 9. GCM PROCEDURE
Remark 9.8.5.
Note that we lack equations for η S , ξ S and the average of a S . The lattercan be fixed by fixing the value of a S (cid:12)(cid:12) SP and observing that a S = a S (cid:12)(cid:12) SP − ˇ a S (cid:12)(cid:12) SP . (9.8.27)In what follows we state a result which ties η S , ξ S to the other GCM conditions in (9.8.13)–(9.8.14). Step 9.
To state the proposition below we split the Ricci coefficients into the followinggroups. Γ S g = (cid:26) ˇ κ S , ϑ S , ζ S , ˇ κ S , r ˇ ρ S , κ S − r S , ρ S + 2 m S ( r S ) (cid:27) , Γ S b = (cid:26) η S , ξ S , ˇ ω S , ω S − m S ( r S ) , β S , α S (cid:27) . Proposition 9.8.6.
The following statements hold true ,1. Under the same assumptions as in Proposition 9.8.4, the Ricci coefficients η S , ξ S , ω S verify the following identities. d/ S ,(cid:63) d/ S ,(cid:63) d/ S d/ S d/ S ,(cid:63) η S = κ S (cid:16) C + 2 d/ S ,(cid:63) d/ S ,(cid:63) d/ S β S (cid:17) − r − S d / C + r − S ( d / S ) ≤ Γ S g + r − S ( d / S ) ≤ (Γ S b · Γ S b ) + l.o.t. , d/ S ,(cid:63) d/ S ,(cid:63) d/ S d/ S d/ S ,(cid:63) ξ = C − κ S (cid:16) C + 2 d/ S ,(cid:63) d/ S ,(cid:63) d/ S β S (cid:17) + r − S ( d / S ) ≤ Γ S g + r − S ( d / S ) ≤ (Γ S b · Γ S b ) + l.o.t. ,d/ S ,(cid:63) ω S = (cid:18) κ S + ω S (cid:19) η S − ( κ S ) − d/ S d/ S ,(cid:63) η S + 14 κ S ξ S −
12 ( κ S ) − C + r − S ( d / S ) ≤ Γ S g + d / S (Γ S b · Γ S b ) , (9.8.28) where, C = e S ( d/ S ,(cid:63) d/ S ,(cid:63) µ S ) ,C = e S ( e S θ κ S ) ,C = e (cid:16) ( d/ S ,(cid:63) d/ S + 2 K S ) d/ S ,(cid:63) d/ S ,(cid:63) κ S ) (cid:17) . (9.8.29) The quadratic terms denoted l.o.t. are lower order both in terms of decay in as wellas in terms of number of derivatives. They also contain only angular derivatives d / S r S here denotes r S the area radius of S . .8. CONSTRUCTION OF GCM HYPERSURFACES and not e S or e S . We also note that the error terms r − S ( d / S ) ≤ Γ S g + r − S ( d / S ) ≤ (Γ S b · Γ S b ) does not in fact contain more than derivatives of ˇ ω S .2. If in addition (9.4.11) of Theorem 9.4.2 hold true then, for k ≤ s max − , (cid:107) d/ S ,(cid:63) η S (cid:107) h k ( S ) (cid:46) r S (cid:107) C (cid:107) h k ( S ) + r (cid:107) C (cid:107) h k ( S ) + ◦ (cid:15)r − S + r − S (cid:107) Γ S g (cid:107) h k ( S ) + (cid:107) Γ S b · Γ S b (cid:107) h k ( S ) + l.o.t. , (cid:107) d/ S ,(cid:63) ξ S (cid:107) h k ( S ) (cid:46) r S (cid:107) C (cid:107) h k ( S ) + r S (cid:107) C (cid:107) h k ( S ) + ◦ (cid:15)r − S + r − S (cid:107) Γ S g (cid:107) h k ( S ) + (cid:107) Γ S b · Γ S b (cid:107) h k ( S ) + l.o.t. , (cid:107) d/ S ,(cid:63) ω (cid:107) h k ( S ) (cid:46) r − S (cid:107) η S (cid:107) h k ( S ) + r − S (cid:107) ξ S (cid:107) h k ( S ) + r (cid:107) C (cid:107) h k ( S ) + r − S (cid:107) Γ S g (cid:107) h k ( S ) + (cid:107) Γ S b · Γ S b (cid:107) h k ( S ) + l.o.t. (9.8.30)
3. If in addition the GCM conditions (9.8.13) hold true along Σ and the estimates (9.4.14) are also verified then, for k ≤ s max − , (cid:13)(cid:13) C (cid:13)(cid:13) h k − ( S ) (cid:46) ◦ (cid:15)r − (cid:18)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12) + r − (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k − ( S ) (cid:19) , (cid:13)(cid:13) C (cid:13)(cid:13) h k − ( S ) (cid:46) ◦ (cid:15)r − (cid:18)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12) + r − (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k − ( S ) (cid:19) , (cid:13)(cid:13) C (cid:13)(cid:13) h k − ( S ) (cid:46) ◦ (cid:15)r − (cid:18)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12) + r − (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k − ( S ) (cid:19) , (9.8.31) where a S was defined in Step 1 and can be expressed in terms of ς S and A S byformula (9.8.22) .Proof. The proof of the first two identities in (9.8.28) were derived in Proposition 7.3.4in connection to the proof of Theorem M4, starting with the following d/ S ,(cid:63) ω S = (cid:18) κ S + 2 ω S (cid:19) η S + e S ( ζ S ) − β S + 12 κξ S + r − Γ S g + Γ S b · Γ S b , d/ S d/ S ,(cid:63) η S = κ S (cid:0) − e ( ζ S ) + β S (cid:1) − e S ( e S θ ( κ S )) + r − S ( d / S ) ≤ Γ S g + r − S d / (Γ S b · Γ S b ) , d/ S d/ S ,(cid:63) ξ S = κ S (cid:0) e ( ζ S ) − β S (cid:1) − e S ( e S θ ( κ S )) + r − S ( d / S ) ≤ Γ S g + r − S d / S (Γ S b · Γ S b ) . (9.8.32) The equations used in the derivation of these identities only require the. transversality conditions(9.4.12). Strictly speaking Proposition 7.3.4 requires the e Ricci and Bianchi identities of a geodesic foliation.It is easy to justify the application of these equations in our context by using the transversality conditionsto generate a geodesic foliation in a neighborhood of Σ . These identities were recorded in Proposition 7.1.12 which was itself a corollary Proposition2.2.19.).Note also that d / (Γ S b · Γ S b ) does not contain derivatives of ˇ ω . CHAPTER 9. GCM PROCEDURE
The last identity in (9.8.28) follows by combining the first two identities in (9.8.32).To prove the estimates for η S in the second part of the proposition we make use of theidentity d/ S ,(cid:63) d/ S = d/ S d/ S ,(cid:63) + 2 K S to deduce, d/ S ,(cid:63) ( d/ S d/ S ,(cid:63) + 2 K S ) d/ S d/ S ,(cid:63) η S = 12 κ S C + κ S d/ S ,(cid:63) ( d/ S d/ S ,(cid:63) + 2 K S ) β S − r − S ( d / S ) C + r − S ( d / S ) ≤ Γ S g + r − S ( d / S ) ≤ (Γ S b · Γ S b ) + l.o.t.i.e., d/ S ,(cid:63) ( d/ S d/ S ,(cid:63) + 2 K S ) (cid:16) d/ S d/ S ,(cid:63) η S − κ S β S (cid:17) = 12 κ S C − r − S ( d / S ) C + r − S ( d / S ) ≤ Γ S g + r − S ( d / S ) ≤ (Γ S b · Γ S b ) + l.o.t.Similarly for ξ S d/ S ,(cid:63) ( d/ S d/ S ,(cid:63) + 2 K S ) (cid:16) d/ S d/ S ,(cid:63) ξ S + κ S β S (cid:17) = 12 C − κ S C + r − S ( d / S ) ≤ Γ S g + r − S ( d / S ) ≤ (Γ S b · Γ S b ) + l.o.t.The desired estimates for η S and ξ S follow then by making use of the coercivity of theoperator d/ S ,(cid:63) ( d/ S d/ S ,(cid:63) + 2 K S ). and the estimate for β = β S in (9.4.14). The estimate for d/ S ,(cid:63) ω S is straightforward from the last identity in (9.8.32).To prove the last part of the proposition we make use of the GCM conditions (9.8.13) onΣ to deduce that ν S ( d/ S ,(cid:63) d/ S ,(cid:63) µ S ) = 0 , ν S ( e S θ κ S ) = 0 , ν S (cid:16) ( d/ S ,(cid:63) d/ S + 2 K S ) d/ S ,(cid:63) d/ S ,(cid:63) κ S ) (cid:17) = 0 . Hence, the quantities C , C , C in (9.8.29) can be expressed in the form C = − a S e S (cid:0) d/ S ,(cid:63) d/ S ,(cid:63) µ S (cid:1) ,C = − a S e S ( e S θ κ S ) ,C = a S e S (cid:16) ( d/ S ,(cid:63) d/ S + 2 K S ) d/ S ,(cid:63) d/ S ,(cid:63) κ S ) (cid:17) . Making use of our commutation formulas of Lemma 2.2.13 and the estimates (9.4.14) and(9.4.11) we easily deduce, (cid:107) e S (cid:0) d/ S ,(cid:63) d/ S ,(cid:63) µ S (cid:1) (cid:107) h k − ( S ) (cid:46) ◦ (cid:15)r − , (cid:107) e S ( e S θ κ S ) (cid:107) h k − ( S ) (cid:46) ◦ (cid:15)r − . .8. CONSTRUCTION OF GCM HYPERSURFACES (cid:13)(cid:13)(cid:13) e S (cid:16) ( d/ S ,(cid:63) d/ S + 2 K S ) d/ S ,(cid:63) d/ S ,(cid:63) κ S ) (cid:17) (cid:13)(cid:13)(cid:13) h k − ( S ) (cid:46) ◦ (cid:15)r − . Writing a S = a S + ˇ a S and making use of product estimates we deduce (cid:13)(cid:13) C (cid:13)(cid:13) h k − ( S ) (cid:46) ◦ (cid:15)r − (cid:18)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12) + r − (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k − ( S ) (cid:19) , (cid:13)(cid:13) C (cid:13)(cid:13) h k − ( S ) (cid:46) ◦ (cid:15)r − (cid:18)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12) + r − (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k − ( S ) (cid:19) , (cid:13)(cid:13) C (cid:13)(cid:13) h k − ( S ) (cid:46) ◦ (cid:15)r − (cid:18)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12) + r − (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k − ( S ) (cid:19) , as stated. Step 10.
Propositions 9.8.4 and 9.8.6 provide us with potential estimates for d/ S ,(cid:63) η S , d/ S ,(cid:63) ξ S , d/ S ,(cid:63) ω S , d/ S ,(cid:63) ς S . To close we also need to control the (cid:96) = 1 modes of η S , ξ S theaverage of ω S and the average of a S . Note that the average of ω S can in fact be derivedform the equation, e S ( κ S ) + 12 κ S κ S − ω S κ S = 2 d/ S η S + 2 ρ S − ϑ S ϑ S + 2( η S ) in terms of A S and η S . Indeed, making use of the GCM condition κ S = r S , ω S = 12 κ S (cid:20) e S ( κ S ) + 12 κ S κ S − d/ S η S − ρ S + 12 ϑ S ϑ S − η S ) (cid:21) = − e ( r S ) − Υ S r S + r S (cid:20) − d/ S η S − ρ S + 12 ϑ S ϑ S − η S ) (cid:21) = − A S + r S (cid:20) − d/ S η S − ρ S + 12 ϑ S ϑ S − η S ) (cid:21) . Thus, recalling the definition of µ S , ω S = − A S + r S µ S − ( η S ) We cannot close the estimates without being also able to estimate the (cid:96) = 1 modes of η S , ξ S , ω S andthe average a S . The quantity ˇ a S can be determined using Proposition 9.8.4. CHAPTER 9. GCM PROCEDURE or, ω S − m S ( r S ) = − A S + r S (cid:32) µ S − m S ( r S ) − η S · η S (cid:33) . (9.8.33) Step 11.
In view of the above we can determine η S , ξ S , ω S , ς S , A S provided that wecontrol the (cid:96) = 1 modes of η S , ξ S and the average of ς S . For this reason we introduce ,along Σ , B S = (cid:90) S η S e Φ , B S = (cid:90) S ξ S e Φ , D S = a S (cid:12)(cid:12)(cid:12) SP + 1 + 2 m S r S . (9.8.34)We are now ready to prove the following Proposition 9.8.7.
Let Σ be a smooth spacelike hypersurface foliated by framed spheres ( S , e S , e S θ .e S ) which verify the GCM conditions (9.8.13) , transversality condition (9.4.12) and the estimates (9.4.11) – (9.4.14) of Theorem 9.4.2. Let u S as in (9.8.16) such that u S + r S is constant on Σ . Extend u S and r S in a neighborhood of Σ such that thetransversality conditions (9.8.17) are verified. As shown above these allow us to define η S , ξ S , ω S , ς S , A S , a S and the constants B S , B S , D S as in (9.8.34) . Finally we assume that, r − (cid:0) | B S | + | B S | (cid:1) + | D S | ≤ ◦ (cid:15) / . (9.8.35) Under these assumptions the following estimates hold true for all k ≤ s max − ,1. The Ricci coefficients η S , ξ S , ω S verify (cid:13)(cid:13) η S (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15) + r − S | B S | , (cid:13)(cid:13) ξ S (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15) + r − S | B S | , (cid:13)(cid:13) ˇ ω S (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15) + r − S (cid:0) | B S | + | B S | (cid:1) , (cid:12)(cid:12)(cid:12) ω S − m S ( r S ) (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) + r − S (cid:0) | B S | + | B S | (cid:1) . (9.8.36)
2. The scalar a S verifies, r − S (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k +1 ( S ) + (cid:12)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) + r − S | B S | + | D S | . (9.8.37) Note that to prove our main theorem we have to construct our hypersurface Σ such that in fact B = B = D = 0. .8. CONSTRUCTION OF GCM HYPERSURFACES
3. We also have (cid:13)(cid:13) A S (cid:13)(cid:13) h k +1 ( S ) (cid:46) ◦ (cid:15) + r − S (cid:0) | B S | + | B S | (cid:1) + | D S | ,r − S (cid:107) ˇ ς S (cid:107) h k +1 ( S ) + (cid:12)(cid:12) ς S − (cid:12)(cid:12) (cid:46) ◦ (cid:15) + r − S (cid:0) | B S | + | B S | (cid:1) + | D S | . (9.8.38)
4. We also have, for all k ≤ s max − (cid:107) e S (ˇ κ S , ϑ S , ζ S , ˇ κ S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − S , (cid:107) e S ( ϑ S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15), (cid:107) e S (cid:0) α S , β S , ˇ ρ S , µ S (cid:1) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − S , (cid:107) e S ( β S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15)r − S , (cid:107) e S ( α S ) (cid:107) h k ( S ) (cid:46) ◦ (cid:15). Proof.
To simplify the exposition below we make the auxiliary bootstrap assumptions, (cid:13)(cid:13) η S (cid:13)(cid:13) h k ( S ) + (cid:13)(cid:13) ξ S (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15) / . (9.8.39)We start with the following lemma. Lemma 9.8.8.
The following estimates hold true r − S (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k +1 ( S ) + (cid:12)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12) D S (cid:12)(cid:12) + (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) + ◦ (cid:15). (9.8.40) Proof.
Since a S = a S + ˇ a S we deduce a S (cid:12)(cid:12) SP = a S + ˇ a S (cid:12)(cid:12) SP . Hence, a S = D S − − m S r S − ˇ a S (cid:12)(cid:12) SP . (9.8.41)We also have (see Proposition 9.8.4) a S = − ς S + Υ S − r S A S . Hence, a S = − ς S + ˇ ς S + Υ S − r S A S = − ς S (cid:32) − ˇ ς S ς S + O (cid:18) ˇ ς S ς S (cid:19) (cid:33) + Υ S − r S A S . Taking the average on S we deduce, a S = − ς S + Υ S − r S A S + O (cid:18) ˇ ς S ς S (cid:19) . (9.8.42)28 CHAPTER 9. GCM PROCEDURE
Also, using (9.8.41), ˇ a S = 2ˇ ς S − r S A S + l.o.t. (9.8.43)where l.o.t. denotes higher order terms in ˇ ς S and ς S −
1. Indeedˇ a S = a S − a S = − ς S + Υ S − r S A S − (cid:18) − ς S + Υ S − r S A S (cid:19) = − ς S + 2 ς S − r S A S = 2ˇ ς S ς S ς S − r S A S = 2ˇ ς S − r S A S + l.o.t.Thus to estimate ˇ a S and a S we first need to estimate A S , ˇ ς S and ς S . Using the equations(see Proposition 9.8.4 ) e S θ ( A S ) = − S r S ( ζ S − η S ) − r S ξ S + ( ζ S − η S ) A S ,A S = 1 ς S (cid:16) ˇ ς S ˇ κ S − ˇ ς S ˇ A S (cid:17) , and the auxiliary assumption we derive, (cid:13)(cid:13) A S (cid:13)(cid:13) h k +1 ( S ) (cid:46) (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) + ◦ (cid:15)r − (cid:16) (cid:107) ˇ ς S (cid:107) h k ( S ) + (cid:12)(cid:12) ς S − (cid:12)(cid:12)(cid:17) . (9.8.44)From the equation e S θ (log ς S ) = ( η S − ζ S ) . we also derive, r − S (cid:107) ˇ ς S (cid:107) h k +1 ( S ) (cid:46) (cid:107) η S (cid:107) h k ( S ) + ◦ (cid:15) + ◦ (cid:15) (cid:12)(cid:12) ς S − (cid:12)(cid:12) . (9.8.45)To estimate ς S − ς S = − a S + Υ S − r S A S = − (cid:18) D S − − m S r S − ˇ a S (cid:12)(cid:12) SP (cid:19) + Υ S − r S A S + l.o.t.= − D S + 2 + ˇ a S (cid:12)(cid:12) SP − r S A S + l.o.t.= − D S + 2 + 2ˇ ς S (cid:12)(cid:12) SP − r S (cid:16) A S + ˇ A S (cid:12)(cid:12) SP (cid:17) + l.o.t.and therefore, 2(1 − ς S ) ς S = − D S + 2ˇ ς S (cid:12)(cid:12) SP − r S (cid:16) A S + ˇ A S (cid:12)(cid:12) SP (cid:17) + l.o.t. .8. CONSTRUCTION OF GCM HYPERSURFACES ς S − D S − ˇ ς S (cid:12)(cid:12) SP + r S (cid:16) A S + ˇ A S (cid:12)(cid:12) SP (cid:17) + l.o.t.where l.o.t. denote higher order terms in ˇ ς S and ς S − (cid:12)(cid:12) ς S − (cid:12)(cid:12) (cid:46) | D S | + (cid:107) ˇ ς S (cid:107) L ∞ ( S ) + r S (cid:107) A S (cid:107) L ∞ ( S ) . Hence, back to (9.8.45) we derive, r − S (cid:107) ˇ ς S (cid:107) h k +1 ( S ) + (cid:12)(cid:12) ς S − (cid:12)(cid:12) (cid:46) | D S | + r S (cid:107) A S (cid:107) L ∞ ( S ) + r (cid:107) η S (cid:107) h k ( S ) + ◦ (cid:15). Combining with (9.8.44) we deduce, (cid:13)(cid:13) A S (cid:13)(cid:13) h k +1 ( S ) (cid:46) (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) + ◦ (cid:15),r − S (cid:107) ˇ ς S (cid:107) h k +1 ( S ) + (cid:12)(cid:12) ς S − (cid:12)(cid:12) (cid:46) r (cid:107) η S (cid:107) h k ( S ) + ◦ (cid:15). (9.8.46)In view of (9.8.43) we also deduce, r − S (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k +1 ( S ) (cid:46) r − S (cid:107) ˇ ς S (cid:107) h k +1 ( S ) + (cid:13)(cid:13) A S (cid:13)(cid:13) h k +1 ( S ) (cid:46) (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) + ◦ (cid:15). From (9.8.41) we further deduce (cid:12)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12) D S (cid:12)(cid:12) + (cid:107) ˇ a S (cid:107) L ∞ ( S ) (cid:46) (cid:12)(cid:12) D S (cid:12)(cid:12) + (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) + ◦ (cid:15). Hence, r − S (cid:13)(cid:13) ˇ a S (cid:13)(cid:13) h k +1 ( S ) + (cid:12)(cid:12)(cid:12) a S + 1 + 2 m S r S (cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12) D S (cid:12)(cid:12) + (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) + ◦ (cid:15) (9.8.47)as stated.In view of the lemma above and the assumption | D S | (cid:46) ◦ (cid:15) / the estimates (9.8.31) become, (cid:13)(cid:13) C (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15)r − S (cid:18) (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) + ◦ (cid:15) / (cid:19) , (cid:13)(cid:13) C (cid:13)(cid:13) h k +3 ( S ) (cid:46) ◦ (cid:15)r − S (cid:18) (cid:107) η S (cid:107) h k +3 ( S ) + (cid:107) ξ S (cid:107) h k +3 ( S ) + ◦ (cid:15) / (cid:19) , (cid:13)(cid:13) C (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15)r − S (cid:18) (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) + ◦ (cid:15) / (cid:19) . (9.8.48)To prove the desired estimate for η S , ξ S , ω S we make use of (9.8.30) and the followinglemma.30 CHAPTER 9. GCM PROCEDURE
Lemma 9.8.9.
The error term E k = r − S (cid:107) Γ S g (cid:107) h k ( S ) + (cid:107) Γ S b · Γ S b (cid:107) h k ( S ) , k ≤ s max − , appearing in (9.8.30) verifies the estimate E k (cid:46) r − S ◦ (cid:15) + r − S ◦ (cid:15) / (cid:16)(cid:13)(cid:13) ( η S , ξ S ) (cid:107) h k ( S ) + (cid:107) ˇ ω S (cid:107) h k +3 ( S ) (cid:17) . Proof.
Since Γ S g contains only terms estimated by (9.4.11), (cid:107) Γ S g (cid:107) h k ( S ) (cid:46) r − S ◦ (cid:15) Γ S b contains ϑ S , which is estimated by (9.4.11), as well as η S , ξ S , ˇ ω S , ω S − m S ( r S ) . Thus, inview of the auxiliary estimates (cid:13)(cid:13) η S , ξ S (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15) / and the fact that the quadratic errorterms contain one less derivative of ˇ ω S , we deduce, (cid:107) Γ S b · Γ S b (cid:107) h k ( S ) (cid:46) r − S ◦ (cid:15) / (cid:32)(cid:13)(cid:13) η S , ξ S (cid:107) h k ( S ) + (cid:107) ˇ ω S (cid:107) h k +3 ( S ) + r S (cid:12)(cid:12)(cid:12)(cid:12) ω S − m S ( r S ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) . In view of equation (9.8.33), ω S − m S ( r S ) = − A S + r S (cid:16) µ S − m S ( r S ) − η S · η S (cid:17) , (cid:12)(cid:12)(cid:12)(cid:12) ω S − m S ( r S ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12) A S (cid:12)(cid:12) + r S (cid:12)(cid:12)(cid:12)(cid:12) µ − m S ( r S ) (cid:12)(cid:12)(cid:12)(cid:12) + | η S | (cid:46) r − S ◦ (cid:15) | D S | + r − ◦ (cid:15) / (cid:0) (cid:107) η S (cid:107) h ( S ) + (cid:107) ξ S (cid:107) h ( S ) (cid:1) + ◦ (cid:15)r − (cid:46) r − S ◦ (cid:15) / (cid:16) (cid:107) η S (cid:107) h ( S ) + (cid:107) ξ S (cid:107) h ( S ) + ◦ (cid:15) (cid:17) . Hence, (cid:107) Γ S b · Γ S b (cid:107) h k ( S ) (cid:46) r − S ◦ (cid:15) / (cid:32)(cid:13)(cid:13) η S , ξ S (cid:107) h k ( S ) + (cid:107) ˇ ω S (cid:107) h k +3 ( S ) + ◦ (cid:15) (cid:33) and, E k = r − S (cid:107) Γ S g (cid:107) h k ( S ) + (cid:107) Γ S b · Γ S b (cid:107) h k ( S ) (cid:46) r − S ◦ (cid:15) + r − S ◦ (cid:15) / (cid:32)(cid:13)(cid:13) η S , ξ S (cid:107) h k ( S ) + (cid:107) ˇ ω S (cid:107) h k +3 ( S ) (cid:33) as stated. .8. CONSTRUCTION OF GCM HYPERSURFACES C , C , C the estimates (9.8.30) of Propo-sition 9.8.6 become, (cid:107) d/ S ,(cid:63) η S (cid:107) h k ( S ) (cid:46) r − S ◦ (cid:15) + r − S ◦ (cid:15) / (cid:32)(cid:13)(cid:13) η S , ξ S (cid:107) h k ( S ) + (cid:107) ˇ ω S (cid:107) h k +3 ( S ) (cid:33) , (cid:107) d/ S ,(cid:63) ξ S (cid:107) h k ( S ) (cid:46) r − S ◦ (cid:15) + r − S ◦ (cid:15) / (cid:32)(cid:13)(cid:13) η S , ξ S (cid:107) h k ( S ) + (cid:107) ˇ ω S (cid:107) h k +3 ( S ) (cid:33) , (cid:107) d/ S ,(cid:63) ω S (cid:107) h k ( S ) (cid:46) r − S (cid:107) η S (cid:107) h k ( S ) + r − S (cid:107) ξ S (cid:107) h k ( S ) + r − S ◦ (cid:15) + r − S ◦ (cid:15) / (cid:32)(cid:13)(cid:13) η S , ξ S (cid:107) h k ( S ) + (cid:107) ˇ ω S (cid:107) h k ( S ) (cid:33) . (9.8.49)From the last equation we derive, (cid:107) ˇ ω S (cid:107) h k ( S ) (cid:46) (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) + ◦ (cid:15). Thus the first two equations in (9.8.49) become r S (cid:107) d/ S ,(cid:63) η S (cid:107) h k ( S ) (cid:46) ◦ (cid:15) + ◦ (cid:15) / (cid:0) (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) (cid:1) ,r S (cid:107) d/ S ,(cid:63) ξ S (cid:107) h k ( S ) (cid:46) ◦ (cid:15) + ◦ (cid:15) / (cid:0) (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) (cid:1) , (9.8.50)from which we deduce, (cid:13)(cid:13) η S (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15) + r − S | B S | , (cid:13)(cid:13) ξ S (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15) + r − S | B S | , (cid:13)(cid:13) ˇ ω S (cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15) + r − S (cid:0) | B S | + | B S | (cid:1) , as stated. We can then go back to the preliminary estimates obtained above for ς S , A S and a S to derive the remaining statements (1-4) of Proposition 9.8.7. To prove the lastpart of the Proposition we make use of the corresponding Ricci and Bianchi equations inthe e S direction. Corollary 9.8.10.
Under the same assumptions as in the proposition above we have themore precise estimates, with d ( S ) = (cid:82) S e , (cid:13)(cid:13)(cid:13)(cid:13) η S − d ( S ) B S e Φ (cid:13)(cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15), (cid:13)(cid:13)(cid:13)(cid:13) ξ S − d ( S ) B S e Φ (cid:13)(cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15). CHAPTER 9. GCM PROCEDURE
Note also that, d ( S ) = ( r S ) (cid:18) π O ( ◦ (cid:15) ) (cid:19) . Proof.
In view of (9.8.50), (9.8.36) and auxiliary assumption (9.8.35) we deduce, (cid:13)(cid:13)(cid:13)(cid:13) η S − (cid:18) (cid:82) S η S e Φ (cid:82) S e (cid:19) e Φ (cid:13)(cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) r (cid:107) d/ S ,(cid:63) η S (cid:107) h k ( S ) (cid:46) ◦ (cid:15) + ◦ (cid:15) / (cid:0) (cid:107) η S (cid:107) h k ( S ) + (cid:107) ξ S (cid:107) h k ( S ) (cid:1) (cid:46) ◦ (cid:15) + ◦ (cid:15) / (cid:16) ◦ (cid:15) + r − (cid:0) | B S | + | B S | (cid:1) (cid:17) (cid:46) ◦ (cid:15). We deduce, (cid:13)(cid:13)(cid:13)(cid:13) η S − B S (cid:82) S e e Φ (cid:13)(cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15). Similarly, (cid:13)(cid:13)(cid:13)(cid:13) ξ S − B S (cid:82) S e e Φ (cid:13)(cid:13)(cid:13)(cid:13) h k ( S ) (cid:46) ◦ (cid:15) as desired. Σ To construct the spacelike hypersurface of Theorem 9.8.1 we proceed as follows.
Step 12.
Let Ψ( s ) , Λ( s ) , Λ( s ) real valued functions that will be carefully chosen later.We look for the hypersurface Σ in the form,Σ = (cid:91) s ≥ ◦ s S [ P ( s )] = (cid:91) s ≥ ◦ s S [Ψ( s ) , s, Λ( s ) , Λ( s )] (9.8.51)where P ( s ) is a curve in the parameter space P given by, P ( s ) = (Ψ( s ) , s, Λ( s ) , Λ( s )) . (9.8.52) .8. CONSTRUCTION OF GCM HYPERSURFACES to start at S = S [ ◦ u, ◦ s, Λ , Λ ] we impose the conditionsΨ( ◦ s ) = ◦ u, Λ( ◦ s ) = Λ , Λ( ◦ s ) = Λ . (9.8.53) Step 13.
We expect Σ to be a perturbation of the timelike surface u + s = c for someconstant c . We thus introduce the notation ψ ( s ) := Ψ( s ) + s − c , so that Ψ( s ) = − s + c + ψ ( s )and expect ψ ( s ) = O ( ◦ δ ). Step 14.
In view of (9.8.15) we can express the collection of spheres Σ in the formΣ = (cid:110) Ξ( s, θ ) , s ≥ ◦ s, θ ∈ [0 , π ] (cid:111) (9.8.54)where the map Ξ( s, θ ) = Ξ(Ψ( s ) , s, θ ) is defined asΞ( s, θ ) := (cid:16) Ψ( s ) + U ( θ, P ( s )) , s + S ( θ, P ( s )) , θ (cid:17) . (9.8.55)At the South Pole, i.e. θ = 0, where U (0 , P ) = S (0 , P ) = 0Ξ( s,
0) = (cid:16) Ψ( s ) , s, (cid:17) . (9.8.56)Clearly, ∂ s Ξ( s, θ ) = (cid:16) Ψ (cid:48) ( s ) + ∂ P U ( θ, P ( s )) P (cid:48) ( s ) , ∂ P S ( θ, P ( s )) P (cid:48) ( s ) , (cid:17) ,∂ θ Ξ( s, θ ) = (cid:16) ∂ θ U ( θ, P ( s )) , ∂ θ S ( θ, P ( s )) , (cid:17) , where, ∂ P U ( · ) P (cid:48) ( s ) = Ψ (cid:48) ( s ) ∂ u U ( · ) + ∂ s U ( · ) + Λ (cid:48) ( s ) ∂ Λ U ( · ) + Λ (cid:48) ( s ) ∂ Λ U ( · ) ,∂ P S ( · ) P (cid:48) ( s ) = Ψ (cid:48) ( s ) ∂ u S ( · ) + ∂ s S ( · ) + Λ (cid:48) ( s ) ∂ Λ S ( · ) + Λ (cid:48) ( s ) ∂ Λ S ( · ) . Given f a function on Σ ∗ we have, dds f (cid:0) Ξ( s, θ ) (cid:1) = (cid:16) Ψ (cid:48) ( s ) + ∂ P U ( θ, P ( s )) P (cid:48) ( s ) (cid:17) ∂ u f + (cid:16) ∂ P S ( θ, P ( s )) P (cid:48) ( s ) (cid:17) ∂ s f = X ∗ f,dds f (cid:0) Ξ( s, θ ) (cid:1) = ∂ θ U ( θ, P ( s )) ∂ s f + ∂ θ S ( θ, P ( s )) ∂ s + ∂ θ f = Y ∗ f, CHAPTER 9. GCM PROCEDURE where X ∗ , Y ∗ are the following tangent vectorfields along Σ ∗ , X ∗ ( s, θ ) : = (cid:16) Ψ (cid:48) ( s ) + ∂ P U ( θ, P ( s )) P (cid:48) ( s ) (cid:17) ∂ u + (cid:16) ∂ P S ( θ, P ( s )) P (cid:48) ( s ) (cid:17) ∂ s ,Y ∗ ( s, θ ) : = ∂ θ U ( θ, P ( s )) ∂ s + ∂ θ S ( θ, P ( s )) ∂ s + ∂ θ , (9.8.57)or, X ∗ ( s, θ ) : = (cid:16) Ψ (cid:48) ( s ) + ˘ A ( s, θ ) (cid:17) ∂ u + (cid:16) B ( s, θ ) P (cid:48) ( s ) (cid:17) ∂ s ,Y ∗ ( s, θ ) : = ˘ C ( s, θ ) ∂ u + ˘ D ( s, θ ) ∂ s + ∂ θ , (9.8.58)where,˘ A ( s, θ ) : = ∂ P U ( θ, P ( s )) P (cid:48) ( s )= ∂ u U ( θ, P ( s ))Ψ (cid:48) ( s ) + ∂ s U ( θ, P ( s )) + ∂ Λ U ( θ, P ( s ))Λ (cid:48) ( s ) + ∂ Λ U ( θ, P ( s ))Λ (cid:48) ( s ) , ˘ B ( s, θ ) : = ∂ P S ( θ, P ( s )) P (cid:48) ( s )= ∂ u S ( θ, P ( s ))Ψ (cid:48) ( s ) + ∂ s S ( θ, P ( s )) + ∂ Λ U ( θ, P ( s ))Λ (cid:48) ( s ) + ∂ Λ S ( θ, P ( s ))Λ (cid:48) ( s ) , ˘ C ( s, θ ) : = ∂ θ U ( θ, P ( s )) , ˘ D ( s, θ ) : = ∂ θ S ( θ, P ( s )) . Step 15.
Define the vectorfield, along the South Pole of each S ⊂ Σ , X ∗ (cid:12)(cid:12)(cid:12) SP h = dds h (cid:0) Ξ( s, (cid:1) . (9.8.59) Lemma 9.8.11.
At the South Pole we have the relations (recall ν S = e S + a S e S ) X ∗ (cid:12)(cid:12) SP = 12 λ ς Ψ (cid:48) ν S (cid:12)(cid:12)(cid:12) SP , (9.8.60) a S (cid:12)(cid:12) SP = 2 λ Ψ (cid:48) ( s ) ς (cid:0) −
12 Ψ (cid:48) ( s ) ς Ω (cid:1) | SP , (9.8.61) or, more precisely, a S (Ψ( s ) , s,
0) = 1Ψ (cid:48) ( s ) 2 λ ς (cid:18) −
12 Ψ (cid:48) ( s ) ς Ω (cid:19) (Ψ( s ) , s, . Here f, f , λ are the transition functions and ς, Ω correspond to the background foliation. .8. CONSTRUCTION OF GCM HYPERSURFACES Proof.
Note that ˘ A ( s,
0) = ˘ B ( s,
0) = ˘ C ( s,
0) = ˘ D ( s,
0) = 0 . Thus, at the South Pole SP, X ∗ ( s,
0) = Ψ (cid:48) ( s ) ∂ u + ∂ s . Recall that ∂ s = e , ∂ u = 12 ς (cid:0) e − Ω e − bγ / e θ (cid:1) , ∂ θ = √ γe θ , or, since b vanishes at the South Pole, X ∗ ( s,
0) = Ψ (cid:48) ς ( e − Ω e ) + e = (1 −
12 Ψ (cid:48) ( s ) ς Ω) e + 12 Ψ (cid:48) ( s ) ςe . On the other hand, since the transition functions f, f vanish at the South Pole, e S = λe , e S = λ − e . Hence, X ∗ ( s,
0) = λ (cid:0) −
12 Ψ (cid:48) ( s ) ς Ω (cid:1) e S + 12 λ − Ψ (cid:48) ( s ) ςe S = 12 λ − Ψ (cid:48) ( s ) ς (cid:18) e S + 2 λ Ψ (cid:48) ( s ) ς (cid:0) −
12 Ψ (cid:48) ( s ) ς Ω (cid:1) e S (cid:19) . In view of the definition of ν S we deduce, X ∗ ( s,
0) = 12 λ − Ψ (cid:48) ( s ) ς ν S (cid:12)(cid:12) SP and , a S ( s,
0) = 2 λ Ψ (cid:48) ( s ) ς (cid:0) −
12 Ψ (cid:48) ( s ) ς Ω (cid:1) as stated. Step 16.
The transition functions ( f, f , λ ) are uniquely determined on S by the resultsof Theorem 9.4.2 in terms of Λ , Λ. The same holds true for all curvature componentsand the Ricci coefficients κ S , ϑ S , ζ S , κ S , ϑ S . One can easily see from the transformation Note that a S ( s,
0) = a S (Ξ( s, CHAPTER 9. GCM PROCEDURE formulas that the values of the e S derivatives of ( f, f , λ ) are determined by the transversalRicci coefficients η S , ξ S , η S . Indeed, schematically, from the transformation formulas for η, ξ, ω in Proposition 9.3.1, e S f = 2( η S − η ) − κf + f ω + F · Γ b + l.o.t. ,e S f = 2( ξ S − ξ ) − f (cid:0) κ + 4 ω ) + F · Γ b + l.o.t. ,e S (log λ ) = 2( ω S − ω ) + Γ b · F + l.o.t. , (9.8.62)where F = ( f, f , log λ ) and l.o.t. denotes terms which are linear in Γ g , Γ b and linear andhigher order in F . Recall also that the e S derivatives of F are fixed by our transversalitycondition (9.4.12) More precisely we have, e S ( f ) = − κf + l.o.t. ,e S ( f ) = 2 e S θ (log λ ) − f κ + 2( ω + 14 κ ) f + l.o.t. ,e S (log λ ) = l.o.t.It follows that η S , ξ S , ω S can be determined by ν S ( f, f , λ ) and the scalar a S . More pre-cisely, ν S ( f ) = 2( η S − η ) −
12 ( κf + a S κf ) + f ω + F · Γ b + l.o.t. ,ν S ( f ) = 2( ξ S − ξ ) − (cid:0) κ + 4 ω )( f − a S f ) + a S (cid:0) e S θ (log λ ) − f κ (cid:1) + F · Γ b + l.o.t.(9.8.63) Step 17.
We derive equations for Λ( s ) = Λ(Ψ( s ) , s, , Λ( s ) = Λ(Ψ( s ) , s,
0) as follows.
Lemma 9.8.12.
We have the following identities c ( s ) 1Ψ (cid:48) ( s ) Λ (cid:48) ( s ) = (cid:90) S ν S ( f ) e Φ − r S Λ( s ) + E ( s ) ,c ( s ) 1Ψ (cid:48) ( s ) Λ (cid:48) ( s ) = (cid:90) S ( s ) ν S ( f ) e Φ − r S Λ( s ) + E ( s ) , (9.8.64) where, c ( s ) = (cid:16) λς (cid:17)(cid:12)(cid:12)(cid:12) SP ( s ) = (cid:16) λς (cid:17) (Ψ( s ) , s, .8. CONSTRUCTION OF GCM HYPERSURFACES and error terms, E ( s ) = 12 (cid:90) S ( s ) (cid:18) κ S − ϑ S − a S ϑ S + 3 r S ( a S + (1 + 2 m S r S ) (cid:19) f e Φ + ( ς S ) − (cid:12)(cid:12)(cid:12) SP (cid:90) S ( s ) (cid:16) ς S − ς S (cid:12)(cid:12)(cid:12) SP (cid:17) (cid:18) ν S ( f ) − r S Λ( s ) (cid:19) e Φ + l.o.t. ,E ( s ) = 12 (cid:90) S ( s ) (cid:18) κ S − ϑ S − a S ϑ S + 3 r S ( a S + (1 + 2 m S r S ) (cid:19) f e Φ + ( ς S ) − (cid:12)(cid:12)(cid:12) SP (cid:90) S ( s ) (cid:16) ς S − ς S (cid:12)(cid:12)(cid:12) SP (cid:17) (cid:18) ν S ( f ) − r S Λ( s ) (cid:19) e Φ + l.o.t.Proof. According to Lemma 9.8.3 we have ν S (cid:18)(cid:90) S h (cid:19) = ( ς S ) − (cid:90) S ς S (cid:0) ν S ( h ) + ( κ S + a S κ S ) h (cid:1) . Thus, applying the vectorfield ν S (cid:12)(cid:12) SP = λς Ψ (cid:48) X ∗ (cid:12)(cid:12) SP to the formulas (9.8.14),1Ψ (cid:48) ( s ) (cid:16) λς (cid:17)(cid:12)(cid:12)(cid:12) SP dds Λ( s ) = ν S (cid:12)(cid:12)(cid:12) SP (Λ) = ν S (Λ) (cid:12)(cid:12) SP = ν S (cid:16) (cid:90) S f e Φ (cid:17)(cid:12)(cid:12)(cid:12) SP = ( ς S ) − (cid:12)(cid:12)(cid:12) SP (cid:90) S ( s ) ς S (cid:16) ν S ( f e Φ ) + ( κ S + a S κ S ) f e Φ (cid:17) . Introducing J ( f ) = e − Φ ν S ( f e Φ ) + ( κ S + a S κ S ) f (9.8.65)we deduce, c ( s ) 1Ψ (cid:48) ( s ) = ( ς S ) − (cid:12)(cid:12)(cid:12) SP (cid:90) S ( s ) ς S J ( f ) e Φ = (cid:90) S ( s ) J ( f ) e Φ + ( ς S ) − (cid:12)(cid:12)(cid:12) SP (cid:90) S ( s ) (cid:16) ς S − ς S (cid:12)(cid:12)(cid:12) SP (cid:17) J ( f ) . On the other hand, since e Φ = ( κ − ϑ ), e Φ = ( κ − ϑ ) J ( f ) = ν S ( f ) + (cid:0) e S Φ + a S e S Φ + κ S + a S κ S (cid:1) f = ν S ( f ) + 12 (cid:0) κ S − ϑ S + a S (3 κ S − ϑ S ) (cid:1) f = ν S ( f ) + 32 (cid:0) κ S + a S κ S (cid:1) − (cid:0) ϑ S + a S ϑ S (cid:1) . CHAPTER 9. GCM PROCEDURE
Since κ S = r S and κ S = κ S + ˇ κ S = − S r S + ˇ κ S we deduce, J ( f ) = ν S ( f ) + 3 r S (cid:0) − Υ S + a S (cid:1) f + 12 (cid:0) κ S − ϑ S − a S ϑ S (cid:1) f = ν S ( f ) + 3 r S (cid:18) − Υ S − (1 + 2 m S r S ) (cid:19) f + 12 (cid:18) κ S − ϑ S − a S ϑ S + 3 r S ( a S + (1 + 2 m S r S ) (cid:19) f = ν S ( f ) − r S Λ( s ) + 12 (cid:18) κ S − ϑ S − a S ϑ S + 3 r S ( a S + (1 + 2 m S r S ) (cid:19) f. We deduce, c ( s ) 1Ψ (cid:48) ( s ) = (cid:90) S ν S ( f ) e Φ − r S Λ( s ) + E ( s )where, E ( s ) = 12 (cid:90) S ( s ) (cid:18) κ S − ϑ S − a S ϑ S + 3 r S ( a S + (1 + 2 m S r S ) (cid:19) f e Φ + ( ς S ) − (cid:12)(cid:12)(cid:12) SP (cid:90) S ( s ) (cid:16) ς S − ς S (cid:12)(cid:12)(cid:12) SP (cid:17) J ( f )= 12 (cid:90) S ( s ) (cid:18) κ S − ϑ S − a S ϑ S + 3 r S ( a S + (1 + 2 m S r S ) (cid:19) f e Φ + ( ς S ) − (cid:12)(cid:12)(cid:12) SP (cid:90) S ( s ) (cid:16) ς S − ς S (cid:12)(cid:12)(cid:12) SP (cid:17) (cid:18) ν S ( f ) − r S Λ( s ) (cid:19) e Φ + l.o.t.The proof for Λ is exactly the same. Step 18.
We make use of the estimates for F = ( f, f , log λ ) and e S ( F ) derived inTheorem 9.4.2 as well as the estimates for a S , ς S , η S , ξ S , ω S derived in Proposition 9.8.7 toevaluate the right hand sides of (9.8.64). Recall that in Proposition 9.8.7 we have madethe auxiliary assumption (9.8.35) i.e. r − S (cid:0) | B S | + | B S | (cid:1) + | D S | ≤ ◦ (cid:15) / . Proposition 9.8.13.
The following equations hold true for the functions Λ( s ) = Λ(Ψ( s ) , s, , Λ( s ) = Λ(Ψ( s ) , s, ,B ( s ) = Λ(Ψ( s ) , s, , B ( s ) = Λ(Ψ( s ) , s, , r ( s ) = r (Ψ( s ) , s, , Note also that r S ( s ) = r S ( s ) = r | SP ( S ( s )) = r ( s ). .8. CONSTRUCTION OF GCM HYPERSURFACES − ψ (cid:48) ( s ) Λ (cid:48) ( s ) = B ( s ) − r ( s ) − Λ( s ) − r ( s ) − Λ( s ) + O ( r − )Λ( s )+ N ( B, B, D, Λ , Λ , ψ )( s ) , − ψ (cid:48) ( s ) Λ (cid:48) ( s ) = B ( s ) − r ( s ) − Λ( s ) + 12 r ( s ) − Λ( s ) + O ( r − ) (cid:0) Λ( s ) + Λ( s ) (cid:1) + N ( B, B, D, Λ , Λ , ψ )( s ) . (9.8.66) The expressions
N, N verify the following properties. • They depend on
B, B, D, Λ , Λ , ψ, F = ( f, f , λ − , the background Ricci coefficients Γ b , Γ g and curvature ˇ R = { α, β, ˇ ρ, β, α } . • N, N vanish at ( B, B, D, Λ , Λ , ψ ) = (0 , , , , , . In fact, | N, N | (cid:46) r ◦ δ. • The linear part in
B, B, D has O ( ◦ (cid:15) ) coefficients, i.e. coefficients which depend onthe quantities Γ b , Γ g , ˇ R, F and Λ , Λ , ψ . • The linear part in Λ , Λ , ψ has O ( ◦ (cid:15) ) coefficients.Proof. To prove the desired result we make use of (9.8.63) to check the following, (cid:90) S ( s ) ν S ( f ) e Φ = 2 B ( s ) − r − Λ( s ) − r − Λ( s ) + O ( r − )Λ( s ) + O ( r ◦ δ ) , (cid:90) S ( s ) ν S ( f ) e Φ = 2 B ( s ) − r − Λ( s ) + r − Λ( s ) + O ( r − ) (cid:0) Λ( s ) + Λ( s ) (cid:1) + O ( r ◦ δ ) . (9.8.67)Combining this with (9.8.64), c ( s ) 1Ψ (cid:48) ( s ) Λ (cid:48) ( s ) = (cid:90) S ν S ( f ) e Φ − r S Λ( s ) + E ( s ) ,c ( s ) 1Ψ (cid:48) ( s ) Λ (cid:48) ( s ) = (cid:90) S ( s ) ν S ( f ) e Φ − r S Λ( s ) + E ( s ) , and the following estimates for the error terms E, E , | E ( s ) | + | E ( s ) | (cid:46) r ◦ δ, (9.8.68)40 CHAPTER 9. GCM PROCEDURE we deduce,1Ψ (cid:48) ( s ) Λ (cid:48) ( s ) = 1 c ( s ) (cid:18) B ( s ) − r − Λ( s ) − r − Λ( s ) + O ( r − )Λ( s ) + O ( r ◦ δ ) (cid:19) , (cid:48) ( s ) Λ (cid:48) ( s ) = 1 c ( s ) (cid:18) B ( s ) − r − Λ( s ) + r − Λ( s ) + O ( r − ) (cid:0) Λ( s ) + Λ( s ) (cid:1) + O ( r ◦ δ ) (cid:19) . According to our assumptions ς = 1 + O ( ◦ (cid:15) ). Also according to Theorem 9.4.2 λ =1 + O ( r − ◦ (cid:15) ). Thus, c ( s ) = (cid:16) λς (cid:17)(cid:12)(cid:12)(cid:12) SP ( s ) = 2(1 + Or − ( ◦ (cid:15) )1 + O ( ◦ (cid:15) ) = 2 + O ( ◦ (cid:15) ) . Hence, 1Ψ (cid:48) ( s ) Λ (cid:48) ( s ) = 12 (cid:16) B ( s ) − r − Λ( s ) − r − Λ( s ) + O ( r − )Λ( s ) + O ( r ◦ δ ) (cid:17) = B ( s ) − r − Λ( s ) − r − Λ( s ) + O ( r − )Λ( s ) + O ( r ◦ δ ) . Setting Ψ( s ) = − s + ψ ( s ) + c and recalling the structure of the error terms we havedenoted by O ( r ◦ δ )1 − ψ (cid:48) ( s ) Λ (cid:48) ( s ) = B ( s ) − r − Λ( s ) − r − Λ( s ) + O ( r − )Λ( s ) + N ( B, B, D, Λ , Λ , ψ )( s )where N verifies the properties mentioned in the proposition. In the same manner wederive 1 − ψ (cid:48) ( s )) Λ (cid:48) ( s ) = B ( s ) − r − Λ( s ) + 12 r − Λ( s ) + O ( r − ) (cid:0) Λ( s ) + Λ( s ) (cid:1) + N ( B, B, D, Λ , Λ , ψ )( s )as stated in the proposition.It remains to check (9.8.67) and (9.8.68) According to (9.8.63) and our assumptions onthe Ricci coefficients κ, κ, ω , we have along the sphere S ν S ( f ) = 2( η S − η ) − (cid:18) r f − a S r f (cid:19) + f mr + F · Γ b + l.o.t.= 2( η S − η ) − r − f + r − (cid:18) mr + a S (1 − mr (cid:19) f + F · Γ . .8. CONSTRUCTION OF GCM HYPERSURFACES (cid:12)(cid:12)(cid:12)(cid:12) a S + (cid:18) m S r S (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) + r − (cid:0) | B S | + | B S | (cid:1) + | D S | (cid:46) ◦ (cid:15) / . Thus, ν S ( f ) = 2( η S − η ) − r − f − r − (cid:18) − mr − m r (cid:19) f + r − O (cid:18) ◦ δ ◦ (cid:15) / (cid:19) . Since r and r S are comparable along S , i.e. | r − r S | ≤ ◦ δ , we deduce, recalling the definitionof B , (cid:90) S ( s ) ν S ( f ) e Φ = 2 B ( s ) − (cid:90) S ( s ) ηe Φ − r − Λ( s ) − r − (cid:18) − mr − m r (cid:19) Λ( s ) + rO (cid:18) ◦ δ ◦ (cid:15) / (cid:19) . Making use of the assumption (9.8.1) for η as well as Corollary 9.2.5 we easily deduce, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( s ) ηe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ◦ δ. (9.8.69)Hence, (cid:90) S ( s ) ν S ( f ) e Φ = 2 B ( s ) − r − Λ( s ) − r − (cid:0) O ( r − ) (cid:1) Λ( s ) + O ( r ◦ δ )= 2 B ( s ) − r − Λ( s ) − r − Λ( s ) + O ( r − )Λ( s ) + O ( r ◦ δ ) . Similarly, starting with, ν S ( f ) = 2( ξ S − ξ ) − (cid:0) κ + 4 ω )( f − a S f ) + a S (cid:0) e S θ (log λ ) − f κ (cid:1) + F · Γ b + l.o.t.we deduce, (cid:90) S ( s ) ν S ( f ) e Φ = 2 B ( s ) − (cid:90) S ( s ) ξe Φ + r − (cid:18) mr (cid:19) Λ( s ) + r − (cid:18) − mr − m r (cid:19) Λ( s ) − (cid:18) mr (cid:19) (cid:90) S ( s ) e S θ (log λ ) e Φ + rO (cid:18) ◦ δ ◦ (cid:15) / (cid:19) . Making use of the assumption (9.8.1) for ξ , as well as Corollary 9.2.5, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( s ) ξe Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ◦ δ. (9.8.70)42 CHAPTER 9. GCM PROCEDURE
Also, in view of the estimates of Theorem 9.4.2, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ( s ) e S θ (log λ ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r ◦ δ. We deduce, (cid:90) S ( s ) ν S ( f ) e Φ = 2 B ( s ) − r − (1 + O ( r − ))Λ( s ) + r − (cid:0) O ( r − ) (cid:1) Λ( s ) + O ( r ◦ δ )as stated. The estimates for E, E in (9.8.68) can also be easily checked. This ends theproof of Proposition 9.8.13.
Step 19.
We derive an equation for ψ . The main result is stated in the propositionbelow. Proposition 9.8.14.
The function ψ ( s ) = Ψ( s ) + s − c defined in Step 13 verifies thefollowing equation ψ (cid:48) ( s ) = − D ( s ) + O ( D ( s ) ) + M ( s ) (9.8.71) where M ( s ) is a function which depends only on Γ , R of the background foliation, ψ and ( f, f , λ − such that, (cid:12)(cid:12) M ( s ) (cid:12)(cid:12) (cid:46) ◦ δr ( s ) − . Proof.
In view of (9.8.61) and the definition of c ( s ) = (cid:16) λς (cid:17)(cid:12)(cid:12)(cid:12) SP ( s ) we have,Ψ (cid:48) ( s ) = 1 a S (cid:12)(cid:12) SP ( s ) · λ ς (cid:18) −
12 Ψ (cid:48) ς Ω (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) SP ( s )or Ψ (cid:48) ( s ) = (cid:20) λ ς a S + λ Ω (cid:21) (cid:12)(cid:12)(cid:12) SP . (9.8.72)Now, we have 2 λ ς a S + λ Ω = 2 ς a S + Ω + O ( λ − − a S + ς + Ω a S + Ω + O ( λ − . .8. CONSTRUCTION OF GCM HYPERSURFACES ψ (cid:48) ( s ) = Ψ (cid:48) ( s ) + 1 = (cid:34) a S + ς + Ω a S + Ω (cid:35) (cid:12)(cid:12)(cid:12) SP + O ( λ −
1) = (cid:34) a S + ς + Ω a S + Ω (cid:35) (cid:12)(cid:12)(cid:12) SP + O ( r − ◦ δ ) . We have, see (9.8.34), a S (cid:12)(cid:12) SP ( s ) = D ( s ) − − m S r S . Hence, (cid:0) a S + Ω (cid:1)(cid:12)(cid:12) SP ( s ) = D ( s ) − − m S r S + Ω (cid:12)(cid:12) SP ( s ) = D ( s ) − − m S r S − (1 − mr ) + O ( ◦ (cid:15) )= D ( s ) − O ( ◦ (cid:15) ) . In view of the assumption (9.8.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ς + Ω (cid:19) (cid:12)(cid:12)(cid:12) SP − − mr (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) r − ◦ δ we deduce (cid:18) a S + 2 ς + Ω (cid:19) (cid:12)(cid:12) SP ( s ) = a S (cid:12)(cid:12) SP + 1 + 2 mr + O ( r − ◦ δ )= D ( s ) + 2 mr − m S r S + O ( r − ◦ δ )= D ( s ) + O ( r − ◦ δ ) . Hence, ψ (cid:48) ( s ) = (cid:34) a S + ς + Ω a S + Ω (cid:35) (cid:12)(cid:12)(cid:12) SP + O ( r − ◦ δ ) = − D ( s ) + O ( D ( s ) ) + O ( r − ◦ δ )as stated. Step 20.
We combine Propositions 9.8.13 and 9.8.14 to derive the closed system ofequations in Λ , Λ , ψ ,1 − ψ (cid:48) ( s ) Λ (cid:48) ( s ) = B ( s ) − r ( s ) − Λ( s ) − r ( s ) − Λ( s ) + O ( r − )Λ( s )+ N ( B, B, D, Λ , Λ , ψ )( s ) , − ψ (cid:48) ( s ) Λ (cid:48) ( s ) = B ( s ) − r ( s ) − Λ( s ) + 12 r ( s ) − Λ( s ) + O ( r − ) (cid:0) Λ( s ) + Λ( s ) (cid:1) + N ( B, B, D, Λ , Λ , ψ )( s ) ,ψ (cid:48) ( s ) = − D ( s ) + O ( D ( s ) ) + M ( s ) , (9.8.73)44 CHAPTER 9. GCM PROCEDURE with initial conditions ψ ( ◦ s ) = 0 , Λ( ◦ s ) = Λ , Λ( ◦ s ) = Λ . (9.8.74)Recall also that r ( s ) is a smooth function of ψ ( s ).The system (9.8.73) is verified for all choices of (Λ , Λ , Ψ). We now make a suitableparticular choice for (Λ , Λ , Ψ) as follows.Consider in particular the system obtained from(9.8.73) by setting
B, B, D to zero ψ (cid:48) ( s ) = M ( s ) , − ψ (cid:48) ( s )) Λ (cid:48) ( s ) = − r ( s ) − Λ( s ) − r ( s ) − Λ( s ) + O ( r − )Λ( s ) + (cid:101) N (Λ , Λ , ψ )( s ) , − ψ (cid:48) ( s )) Λ (cid:48) ( s ) = − r ( s ) − Λ( s ) + 12 r ( s ) − Λ( s ) + O ( r − ) (cid:0) Λ( s ) + Λ( s ) (cid:1) + (cid:101) N (Λ , Λ , ψ )( s ) , (9.8.75)where, (cid:101) N (Λ , Λ , ψ ) = N (0 , , , Λ , Λ , ψ ) , (cid:101) N (Λ , Λ , ψ ) = N (0 , , , Λ , Λ , ψ ) . We initialize the system at s = ◦ s as in (9.8.74), i.e.,Λ( ◦ s ) = ψ ( ◦ s ) = 0 , Λ , Λ( ◦ s ) = Λ . The system admits a unique solution ψ ( s ) defined in a small neighborhood ◦ I of ◦ s . Thefunction Ψ( s ) = − s + ψ ( s ) + c defines the desired hypersurface Σ . Step 21.
It remain to show that the function
B, B, D vanish on the hypersurface Σ defined above. Since the system (9.8.73) is verified for all functions Λ , Λ , ψ we deduce,along Σ , D = 0 ,B = N ( B, B, D, Λ , Λ , ψ )( s ) − N (0 , , , Λ , Λ , ψ )( s ) ,B = N ( B, B, D, Λ , Λ , ψ )( s ) − N (0 , , , Λ , Λ , ψ )( s ) . In view of the properties of
N, N we deduce, (cid:12)(cid:12)(cid:12) N ( B, B, D, Λ , Λ , ψ )( s ) − N (0 , , , Λ , Λ , ψ )( s ) (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) sup ◦ I (cid:16) | B ( s ) | + | B ( s ) | (cid:17) , (cid:12)(cid:12)(cid:12) N ( B, B, D, Λ , Λ , ψ )( s ) − N (0 , , , Λ , Λ , ψ )( s ) (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15) sup ◦ I (cid:16) | B ( s ) | + | B ( s ) | (cid:17) . .8. CONSTRUCTION OF GCM HYPERSURFACES ◦ I | B ( s ) | + sup ◦ I | B ( s ) | (cid:46) ◦ δ (cid:0) sup ◦ I | B ( s ) | + sup ◦ I | B ( s ) | (cid:1) . Hence
B, B, D vanish identically on Σ . Step 22.
We have, (cid:12)(cid:12)(cid:12) drds − (cid:12)(cid:12)(cid:12) (cid:46) ◦ (cid:15). (9.8.76)Indeed, according to Step 15 and Lemma 9.8.3 we have dds r ( s ) = X ∗ (cid:12)(cid:12)(cid:12) SP r S = 12 λ ς Ψ (cid:48) ν S ( r S ) (cid:12)(cid:12)(cid:12) SP = ( − ψ (cid:48) ( s )) 12 λ ςν S ( r S ) (cid:12)(cid:12)(cid:12) SP = ( − ψ (cid:48) ( s )) (cid:18) λ ς r S ς S ) − ς S ( κ S + a S κ S ) (cid:19) (cid:12)(cid:12)(cid:12) SP . In view of Proposition 9.8.14, with D = 0, (cid:12)(cid:12) ψ (cid:48) (cid:12)(cid:12) (cid:46) r − ◦ δ . We deduce, dds r ( s ) = − (cid:18) λ ς r S ς S ) − ς S ( κ S + a S κ S ) (cid:19) (cid:12)(cid:12)(cid:12) SP + O ( ◦ δ ) . Step 23.
Therefore the functions
B, B, D vanish identically on the hypersurface Σ defined by the function Ψ( s ) = − s + ψ ( s ) + c which accomplishes the main task ofTheorem 9.8.1. More precisely we have produced a local hypersurface Σ , as defined inStep 12, foliated by the function u S , defined in Step 2 and extended in Step 3, such thatthe items 2-5 of the theorem are verified. The estimates in items 6-7 are an immediateconsequence of Proposition 9.8.7. It only remains to prove the smoothness of the functionΞ( s, θ ) in (9.8.54), Step 14 and the estimates for F = ( f, f , log λ ) in the last part of thetheorem. To check the differentiability properties recall that, ∂ s Ξ( s, θ ) = (cid:16) Ψ (cid:48) ( s ) + ∂ P U ( θ, P ( s )) P (cid:48) ( s ) , ∂ P S ( θ, P ( s )) P (cid:48) ( s ) , (cid:17) ,∂ θ Ξ( s, θ ) = (cid:16) ∂ θ U ( θ, P ( s )) , ∂ θ S ( θ, P ( s )) , (cid:17) , where, ∂ P U ( · ) P (cid:48) ( s ) = Ψ (cid:48) ( s ) ∂ u U ( · ) + ∂ s U ( · ) + Λ (cid:48) ( s ) ∂ Λ U ( · ) + Λ (cid:48) ( s ) ∂ Λ U ( · ) ,∂ P S ( · ) P (cid:48) ( s ) = Ψ (cid:48) ( s ) ∂ u S ( · ) + ∂ s S ( · ) + Λ (cid:48) ( s ) ∂ Λ S ( · ) + Λ (cid:48) ( s ) ∂ Λ S ( · ) . Thus to prove the smoothness of Ξ we need to appeal to the smoothness of
U, S withrespect to the parameters Λ , Λ and u, s . Though tedious, this can be easily done, by46
CHAPTER 9. GCM PROCEDURE appealing back to the coupled system of equations, (9.4.2), (9.4.3) and (9.4.4) (9.4.15),as in the proof of Theorem 9.4.2 and studying its dependence on these parameters.
Step 24.
It only remains to derive the estimates (9.8.11) for the transition functions F = ( f, f , log λ ). To start with we have, in view of the construction of Σ and theestimates for F = ( f, f , log λ ) of Theorem 9.4.2, for every S ⊂ Σ (cid:107) F (cid:107) h s max+1 ( S ) (cid:46) ◦ δ. (9.8.77)To derive the remaining tangential derivatives of F along Σ we the commute the GCMsystem (9.4.2) of Proposition 9.4.1 with respect to ν = ν S = e S + a S e S and then proceed, asin the proof of the apriori estimates of Theorem 9.4.4 to derive recursively the estimates,for K = k max + 1, (cid:107) ν l ( F ) (cid:107) h K − l ( S ) (cid:46) ◦ δ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν l ( f ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν l ( f ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + ◦ δ (cid:13)(cid:13) ν ≤ l − a S (cid:13)(cid:13) h K − l +1 ( S ) + (cid:107) ν ≤ l − F (cid:107) h K − l +1 ( S ) . (9.8.78)We already have estimates for the (cid:96) = 1 modes of F = ( f, f ). To estimate the (cid:96) = 1modes of ν l ( f, f ), l ≥
1, we make use of the equations (9.8.63) and the vanishing of the (cid:96) = 1 modes of η S , ξ S along Σ to derive, recursively, for all 1 ≤ l ≤ K , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν l ( f ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S ν l ( f ) e Φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ◦ δ + ◦ δ (cid:13)(cid:13)(cid:13) ν ≤ l − (cid:16) a, Ω S , ς S , ξ S , η S , ˇ ω S (cid:17)(cid:13)(cid:13)(cid:13) h K − l +1 ( S ) + (cid:107) ν ≤ l − ( F ) (cid:107) h K − l +1 ( S ) . (9.8.79)We can then proceed as in the proof of Proposition 9.8.7 derive, recursively, the estimates (cid:13)(cid:13)(cid:13) ν ≤ l − (cid:16) a S , Ω S , ς S , ξ S , η S , ˇ ω S (cid:17)(cid:13)(cid:13)(cid:13) h K − l +1 ( S ) (cid:46) l (cid:88) j =0 (cid:107) ν j ( F ) (cid:107) h K − j ( S ) . (9.8.80)Combining (9.8.78) (9.8.79) (9.8.80), we obtain (cid:107) ν l ( F ) (cid:107) h K − l ( S ) (cid:46) ◦ δ + l − (cid:88) j =0 (cid:107) ν j ( F ) (cid:107) h K − j ( S ) , which, together with (9.8.77), yields the desired estimate for all tangential derivatives K (cid:88) j =0 (cid:107) ν j ( F ) (cid:107) h K − j ( S ) (cid:46) ◦ δ. .8. CONSTRUCTION OF GCM HYPERSURFACES e S ( F ),due to the transversality conditions (9.4.12). The e S derivatives can then be derived from ν S = e S + a S e S and the estimates for a S . This ends the proof of Theorem 9.8.1. Step 25.
To prove Corollary 9.8.2 we start with the rigidity statement of Corollary9.4.5. Making use of the assumption (cid:82) S f e Φ = O ( ◦ δ ) , (cid:82) S f e Φ = O ( ◦ δ ) we infer that theestimate (9.8.77) holds true for S . We then proceed exactly as in Step 24 to derive theestimates (9.8.78) (9.8.79) (9.8.80) for our distinguished sphere S . Note that S can beviewed as a deformation of the unique background sphere sharing the same south pole.48 CHAPTER 9. GCM PROCEDURE hapter 10REGGE-WHEELER TYPEEQUATIONS
The goal of this chapter is to prove Theorem 5.3.4 and Theorem 5.3.5 concerning theweighted estimates for the solution ψ to (cid:3) ψ + V ψ = N, V = κκ. Recall that these theorems where used in Chapter 5 to prove Theorem M1.The structure of the chapter is as follows. • In section 10.1, we prove basic Morawetz estimates for ψ . • In section 10.2, we prove r p -weighted estimates in the spirit of Dafermos-Rodnianski[20] for ψ . In particular, we obtain as an immediate corollary the proof of Theorem5.3.4 in the case s = 0 (i.e. without commutating the equation of ψ with derivatives). • In section 10.3, we use a variation of the method of [4] to derive slightly strongerweighted estimates and prove Theorem 5.3.5 in the case s = 0 (i.e. without com-mutating the equation of ˇ ψ with derivatives). • In section 10.4, commuting the equation of ψ with derivatives, we complete the proofof Theorem 5.3.4 by controlling higher order derivatives of ψ , i.e. for s ≤ k small + 30.Also, commuting the equation of ˇ ψ with derivatives, we complete the proof ofTheorem 5.3.5 by controlling higher order derivatives of ˇ ψ , i.e. for s ≤ k small + 29.64950 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Recall • the definitions in section 5.1.1 of ( trap ) M , ( trap (cid:14) ) M , τ , Σ( τ ) and ( trap ) Σ, • the main quantities involved in the energy and Morawetz estimates, e.g. E [ ψ ]( τ ),Mor[ ψ ]( τ , τ ), Morr[ ψ ]( τ , τ ), F [ ψ ]( τ , τ ), J δ [ ψ, N ]( τ , τ ) and ˙ B sp ; R [ ψ ]( τ , τ ), in-troduced in section 5.1.4.The following theorem claims basic Morawetz estimates for the solution ψ of the waveequation (5.3.5). Theorem 10.1.1 (Morawetz) . Let ψ a reduced 2-scalar solution to (cid:3) ψ + V ψ = N, V = κκ. Let δ > be a fixed small constant verifying < (cid:15) (cid:28) δ . The following estimates hold truein M ( τ , τ ) , ≤ τ < τ ≤ τ ∗ , E [ ψ ]( τ ) + Morr [ ψ ]( τ , τ ) + F [ ψ ]( τ , τ ) (cid:46) E [ ψ ]( τ ) + J δ [ ψ, N ]( τ , τ )+ O ( (cid:15) ) ˙ B sδ ; 4 m [ ψ ]( τ , τ ) . (10.1.1) Remark 10.1.2.
Note that the bulk term ˙ B sδ ; 4 m [ ψ ]( τ , τ ) cannot yet be absorbed on theleft hand side of the inequality. To do that we will rely on the r p weighted estimates ofTheorem 10.2.1. Remark 10.1.3.
In addition to (cid:15) and δ , the proof of Theorem 10.1.1 will involve severalsmallness constants: C − , (cid:98) δ , δ , δ H , (cid:15) H , Λ − H and Λ − . These smallness constants will bechosen such that < (cid:15) (cid:28) (cid:98) δ, δ H , (cid:15) H , Λ − H , Λ − (cid:28) δ (cid:28) C − . (10.1.2) In addition, (cid:98) δ , (cid:15) H , Λ − H and Λ − will in fact be chosen towards the end of the proof asexplicit powers of δ H , see (10.1.62) , (10.1.64) and Proposition 10.1.30. The goal of this section is to prove Theorem 10.1.1. This will be achieved in section10.1.15.
To prove Theorem 10.1.1, we proceed as follows • In section 10.1.2, we introduce a simplified set of assumptions of the Ricci coefficientswhich is sufficient in order to prove Theorem 10.1.1. • In section 10.1.3, we discuss notations concerning functions depending on m and r . • In section 10.1.4, we compute the deformation tensor of the vectorfields R , T , and X = f ( r, m ) R . • In section 10.1.5, we introduce the basic integral identities for wave equations thatwill be used repeatedly in the proof of Theorem 10.1.1. • In section 10.1.6, we derive the main Morawetz identity. • In section 10.1.7, we derive a first estimate. This estimate is insufficient du to – a lack of positivity of the bulk in to the region 3 m ≤ r ≤ m , – a log divergence of a suitable choice of vectorfield at r = 2 m , – a degeneracy at r = 2 m . • In section 10.1.8, we add a correction and rely on a Poincar´e inequality to obtain apositive estimate also on the region 3 m ≤ r ≤ m . • In section 10.1.9, we perform a cut-off to remove above mentioned log divergence at r = 2 m . • In section 10.1.10, we introduce the red shift vectorfield to remove the above men-tioned degeneracy at r = 2 m . • In section 10.1.11, we combine the previous estimates with the redshift vectorfieldto obtain a bulk term suitable on the whole spacetime M . • In section 10.1.12, we prove the positivity of the boundary terms arising from addinga large multiple of the energy estimate to the Morawetz estimate. • In section 10.1.13, combining the good properties of the bulk and of the boundaryterms established so far, we obtain a first Morawetz estimate providing in particularthe control of the the quantity Mor[ ψ ]. • In section 10.1.14, we analyse an error term appearing in the right-hand side of theabove mentioned Morawetz estimate.52
CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS • Finally, in section 10.1.15, we add a correction to upgrade the control of Mor[ ψ ] tothe control of the quantity Morr[ ψ ], hence concluding the proof of Theorem 10.1.1. To prove Theorem 10.1.1, it suffices to make a simplified set of assumptions. Define u trap = (cid:40) τ for r ∈ [ m , m ] , r / ∈ [ m , m ] . (10.1.3)For k = 0 ,
1, we assume the following.
Mor1.
The renormalized Ricci coefficients ˇΓ ≤ k verify on M = ( int ) M ∪ ( ext ) M , | ˇΓ ≤ k | (cid:46) (cid:15)r − u − − δ dec trap , (cid:12)(cid:12)(cid:12) d ≤ k (cid:16) ω + mr , ξ (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap . (10.1.4) Mor2.
The Gauss curvature K of S and ρ verify, (cid:12)(cid:12)(cid:12) d ≤ k (cid:16) ρ + 2 mr (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap , (cid:12)(cid:12)(cid:12) d ≤ k (cid:16) K − r (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap . (10.1.5) Mor3.
We also assume | m − m | (cid:46) (cid:15)m , | d ≤ k ( e m, r e m ) | (cid:46) (cid:15) u − − δ dec trap . (10.1.6) Remark 10.1.4.
Note that in the case when the bootstrap constant (cid:15) = 0 , i.e.in Schwarzschild,the assumptions made above are consistent with the behavior relative to the regular frame(near horizon) e = Υ − ∂ t − ∂ r , e = ∂ t + Υ ∂ r . m and r In order to prove Theorem 10.1.1, we will adapt the derivation of the Morawetz estimatefor the wave equation in Schwarzschild. In particular, we will need to consider various m and r . Now, m is now a scalar function unlike the Schwarzschild case whereit is constant. To take this into account, we will rely on the following lemma. Lemma 10.1.5.
Let f = f ( r, m ) a C function of r and m . Then, we have e (cid:0) f ( r, m ) (cid:1) = ∂ r f ( r, m ) e ( r ) + O ( (cid:15)r − u − − δ dec trap | ∂ m f | ) ,e (cid:0) f ( r, m ) (cid:1) = ∂ r f ( r, m ) e ( r ) + O ( (cid:15)u − − δ dec trap | ∂ m f | ) ,e (cid:0) e (cid:0) f ( r, m ) (cid:1)(cid:1) = ∂ r f ( r, m ) e ( r ) e ( r ) + ∂ r f ( r, m ) e ( e ( r ))+ O ( (cid:15)r − u − − δ dec trap ( r | ∂ r ∂ m f | + | ∂ m f | )) ,e (cid:0) e (cid:0) f ( r, m ) (cid:1)(cid:1) = ∂ r f ( r, m ) e ( r ) e ( r ) + ∂ r f ( r, m ) e ( e ( r ))+ O ( (cid:15)r − u − − δ dec trap ( r | ∂ r ∂ m f | + | ∂ m f | )) ,e θ (cid:0) f ( r, m ) (cid:1) = 0 . Proof.
Straightforward verification using (10.1.6).
Remark 10.1.6.
Note that in the sequel, ∂ r f will not denote a spacetime coordinatevectorfield applied to f , but instead the partial derivative with respect to the variable r ofthe function f ( r, m ) . R, T , X
Recall the definition (5.1.10) of the regular vectorfields , T = 12 ( e + Υ e ) , R = 12 ( e − Υ e ) . Note that, − g ( T, T ) = g ( R, R ) = Υ , g ( T, R ) = 0 . Note also that, R ( r ) = 1 − mr + O ( (cid:15)u − − δ dec trap ) , T ( r ) = O ( (cid:15)u − − δ dec trap ) . Lemma 10.1.7.
The following hold true. In Schwarzschild, in standard coordinates, we have T = ∂ t , R = Υ ∂ r which are regular near thehorizon. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
1. The components of the deformation tensor of R = ( e − Υ e ) are given by, (cid:12)(cid:12)(cid:12) ( R ) π + 4 mr (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap , (cid:12)(cid:12)(cid:12) ( R ) π ( e A , e B ) − r Υ δ AB (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap , (cid:12)(cid:12)(cid:12) ( R ) π (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap , (cid:12)(cid:12)(cid:12) ( R ) π θ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap , (cid:12)(cid:12)(cid:12) ( R ) π θ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap . Moreover, (cid:12)(cid:12)(cid:12) ( R ) π (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap .
2. If V = − κ κ we have, e ( V ) = 8 r (cid:18) − mr (cid:19) + O ( (cid:15) ) r − u − − δ dec trap ,e ( V ) = − r (cid:18) − mr (cid:19) + O ( (cid:15) ) r − u − − δ dec trap , (10.1.7) and, R ( V ) = − r (cid:18) − mr (cid:19) + O ( (cid:15) ) r − u − − δ dec trap ,T ( V ) = O ( (cid:15) ) r − u − − δ dec trap .
3. All components of the deformation tensor of T = ( e + Υ e ) can be bounded by O ( (cid:15)r − u − − δ dec trap ) . Moreover, (cid:12)(cid:12)(cid:12) ( T ) π (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap . Proof.
We have ( R ) π = g ( D ( e − Υ e ) , e ) = 2 e (Υ) + 4Υ ω, ( R ) π = 12 g ( D ( e − Υ e ) , e ) + 12 g ( D ( e − Υ e ) , e )= e (Υ) − ω + 2 ω, ( R ) π = g ( D ( e − Υ e ) , e ) = − ω, ( R ) π AB = 12 g ( D A ( e − Υ e ) , e B ) + 12 g ( D B ( e − Υ e ) , e A ) , = (1+3) χ AB − Υ (1+3) χ AB = 12 ( κ − Υ κ ) δ AB + (1+3) (cid:98) χ AB − Υ (1+3) (cid:98) χ AB . Note that, e (Υ) = e (cid:18) − mr (cid:19) = 2 mr e ( r ) − e mr = mr ( κ + A ) + O ( (cid:15)r − u − − δ dec trap )= mr κ + O ( (cid:15)r − u − − δ dec trap ) = − mr + O ( (cid:15)r − u − − δ dec trap ) ,e (Υ) = e (cid:18) − mr (cid:19) = 2 mr e ( r ) − e mr = mr ( κ + A ) + O ( (cid:15)r − u − − δ dec trap )= mr κ + O ( (cid:15)r − u − − δ dec trap ) = 2 mr Υ + O ( (cid:15)r − u − − δ dec trap ) . Thus ( R ) π = O ( (cid:15)r − u − − δ dec trap ) , ( R ) π = O ( (cid:15)r − u − − δ dec trap ) , ( R ) π = − mr + O ( (cid:15)r − u − − δ dec trap ) , ( R ) π AB = 2Υ r δ AB + O ( (cid:15)r − u − − δ dec trap ) . Also, in view of, (cid:12)(cid:12)(cid:12) ξ, ξ, η, η, ζ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap , we deduce, (cid:12)(cid:12)(cid:12) ( R ) π θ , ( R ) π θ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − u − − δ dec trap as desired.56 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
To prove the second part of the lemma we write, e ( V ) = − e ( κ ) κ − κe ( κ )= − (cid:18) − κκ + 2 ωκ + 2 ρ + O ( (cid:15)r − u − − δ dec trap ) (cid:19) κ − κ (cid:18) − κ − ω κ + O ( (cid:15)r − u − − δ dec trap ) (cid:19) = ( κκ − ρ ) κ + O ( (cid:15)r − u − − δ dec trap ) . On the other hand, κκ − ρ = − (cid:18) r + O ( (cid:15) ) r − u − − δ dec trap (cid:19) (cid:18) r + O ( (cid:15) ) r − u − − δ dec trap (cid:19) + 4 mr + O ( (cid:15)r − u − − δ dec trap )= − r (cid:18) − mr (cid:19) + O ( (cid:15)r − u − − δ dec trap ) . Hence, e ( V ) = ( κκ − ρ ) κ + O ( (cid:15)r − u − − δ dec trap )= 8 r (cid:18) − mr (cid:19) + O ( (cid:15)r − u − − δ dec trap )and similarly for e ( V ). Thus, R ( V ) = 12 ( e − Υ e ) V = − r (cid:18) − mr (cid:19) + O ( (cid:15)r − u − − δ dec trap ) ,T ( V ) = 12 ( e + Υ e ) V = O ( (cid:15) ) r − u − − δ dec trap , as desired.To prove the last part of the lemma we write, ( T ) π = g ( D ( e + Υ e ) , e ) = − e (Υ) − ω, ( T ) π = 12 g ( D ( e + Υ e ) , e ) + 12 g ( D ( e + Υ e ) , e )= − e (Υ) + 2Υ ω + 2 ω, ( T ) π = g ( D ( e + Υ e ) , e ) = − ω, ( T ) π AB = 12 g ( D A ( e + Υ e ) , e B ) + 12 g ( D B ( e + Υ e ) , e A ) , = (1+3) χ AB + Υ (1+3) χ AB = 12 ( κ + Υ κ ) δ AB + (1+3) (cid:98) χ AB + Υ (1+3) (cid:98) χ AB , and the proof continues as above in view of our assumptions. X = f ( r, m ) R and ( X ) π its deformation tensor. We have the followinglemma. Lemma 10.1.8.
Let X = f ( r, m ) R and ( X ) π its deformation tensor. We have, ( X ) π = ( X ) ˙ π + (cid:15) ( X ) ¨ π where • The only nonvanishing components of ( X ) ˙ π are ( X ) ˙ π = 2 ∂ r f, ( X ) ˙ π = 2 ∂ r f Υ , ( X ) ˙ π = − mr f − ∂ r f Υ , ( X ) ˙ π AB = f r δ AB . • All components of ( X ) ¨ π verify, (cid:12)(cid:12)(cid:12) ( X ) ¨ π (cid:12)(cid:12)(cid:12) (cid:46) r − u − − δ dec trap ( | f | + r | ∂ m f | + r | ∂ r f | ) . Moreover, (cid:12)(cid:12)(cid:12) ( X ) ¨ π (cid:12)(cid:12)(cid:12) (cid:46) r − u − − δ dec trap ( | f | + r | ∂ m f | + r | ∂ r f | ) . Proof.
Clearly, ( X ) π µν = f ( R ) π µµ + e µ f R ν + e ν f R µ . Therefore, since g ( R, e ) = − , g ( R, e ) = Υ and, (cid:12)(cid:12)(cid:12) e ( r ) − Υ , e ( r ) + 1 (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) u − − δ dec trap , Recall from Remark 10.1.6 that ∂ r f does no denote a spacetime coordinate vectorfield applied to f ,but instead the partial derivative with respect to the variable r of the function f ( r, m ). CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS and using Lemma 10.1.5, we deduce, ( X ) π = f ( R ) π − e ( f ) = f ( R ) π − ∂ r f e ( r ) − ∂ m f e ( m )= 2 ∂ r f + O (cid:16) (cid:15)r − u − − δ dec trap ( | f | + r | ∂ m f | + r | ∂ r f | ) (cid:17) ( X ) π = f ( R ) π + 2Υ e ( f ) = f ( R ) π + 2Υ ∂ r f e ( r ) + 2Υ ∂ m f e ( m )= 2 ∂ r f Υ + O (cid:16) ( (cid:15)r − u − − δ dec trap ( | f | + r | ∂ m f | + r | ∂ r f | ) (cid:17) ( X ) π = f ( R ) π + e ( f )Υ − e ( f )= f ( R ) π + ( ∂ r f e ( r ) + ∂ m f e ( m ))Υ − ( ∂ r f e ( r ) + ∂ m f e ( m ))= − mr f − ∂ r f Υ + O (cid:16) (cid:15)r − u − − δ dec trap ( | f | + r | ∂ m f | + r | ∂ r f | ) (cid:17) . This concludes the proof of the lemma.
We recall, see section 2.4.1, that wave equations for ψ ∈ s ( M ) of the form (cid:3) ψ = V ψ + N [ ψ ] , V = − κκ, (10.1.8)can be lifted to the spacetime version,˙ (cid:3) Ψ + V Ψ = N [Ψ] (10.1.9)where Ψ ∈ S ( M ) and N [Ψ] ∈ S ( M ) are defined according to Proposition 2.4.5. In fact,Ψ θθ = − Ψ ϕϕ = ψ, Ψ θϕ = 0 .N θθ [Ψ] = N ϕϕ [Ψ] = N ( ψ ) , N [Ψ] θϕ = 0 . All estimates for (10.1.9) derived in this section can be easily transferred to estimates for(10.1.8) and vice versa.Consider wave equations of the form,˙ (cid:3) g Ψ = V Ψ + N (10.1.10)with Ψ ∈ S ( M ) and N a given symmetric traceless tensor, i.e. N ∈ S ( M ). Proposition 10.1.9.
Assume Ψ ∈ S ( M ) verifies (10.1.9) . Then,
1. The energy momentum tensor Q = Q [Ψ] given by, Q µν : = ˙ D µ Ψ · ˙ D ν Ψ − g µν (cid:16) ˙ D λ Ψ · ˙ D λ Ψ + V Ψ · Ψ (cid:17) = ˙ D µ Ψ · ˙ D ν Ψ − g µν L (Ψ) verifies, D ν Q µν = ˙ D µ Ψ · N [Ψ] + ˙ D ν Ψ A R ABνµ Ψ B − D µ V Ψ · Ψ .
2. The null components of Q are given by, Q = | e Ψ | , Q = | e Ψ | , Q = |∇ / Ψ | + V | Ψ | , and, g µν Q µν = −L (Ψ) − V | Ψ | . Also, |L (Ψ) | (cid:46) | e Ψ | | e Ψ | + |∇ / Ψ | + V | Ψ | and |Q AB | ≤ | e Ψ || e Ψ | + |∇ / Ψ | + | V || Ψ | , |Q A | ≤ | e Ψ ||∇ / Ψ | , |Q A | ≤ | e Ψ ||∇ / Ψ | .
3. Introducing (cid:98) Q := Q − V | Ψ | = |∇ / Ψ | we have, − (cid:98) Q + Q θθ + Q ϕϕ = −L (Ψ) .
4. Let X = ae + be . Then, since R AB = 0 in an axially symmetric polarizedspacetime, D µ ( Q µν X ν ) = 12 Q · ( X ) π + X (Ψ) · N [Ψ] − X ( V )Ψ · Ψ . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
5. Let X = ae + be as above, w a scalar function and M a one form. Define, P µ = P µ [ X, w, M ] = Q µν X ν + 12 w Ψ ˙ D µ Ψ − | Ψ | ∂ µ w + 14 | Ψ | M µ . Then, D µ P µ [ X, w, M ] = 12
Q · ( X ) π − X ( V )Ψ · Ψ + 12 w L [Ψ] − | Ψ | (cid:3) g w + 14 ˙ D µ ( | Ψ | M µ ) + (cid:18) X (Ψ) + 12 w Ψ (cid:19) · N [Ψ] . (10.1.11) Proof.
See sections D.1.4 and D.2 in the appendix.
Notation.
For convenience we introduce the notation, E [ X, w, M ](Ψ) := D µ P µ [ X, w, M ] − (cid:18) X (Ψ) + 12 w Ψ (cid:19) · N [Ψ] . (10.1.12)Thus equation (10.1.11) becomes, E [ X, w, M ](Ψ) = 12
Q · ( X ) π − X ( V )Ψ · Ψ + 12 w L [Ψ] − | Ψ | (cid:3) g w + 14 ˙ D µ ( | Ψ | M µ ) . (10.1.13)When M = 0 we simply write E [ X, w ](Ψ).
Lemma 10.1.10.
Let f ( r, m ) a function of r and m , and let X a vectorfield defined by X = f ( r, m ) R . Then, we have , Q · ( X ) ˙ π = f (cid:18) − mr + 2Υ r (cid:19) |∇ / Ψ | + 2 ∂ r f | R Ψ | − (cid:18) r f + Υ ∂ r f (cid:19) L − mr f V | Ψ | . where ( X ) ˙ π has been defined in Lemma 10.1.8. Recall from Remark 10.1.6 that ∂ r f does no denote a spacetime coordinate vectorfield applied to f ,but instead the partial derivative with respect to the variable r of the function f ( r, m ). Proof.
In view of Lemma 10.1.8, we have
Q · ( X ) ˙ π = 12 Q ˙ π + 14 Q ˙ π + 14 Q ˙ π + Q AB ˙ π AB = − mr f Q − ∂ r f Υ Q + 12 Q ∂ r f + 12 Q Υ ∂ r f + 2Υ r f δ AB Q AB = − mr f Q + 2Υ r f δ AB Q AB + 12 ∂ r f (cid:0) Q − Q + Υ Q (cid:1) Note that, (cid:0) Q − Q + Υ Q (cid:1) = 4 Q RR and, since g µν Q µν = −L (Ψ) − V | Ψ | , δ AB Q AB = Q − L − V | Ψ | = (cid:98) Q − L . Hence,
Q · ( X ) ˙ π = − mr f Q + 2Υ r f (cid:16) (cid:98) Q − L (cid:17) + 2 ∂ r f Q RR = − mr f (cid:16) (cid:98) Q + V | Ψ | (cid:17) + 2Υ r f (cid:16) (cid:98) Q − L (cid:17) + 2 ∂ r f Q RR = f (cid:18) − mr + 2Υ r (cid:19) (cid:98) Q + 2 ∂ r f Q RR − r f L − mr f V | Ψ | . Finally, Q RR = | R Ψ | − g ( R, R ) L = | R Ψ | −
12 Υ L . Hence,
Q · ( X ) ˙ π = 2 f (cid:18) − mr + Υ r (cid:19) (cid:98) Q + 2 ∂ r f (cid:18) | R Ψ | −
12 Υ L (cid:19) − r f L − mr f V | Ψ | = 2 f (cid:18) − mr + Υ r (cid:19) |∇ / Ψ | + 2 ∂ r f | R Ψ | − (cid:18) r f + ∂ r f Υ (cid:19) L − mr f V | Ψ | . This concludes the proof of the lemma.We shall also make use of the following lemma.
Lemma 10.1.11. If f = f ( r, m ) , then (cid:3) g ( f ( r, m )) = r − ∂ r ( r Υ ∂ r f ) + O ( (cid:15)r − u − − δ dec trap ) (cid:104) r | ∂ r f ( r, m ) | + r | ∂ r f ( r, m ) | + r | ∂ r ∂ m f ( r, m ) | + | ∂ m f ( r, m ) | (cid:105) . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Proof.
Recall from Lemma 2.4.1 that, for a general scalar f , (cid:3) g f = −
12 ( e e + e e ) f + (cid:52) / f + (cid:18) (1+3) ω − (1+3) tr χ (cid:19) e f + (cid:18) (1+3) ω − (1+3) tr χ (cid:19) e f. Recall that, (1+3) tr χ = 2 χ − ϑ, (1+3) tr χ = 2 χ − ϑ, (1+3) ω = ω, (1+3) ω = ω and (cid:52) / f = e θ e θ f + ( e θ Φ) e θ f. Using Lemma 10.1.5, we deduce, for a function f = f ( r, m ), (cid:3) g f = −
12 ( e e + e e ) f + (cid:18) ω − κ (cid:19) e f + (cid:18) ω − κ (cid:19) e f = − ∂ r f ( r, m ) e ( r ) e ( r ) − ∂ r f ( r, m ) ( e e ( r ) + e e ( r )) − κ∂ r f ( r, m ) e r + (cid:18) ω − κ (cid:19) ∂ r f ( r, m ) e ( r ) + O ( (cid:15)r − u − − δ dec trap ) (cid:104) r | ∂ r f ( r, m ) | + r | ∂ r f ( r, m ) | + r | ∂ r ∂ m f ( r, m ) | + | ∂ m f ( r, m ) | (cid:105) = − ∂ r f ( r, m ) (cid:16) − Υ + O ( (cid:15)u − − δ dec trap (cid:17) + ∂ r f ( r, m ) mr + ∂ r f ( r, m ) Υ r + r − mr ∂ r f ( r, m )+ O ( (cid:15)r − u − − δ dec trap ) (cid:104) r | ∂ r f ( r, m ) | + r | ∂ r f ( r, m ) | + r | ∂ r ∂ m f ( r, m ) | + | ∂ m f ( r, m ) | (cid:105) = Υ ∂ r f ( r, m ) + ∂ r f ( r, m ) (cid:18) r − mr (cid:19) + O ( (cid:15)r − u − − δ dec trap ) (cid:104) r | ∂ r f ( r, m ) | + r | ∂ r f ( r, m ) | + r | ∂ r ∂ m f ( r, m ) | + | ∂ m f ( r, m ) | (cid:105) = r − ∂ r ( r Υ ∂ r f )+ O ( (cid:15)r − u − − δ dec trap ) (cid:104) r | ∂ r f ( r, m ) | + r | ∂ r f ( r, m ) | + r | ∂ r ∂ m f ( r, m ) | + | ∂ m f ( r, m ) | (cid:105) as desired.According to equation (10.1.13) we have, E [ X, w ](Ψ) = 12
Q · ( X ) π − X ( V ) | Ψ | + 12 w L (Ψ) − | Ψ | (cid:3) g w. In the next proposition we choose X to be of the form X = f ( r, m ) R and make a choiceof w as a function of f . Proposition 10.1.12.
Assume X = f ( r, m ) R and w ( r, m ) = r − Υ ∂ r ( r f ) . Then, E [ X, w ](Ψ) = ˙ E [ X, w ] + E (cid:15) [ X, w ] where, with (cid:98) Q := Q − V | Ψ | = |∇ / Ψ | , ˙ E [ f R, w ](Ψ) = 1 r (cid:18) − mr (cid:19) f (cid:98) Q + ∂ r f | R (Ψ) | − r − ∂ r ( r Υ ∂ r w ) | Ψ | + 4Υ r − mr f | Ψ | , E (cid:15) [ f R, w ](Ψ) = (cid:15) Q · ( X ) ¨ π + O (cid:16) (cid:15)r − u − − δ dec trap (cid:0) | f | + r | ∂ r w | + r | ∂ r w | + r | ∂ r ∂ m w | + r | ∂ m w | (cid:1)(cid:17) | Ψ | . (10.1.14) Proof.
According to Lemma 10.1.8 and equation (10.1.13) we have, E [ X, w ](Ψ) = 12
Q · ( ( X ) ˙ π + (cid:15) ( X ) ¨ π ) − X ( V ) | Ψ | + 12 w L (Ψ) − | Ψ | (cid:3) g w. Hence, in view of lemmas 10.1.8 and 10.1.10, E [ X, w ](Ψ) − (cid:15) Q · ( X ) ¨ π = 12 Q · ( X ) ˙ π − X ( V ) | Ψ | + 12 w L (Ψ) − | Ψ | (cid:3) g w = f (cid:18) − mr + Υ r (cid:19) |∇ / Ψ | + ∂ r f | R Ψ | − (cid:18) Υ r f + 12 Υ ∂ r f (cid:19) L (Ψ) − mr f V | Ψ | − X ( V ) | Ψ | + 12 w L (Ψ) − | Ψ | (cid:3) g w. Thus, assuming w = r − Υ ∂ r ( r f ) = r f + ∂ r f Υ, E [ X, w ](Ψ) − (cid:15) Q · ( X ) ¨ π = r − f (cid:18) − mr (cid:19) |∇ / Ψ | + ∂ r f | R Ψ | − (cid:18) mr f V + 12 X ( V ) + 14 (cid:3) g w (cid:19) | Ψ | . Note that, in view of Lemma 10.1.7, X ( V ) = f R ( V ) = − f r − mr + O ( (cid:15)r − u − − δ dec trap | f | )64 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS and, mr f V + 12 X ( V ) = f (cid:18) mr Υ − r − mr (cid:19) + O ( (cid:15)r − u − − δ dec trap | f | )= − f Υ r − mr + O ( (cid:15)r − u − − δ dec trap | f | ) . Note also that, in view of Lemma 10.1.11 (cid:3) g ( w ) = r − ∂ r ( r Υ ∂ r w ) + O ( (cid:15)r − u − − δ dec trap ) (cid:104) r | ∂ r w | + r | ∂ r w | + r | ∂ r ∂ m w | + | ∂ m w | (cid:105) . Thus, mr f V + 12 X ( V ) + 14 (cid:3) g w = − r − mr f + 14 r − ∂ r ( r Υ ∂ r w )+ O ( (cid:15)r − u − − δ dec trap ) (cid:104) | f | + r | ∂ r w | + r | ∂ r w | + r | ∂ r ∂ m w | + r | ∂ m w | (cid:105) and hence E [ X, w ](Ψ) − (cid:15) Q · ( X ) ¨ π = r − f (cid:18) − mr (cid:19) |∇ / Ψ | + ∂ r f | R Ψ | − | Ψ | r − ∂ r ( r Υ ∂ r w )+ 4Υ r − mr f | Ψ | + O (cid:16) (cid:15)r − u − − δ dec trap (cid:0) | f | + r | ∂ r w | + r | ∂ r w | + r | ∂ r ∂ m w | + r | ∂ m w | (cid:1)(cid:17) | Ψ | as desired. We concentrate our attention on the principal term˙ E [ f R, w ](Ψ) = 1 r (cid:18) − mr (cid:19) f (cid:98) Q + ∂ r f | R (Ψ) | − r − ∂ r ( r Υ ∂ r w ) | Ψ | + 4Υ r − mr f | Ψ | and choose f = f ( r, m ) such that the right hand side is positive definite.Consider the quadratic forms,˙ E (Ψ) : = A (cid:98) Q + B | R Ψ | + r − W | Ψ | , ˙ E (Ψ) : = ˙ E (Ψ) + 4Υ r − mr f | Ψ | , (10.1.15) A := r − f (cid:18) − mr (cid:19) , B := ∂ r f, W := − ∂ r ( r Υ ∂ r w ) . (10.1.16)The goal is to show that there exist choices of f, w verifying the condition of Proposition10.1.12, i.e. w = r − ∂ r ( r f ), which makes ˙ E (Ψ) positive definite, for all smooth S -valued tensorfields Ψ defined in the region r ≥ m (1 − δ H ), which decay reasonable fastat infinity. We look first for choices of f, w such that the coefficient A, B, W are non-negative. Note in particular that f must be increasing as a function of r and f = 0 on r = 3 m . Following J. Stogin [53] we choose w first to ensure that W is non-negative andthen choose f , compatible with the equation, ∂ r ( r f ) = r Υ w, f = 0 on r = 3 m. (10.1.17)To ensure that A = r − f ( r − m ) is positive we need a non-negative w which verifies(modulo error terms ) W = − ∂ r ( r Υ ∂ r w ) ≥
0. It is more difficult to choose w such that B = ∂ r f is also non-negative.Stogin defines w based on the following lemma. Lemma 10.1.13.
The scalar function w defined by w ( r, m ) = (cid:40) m , if r ≤ m, r , if r ≥ m, is C , non-negative and such that W = − ∂ r ( r Υ ∂ r w ) verifies, W ( r, m ) = (cid:40) , if r < m, mr (cid:0) − mr (cid:1) , if r > m. (10.1.18) Proof.
For r ≥ m , we have w ( r, m ) = 2Υ r , ∂ r w ( r, m ) = − r + 8 mr , ∂ r w ( r, m ) = 4 r − mr . In particular, we have w = 14 m , ∂ r w = 0 at r = 4 m i.e. terms which vanish in Schwarzschild. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS so that w is indeed C . Furthermore, we also have r − ∂ r ( r Υ ∂ r w ) = Υ ∂ r w ( r ) + ∂ r w ( r ) (cid:18) r − mr (cid:19) = Υ (cid:18) r − mr (cid:19) + (cid:18) − r + 8 mr (cid:19) (cid:18) r − mr (cid:19) = − mr (cid:18) − mr (cid:19) so that, for r ≥ m , W = − ∂ r ( r Υ ∂ r w ) = mr (cid:18) − mr (cid:19) as desired.Once w is defined we can evaluate f as follows. Lemma 10.1.14.
Let w ( r, m ) defined as in Lemma 10.1.13. Then, the function f ( r, m ) given by, r f ( r, m ) := (cid:40) m log (cid:0) r − mm (cid:1) + ( r − m ) r +6 mr +30 m m , for r ≤ m,C ∗ m + r − (4 m ) , for r ≥ m, (10.1.19) with the constant C ∗ given by C ∗ := 2 log(2) + 356 , C ∗ ∼ . , is C and satisfies (10.1.17) , i.e. we have ∂ r ( r f ) = r Υ − w, f = 0 on r = 3 m. Proof.
By direct check , we have for r ≤ m∂ r ( r f )( r, m ) = 2 m ( r − m ) + r + 6 mr + 30 m m + ( r − m ) 2 r + 6 m m = r m ( r − m )= r Υ 14 m C ∗ is chosen so that f is continuous across r = 4 m . Recall from Remark 10.1.6 that ∂ r f does no denote a spacetime coordinate vectorfield applied to f ,but instead the partial derivative with respect to the variable r of the function f ( r, m ). r ≥ m ∂ r ( r f )( r, m ) = 2 r, as well as f = 0 on r = 3 m so that, in view of the definition of w ( r ) in Lemma 10.1.13,we infer ∂ r ( r f ) = r Υ − w, f = 0 on r = 3 m as desired. Note also that w being C , f is thus indeed C .Next, we derive a lower bound on ∂ r f for r ≤ m . Lemma 10.1.15.
We have for all r and mr ∂ r f ≥ m . Also, there exists a constant
C > such that for all r and m (cid:18) − mr (cid:19) f ≥ C − (cid:18) − mr (cid:19) . Proof.
We have ∂ r ( r ∂ r f ) = ∂ r ( r∂ r ( r f )) − ∂ r ( r f ) . Using the identity ∂ r ( r f ) = r Υ − w , we infer ∂ r ( r ∂ r f ) = ∂ r ( r Υ − w ) − r Υ − w. For r ≤ m , we have w = (4 m ) − and hence ∂ r ( r ∂ r f ) = 14 m (cid:0) ∂ r ( r Υ − ) − r Υ − (cid:1) = 14 m r ∂ r (Υ − )= − r In particular, r ∂ r f is decreasing in r on r ≤ m and hence r ∂ r f ≥ (4 m ) ∂ r f ( r = 4 m, m ) on r ≤ m. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
On the other hand, we have, in view of the definition 10.1.19 of f∂ r ( r f )( r = 4 m, m ) = (4 m ) ∂ r f ( r = 4 m, m ) + 8 mf ( r = 4 m, m )= (4 m ) ∂ r f ( r = 4 m, m ) + m C ∗ and hence (4 m ) ∂ r f ( r = 4 m, m ) = (cid:18) − C ∗ (cid:19) m so that r ∂ r f ≥ − C ∗ ) m on r ≤ m. Since C ∗ ∼ . <
8, we deduce r ∂ r f ≥ m on r ≤ m. Also, for r ≥ m , we have f = 1 − (16 − C ∗ ) m r so that ∂ r f = 2(16 − C ∗ ) m r . Since C ∗ ∼ . <
8, we deduce r ∂ r f ≥ m on r ≥ m which together with the case r ≤ m above yields for all r and m the desired estimate for ∂ r f r ∂ r f ≥ m . In particular, ∂ r f > f = 0 on r = 3and converges to 1 as r → + ∞ . We deduce the existence of a constant C > (cid:18) − mr (cid:19) f ≥ C − (cid:18) − mr (cid:19) as desired. Proposition 10.1.16.
There exist functions f ∈ C , w ∈ C verifying the relation w = r − Υ ∂ r ( r f ) and such that, r f = (cid:40) m log (cid:0) r − mm (cid:1) + ( r − m ) r +6 mr +30 m m , for r ≤ m,C ∗ m + r − (4 m ) , for r ≥ m, (10.1.20) where C ∗ is a constant satisfying < C ∗ < . In particular, f = (cid:40) m r log (cid:0) r − mm (cid:1) + O ( r − mm ) , for r ≤ m, O ( m r ) , for r ≥ m, (10.1.21) and, for some C > and all r ≥ m (cid:18) − mr (cid:19) f ≥ C − (cid:18) − mr (cid:19) , ∂ r f ≥ m r . (10.1.22) Also, w is given by w = (cid:40) m , for r ≤ m, r (cid:0) − mr (cid:1) , for r ≥ m. (10.1.23) Moreover W = − ∂ r ( r Υ ∂ r w ) verifies, W = (cid:40) , if r < m, mr (cid:0) − mr (cid:1) , if r > m, (10.1.24) and, ˙ E [ f R, w ](Ψ) = ∂ r f | R (Ψ) | + r − W | Ψ | + r − (cid:18) − mr (cid:19) f (cid:98) Q , ˙ E [ f R, w ](Ψ) = E [ f R, w ](Ψ) + 4Υ r − mr f | Ψ | . (10.1.25) Recall also that, (cid:98) Q = |∇ / Ψ | . Remark 10.1.17.
The estimates obtained so far have two major deficiencies1. The quadratic form ˙ E [ f R, w ](Ψ) + 4Υ r − mr f | Ψ | fails to be positive definite in theregion m ≤ r ≤ m because of the potential term Υ r − mr f | Ψ | .2. The function f blows up logarithmically at r = 2 m in ( int ) M .In the next section we deal with the first issue. We handle the second problem in thefollowing two sections. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS ( ext ) M Note that the term 4 f Υ r − mr is negative for 3 m ≤ r ≤ m and positive everywhere else.An improvement can be obtained by using the following Poincar´e inequality. Lemma 10.1.18.
We have for Ψ ∈ S ( M ) , (cid:90) S |∇ / Ψ | ≥ r − (cid:16) − O ( (cid:15) ) (cid:17) (cid:90) S Ψ da S . (10.1.26) Proof.
See Proposition 2.1.32.According to Proposition 10.1.16 we deduce, (cid:90) S ˙ E [ f R, w ](Ψ) ≥ (cid:90) S ˙ E − O ( (cid:15)r − ) (cid:90) S Ψ da S , ˙ E := ∂ r f | R (Ψ) | + r − W | Ψ | + 2 r − (cid:18) − mr (cid:19) f | Ψ | + 4Υ r − mr f | Ψ | , (10.1.27)with W defined in (10.1.24). It is easy to see however that ˙ E still fails to be positivefor 3 m < r < m . To achieve positivity we also need to modify the original energydensity E [ f R, w ](Ψ) by considering instead the modified energy density E [ f R, w, M ](Ψ)(see (10.1.11) and notation (10.1.12)) with M = 2 hR for a function h = h ( r, m ) supportedfor r ≥ m and constant for r ≥ m . E [ f R, w, M ](Ψ) = E [ f R, w ](Ψ) + 14 ˙ D µ ( | Ψ | M µ ) = E [ f R, w ](Ψ) + 14 ( D µ M µ ) | Ψ | + 12 Ψ M (Ψ)= E [ f R, w ](Ψ) + 12 D µ ( hR µ ) | Ψ | + h Ψ R (Ψ) . To take into account the additional terms in the modified E [ f R, w, M ](Ψ) we first derivethe following. Lemma 10.1.19.
Let h ( r, m ) a C function of r and m . We have, D µ ( hR µ ) = r − ∂ r (Υ r h ) + O (cid:16) (cid:15) r − u − − δ dec trap (cid:0) r | ∂ r h | + | h | + r | ∂ m h | (cid:1)(cid:17) . (10.1.28) Proof.
In view of Lemma 10.1.7, which computes the components of ( R ) π , as well as R ( h ), we calculate, D µ ( hR µ ) = R ( h ) + h ( D µ R µ ) = 12 ( e ( h ) − Υ e ( h )) + h
12 tr ( ( R ) π )= 12 ( e ( r ) − Υ e ( r )) ∂ r h + O ( (cid:15) u − − δ dec trap | ∂ m h | ) + 12 h (cid:0) − ( R ) π + ( R ) π θθ + ( R ) π ϕϕ (cid:1) = Υ ∂ r h + 12 (cid:18) mr + 4 Υ r (cid:19) h + O (cid:16) (cid:15) u − − δ dec trap (cid:0) | ∂ r h | + r − | h | + | ∂ m h | (cid:1)(cid:17) = r − ∂ r (Υ r h ) + O (cid:16) (cid:15) u − − δ dec trap (cid:0) | ∂ r h | + r − | h | + | ∂ m h | (cid:1)(cid:17) as desired.In view of the lemma we write, E [ f R, w, hR ](Ψ) = ˙ E [ f R, w, hR ](Ψ) + E (cid:15) [ f R, w, hR ](Ψ) , ˙ E [ f R, w, hR ](Ψ) : = ˙ E [ f R, w ](Ψ) + 12 r − ∂ r (Υ r h ) | Ψ | + h Ψ R (Ψ) , E (cid:15) [ f R, w, hR ](Ψ) : = E (cid:15) [ f R, w ](Ψ) + O (cid:16) r − (cid:15) u − − δ dec trap (cid:0) r | ∂ r h | + | h | + r | ∂ m h | (cid:1)(cid:17) | Ψ | . (10.1.29)The main result of this section is stated below. Proposition 10.1.20.
There exists a function h = h ( r, m ) with bounded derivative h (cid:48) ,supported in r ≥ m such that h = O ( r − ) , h (cid:48) = O ( r − ) for for r ≥ m such that, E [ f R, w, hR ](Ψ) = ˙ E [ f R, w, hR ](Ψ) + E (cid:15) [ f R, w, hR ](Ψ) , E (cid:15) [ f R, w, hR ](Ψ) = (cid:15) Q · ( X ) ¨ π + O (cid:16) (cid:15)r − u − − δ dec trap ( | f | + 1) (cid:17) | Ψ | , (10.1.30) and, for sufficiently large universal constant C > , in the region r ≥ m , (cid:90) S ˙ E [ f R, w, hR ](Ψ) ≥ C − (cid:90) S (cid:32) m r | R (Ψ) | + r − (cid:18) − mr (cid:19) |∇ / Ψ | + mr | Ψ | (cid:33) . (10.1.31) Proof.
We first derive the weaker inequality, (cid:90) S ˙ E [ f R, w, hR ](Ψ) ≥ C − (cid:90) S (cid:18) m r | R (Ψ) | + mr | Ψ | (cid:19) on r ≥ m CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS by making full use of the Poincar´e inequality above, i.e., (cid:90) S r − (cid:18) − mr (cid:19) f ( r, m ) |∇ / Ψ | ≥ (cid:90) S (2 − O ( (cid:15) )) r − (cid:18) − mr (cid:19) f ( r, m ) | Ψ | . The result will the easily follow by writing instead, with a sufficiently small µ > (cid:90) S r − (cid:18) − mr (cid:19) f ( r, m ) |∇ / Ψ | = µ (cid:90) S r − (cid:18) − mr (cid:19) f ( r, m ) |∇ / Ψ | + (1 − µ ) (cid:90) S r − (cid:18) − mr (cid:19) f ( r, m ) |∇ / Ψ | ≥ µ (cid:90) S r − (cid:18) − mr (cid:19) f ( r, m ) |∇ / Ψ | + (1 − µ ) (cid:90) S r − (cid:18) − mr (cid:19) f ( r, m ) | Ψ | and then proceeding exactly as below.We start with,˙ E [ f R, w, hR ](Ψ) = ˙ E [ f R, w ](Ψ) + 12 r − ∂ r (Υ r h ) | Ψ | + h Ψ R (Ψ) . Recalling the definition of ˙ E in (10.1.27),˙ E := ∂ r f | R (Ψ) | + r − W | Ψ | + 2 r − (cid:18) − mr (cid:19) f | Ψ | + 4Υ r − mr f | Ψ | and setting,˙ E := ˙ E + 12 r − (Υ r h ) (cid:48) | Ψ | + h Ψ R (Ψ) (10.1.32)= ∂ r f | R (Ψ) | + 2 r − (cid:18) − mr (cid:19) f | Ψ | + 4Υ r − mr f | Ψ | + r − W | Ψ | + 12 r − (Υ r h ) (cid:48) | Ψ | + h Ψ R (Ψ)we deduce, from (10.1.27) (cid:90) S ˙ E [ f R, w, hR ](Ψ) ≥ (cid:90) S ˙ E − O ( (cid:15)r − ) (cid:90) S | Ψ | . We now substitute, h = 4Υ r − (cid:101) h. r − ∂ r (Υ r h ) | Ψ | + h Ψ R (Ψ) = 12 r − ∂ r (4Υ r − (cid:101) h ) | Ψ | + 4Υ r − (cid:101) h Ψ R (Ψ)= 12 r − ∂ r (4Υ r − ) (cid:101) h | Ψ | + 2 r − Υ ∂ r (cid:101) h | Ψ | + 4Υ r − (cid:101) h Ψ R (Ψ)or, since r − ∂ r (4Υ r − ) = − r − Υ r − mr ,12 r − ∂ r (Υ r h ) | Ψ | + h Ψ R (Ψ) = − r − Υ r − mr (cid:101) h | Ψ | + 2 r − Υ ∂ r (cid:101) h | Ψ | + 4Υ r − (cid:101) h Ψ R (Ψ) . Thus we have,˙ E = ∂ r f | R (Ψ) | + 2 r − (cid:18) − mr (cid:19) f | Ψ | + 4Υ r − mr ( f − r − (cid:101) h ) | Ψ | + 2 r − Υ ∂ r (cid:101) h | Ψ | + 4Υ r − (cid:101) h Ψ R (Ψ) + r − W | Ψ | . We also express,4Υ r − (cid:101) h Ψ R (Ψ) = 2 (cid:101) hr ( R (Ψ) + Υ r − Ψ) − (cid:101) hr | R (Ψ) | − (cid:101) hr Υ | Ψ | and therefore,˙ E = ( ∂ r f − r − (cid:101) h ) | R (Ψ) | + 2 (cid:101) hr ( R (Ψ) + Υ r − Ψ) + r − W | Ψ | + (cid:20) r − (cid:18) − mr (cid:19) f + 4Υ r − mr ( f − r − (cid:101) h ) + 2 r − Υ ∂ r (cid:101) h − r − Υ (cid:101) h (cid:21) | Ψ | . We choose (cid:101) h ( r, m ) as the following continuous and piecewise C function, (cid:101) h = , r ≤ m ,δ (cid:101) h (cid:0) m − r (cid:1) , m ≤ r ≤ m ,δ (cid:101) h ( r − m ) , m ≤ r ≤ m,r f, m ≤ r ≤ m, (4 m ) f (4 m, m ) , r ≥ m. where the constant δ (cid:101) h > CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Case 1 ( m ≤ r ≤ m ). In view of the definition of (cid:101) h and since W = 0, we deduce,˙ E = ∂ r f | R (Ψ) | + (cid:34) r − (cid:18) − mr (cid:19) f + 4Υ r − mr ( f − r − (cid:101) h )+2 δ (cid:101) h r − Υ (cid:16) m ≤ r ≤ m − m ≤ r ≤ m (cid:17)(cid:35) | Ψ | + δ (cid:101) h O (1)Ψ R (Ψ)1 m ≤ r ≤ m . In view of (10.1.22), we may assume, choosing for δ (cid:101) h > f − ˜ h ≤ − | f | on r ≤ m. (10.1.33)We infer, using also that f < r ≤ m ,˙ E ≥ ∂ r f | R (Ψ) | + (cid:34) r − (cid:18) − mr (cid:19) f + 2Υ r − mr f +2 δ (cid:101) h r − Υ (cid:16) m ≤ r ≤ m − m ≤ r ≤ m (cid:17)(cid:35) | Ψ | + δ (cid:101) h O (1)Ψ R (Ψ)1 m ≤ r ≤ m . Since we have ∂ r f (cid:38) , r − (cid:18) − mr (cid:19) + 4Υ r − mr (cid:46) − , f (cid:46) − (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) on r ≤ m, where we have used in particular Lemma 10.1.15 and Proposition 10.1.16, we infer˙ E (cid:38) | R (Ψ) | + (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) + δ (cid:101) h m ≤ r ≤ m − O (1) δ (cid:101) h m ≤ r ≤ m (cid:19) | Ψ | − δ (cid:101) h O (1)Ψ R (Ψ)1 m ≤ r ≤ m ≥ | R (Ψ) | + (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) + δ (cid:101) h (cid:16) − O (1) δ (cid:101) h (cid:17) m ≤ r ≤ m − O (1) δ (cid:101) h m ≤ r ≤ m (cid:19) | Ψ | . Thus, for δ (cid:101) h > C > E ≥ C − (cid:2) | R (Ψ) | + | Ψ | (cid:3) on 5 m ≤ r ≤ m. (10.1.34) Case 2 (3 m ≤ r ≤ m ). Since (cid:101) h = r f and W = 0, using in particular (cid:101) h ≥ m ≤ r ≤ m , we deduce,˙ E ≥ ( ∂ r f − r − ( r f )) | R (Ψ) | + (cid:20) r − (cid:18) − mr (cid:19) f + 2 r − Υ ∂ r ( r f ) − r − Υ ( r f ) (cid:21) | Ψ | = ( ∂ r f − r − f ) | R (Ψ) | + (cid:20) r − (cid:18) − mr (cid:19) f + 2 r − Υ (2 rf + r ∂ r f ) − r − Υ f (cid:21) | Ψ | = ( ∂ r f − r − f ) | R (Ψ) | + (cid:20) r − (cid:18) − mr (cid:19) f + 2 r − Υ ∂ r f + 2 r − Υ f (cid:21) | Ψ | . Lemma 10.1.21.
In the interval [3 m, m ] we have, ∂ r f − r − f > . Proof.
Recall from Proposition 10.1.16 that w = r − Υ ∂ r ( r f ) = m in the interval[3 m, m ]. Using also f = 0 on r = 3 m , we deduce ∂ r ( r f ) = r Υ 14 m .
We compute ∂ r (cid:18) r f − ( r − m ) r m Υ (cid:19) = − ( r − m )4 m ∂ r (cid:18) r Υ (cid:19) = − ( r − m )( r − m )2 m Υ ≤ m ≤ r ≤ m, so that the differentiated quantity decays in r on [3 m, m ]. Since it vanishes on r = 3 m ,we infer f ≤ ( r − m )4 m Υ on 3 m ≤ r ≤ m. Thus, we deduce, using again ∂ r ( r f ) = r Υ 14 m , ∂ r f − r f = r − (cid:16) ∂ r ( r f ) − rf (cid:17) = 14 m Υ − r f ≥ m Υ − ( r − m ) rm Υ ≥ m Υ (cid:18) − (cid:18) − mr (cid:19)(cid:19) > m ≤ r < m. On the other hand, we have by direct check at r = 4 m , using (10.1.20), (cid:18) ∂ r f − r f (cid:19) r =4 m = 12 m − m f r =4 m = 12 m (cid:18) − C ∗ (cid:19) > C ∗ <
8. Hence, we infer ∂ r f − r − f > m ≤ r ≤ m as desired.76 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
We thus conclude, for some
C >
0, in the interval [3 m, m ]˙ E ≥ C − (cid:2) | R (Ψ) | + | Ψ | (cid:3) . (10.1.35) Case 3 ( r ≥ m ). Since (cid:101) h is constant and positive on r ≥ m , we deduce,˙ E ≥ ( ∂ r f − r − (cid:101) h ) | R (Ψ) | + (cid:20) r − (cid:18) − mr (cid:19) f + 4Υ r − mr ( f − r − (cid:101) h ) − r − Υ (cid:101) h + r − W (cid:21) | Ψ | . We examine the first term. In view of the formula for f for r ≥ m , see (10.1.20), ∂ r f = 2 r (16 − C ∗ ) m , (cid:101) h = (4 m ) f (4 m, m ) = C ∗ m and hence ∂ r f − r − (cid:101) h = 2(16 − C ∗ ) m r and hence, since C ∗ <
8, we have ∂ r f − r − (cid:101) h (cid:38) m r for r ≥ m. It remains to analyze the sign of2 r − (cid:18) − mr (cid:19) f + 4Υ r − mr ( f − r − (cid:101) h ) − r − Υ (cid:101) h = (cid:20) r − (cid:18) − mr (cid:19) + 4Υ r − mr (cid:21) ( f − r − (cid:101) h ) + (cid:20) r − (cid:18) − mr (cid:19) − r − Υ (cid:21) r − (cid:101) h. The first term, which can be written in the form, (cid:20) r − (cid:18) − mr (cid:19) + 4Υ r − mr (cid:21) r − (cid:16) r f ( r, m ) − (4 m ) f (4 m, m ) (cid:17) is manifestly positive for r ≥ m . To evaluate the sign of the second term we calculate,2 r − (cid:18) − mr (cid:19) − r − Υ = 2 mr − ( r − m ) . Thus, for r ≥ m ,2 r − (cid:18) − mr (cid:19) f + 4Υ r − mr ( f − r − (cid:101) h ) − r − Υ (cid:101) h ≥ . W = mr (cid:0) − mr (cid:1) , we have r − W (cid:38) r . Thus, in view of the above, we have for some
C > r ≥ m ,˙ E ≥ C − (cid:2) r | R (Ψ) | + r | Ψ | (cid:3) . (10.1.36)Gathering (10.1.34), (10.1.35) and (10.1.36), we infer for some C > E ≥ C − (cid:2) r | R (Ψ) | + r | Ψ | (cid:3) on r ≥ m . Recalling (cid:90) S ˙ E [ f R, w, hR ](Ψ) ≥ (cid:90) S ˙ E − O ( (cid:15)r − ) (cid:90) S | Ψ | , we infer (cid:90) S ˙ E [ f R, w, hR ](Ψ) ≥ C − (cid:90) S (cid:20) r | R (Ψ) | + 1 r | Ψ | (cid:21) − O ( (cid:15)r − ) (cid:90) S | Ψ | and hence, for (cid:15) > (cid:90) S ˙ E [ f R, w, hR ](Ψ) ≥ C − (cid:90) S (cid:20) r | R (Ψ) | + 1 r | Ψ | (cid:21) on r ≥ m E (cid:15) [ f R, w, hR ](Ψ) = E (cid:15) [ f R, w ](Ψ) + O (cid:16) r − (cid:15) u − − δ dec trap ( r | ∂ r h | + | h | + r | ∂ m h | ) (cid:17) | Ψ | = (cid:15) Q ( X ) ¨ π + O ( (cid:15)r − u − − δ dec trap ( | f | + r | ∂ r w | + r | ∂ r w | + r | ∂ r ∂ m w | + r | ∂ m w | )) | Ψ | + O (cid:16) r − (cid:15) u − − δ dec trap ( r | ∂ r h | + r | h | + r | ∂ m h | ) (cid:17) | Ψ | . Recall that, w = (cid:40) m , for r ≤ m, r (cid:0) − mr (cid:1) , for r ≥ m, CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS and h = 4Υ r − (cid:101) h , with (cid:101) h = , r ≤ m ,δ (cid:101) h (cid:0) m − r (cid:1) , m ≤ r ≤ m ,δ (cid:101) h ( r − m ) , m ≤ r ≤ m,r f, m ≤ r ≤ m, (4 m ) f (4 m, m ) , r ≥ m. We deduce, E (cid:15) [ f R, w, hR ](Ψ) = (cid:15) Q ( X ) ¨ π + O (cid:16) (cid:15)r − u − − δ dec trap ( | f | + 1) (cid:17) which concludes the proof of Proposition 10.1.20. ( int ) M So far we have found a triplet ( X = f R, w = r − Υ ∂ r ( r f ) , M = 2 hR ) with f defined inProposition 10.1.16 and h in Proposition 10.1.20 allowing for the lower bound (10.1.31)on (cid:82) S ˙ E [ f R, w, M ](Ψ). The main problem which remains to be addressed is that1. f blows up logarithmically near r = 2 m .2. The lower bound for (cid:82) S ˙ E [ f R, w, hR ](Ψ) does not control e (Ψ) near r = 2 m .In this section, we deal with the first problem, while the second problem will be treatedin section 10.1.10. To correct for the first problem, i.e. the fact that f blows up logarith-mically near r = 2 m , we have to modify our choice of f and w there. Introducing u := r f, we have, f = r − u, w = r − Υ ∂ r u. (10.1.37) Warning.
The auxiliary function u introduced here, and used only in this section, hasof course nothing to do with our previously defined optical function on ( ext ) M . Definition 10.1.22.
For a given (cid:98) δ > we define the following functions of ( r, m ) u (cid:98) δ := − m (cid:98) δ F (cid:32) − (cid:98) δm u (cid:33) , f (cid:98) δ := r − u (cid:98) δ ,w (cid:98) δ := r − Υ ∂ r u (cid:98) δ , W (cid:98) δ := − ∂ r (cid:0) r Υ ∂ r w (cid:98) δ (cid:1) , where F : R −→ R is is a fixed, increasing, smooth function such that F ( x ) = (cid:40) x for x ≤ , for x ≥ . We now derive useful properties satisfied by f (cid:98) δ , w (cid:98) δ and W (cid:98) δ . Lemma 10.1.23.
Let f (cid:98) δ , w (cid:98) δ and W (cid:98) δ introduced in definition 10.1.22. Then, f (cid:98) δ ∈ C ( r > , w (cid:98) δ ∈ C ( r > , and we have for (cid:98) δ > sufficiently small f (cid:98) δ = f w (cid:98) δ = w, W (cid:98) δ = W for r ≥ m . Also, we have for all r > r − f (cid:98) δ (cid:18) − mr (cid:19) ≥ C − r − (cid:18) − mr (cid:19) (10.1.38) and ∂ r ( f (cid:98) δ ) ≥ m r . (10.1.39) Proof.
Note first that w (cid:98) δ = r − Υ ∂ r u (cid:98) δ = r − Υ F (cid:48) (cid:32) − (cid:98) δm u (cid:33) ∂ r u = wF (cid:48) (cid:32) − (cid:98) δm u (cid:33) . In view of the definition of u (cid:98) δ , f (cid:98) δ , w (cid:98) δ and W (cid:98) δ , we have u (cid:98) δ = u, f (cid:98) δ = f, w (cid:98) δ = w, W (cid:98) δ = W for u ≥ − m (cid:98) δ ,u (cid:98) δ = − m (cid:98) δ , f (cid:98) δ = − m (cid:98) δr , w (cid:98) δ = 0 , W (cid:98) δ = 0 for u ≤ − m (cid:98) δ . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Also, according to (10.1.21) u = (cid:40) m log r − mm + O ( m ( r − m )) , for r ≤ m,r + O ( m ) , for r ≥ m, and hence, for (cid:98) δ > (cid:110) r ≥ m + e − (cid:98) δ (cid:111) ∪ (cid:26) u ≥ − m (cid:98) δ (cid:27) , (cid:110) r ≤ m + e − (cid:98) δ (cid:111) ⊂ (cid:26) u ≤ − m (cid:98) δ (cid:27) . This yields f (cid:98) δ = f w (cid:98) δ = w, W (cid:98) δ = W for r ≥ m . Also, we have f (cid:98) δ = (cid:40) − m (cid:98) δr , for r ≤ m + e − (cid:98) δ ,f, for r ≥ m + e − (cid:98) δ , and f (cid:98) δ (cid:38) (cid:98) δ on 2 m + e − (cid:98) δ ≤ r ≤ m + e − (cid:98) δ , and thus, there exists C > r > r − f (cid:98) δ (cid:18) − mr (cid:19) ≥ C − r − (cid:18) − mr (cid:19) which is (10.1.38).For u ≤ − m (cid:98) δ , ∂ r ( f (cid:98) δ ) = ∂ r ( r − u (cid:98) δ ) = − r − u (cid:98) δ + r − ∂ r ( u (cid:98) δ ) = 4 m (cid:98) δ r − . For − m (cid:98) δ ≤ u ≤ − m (cid:98) δ ∂ r ( f (cid:98) δ ) = ∂ r ( r − u (cid:98) δ ) = − r − u (cid:98) δ + r − ∂ r ( u (cid:98) δ )= − r − u (cid:98) δ + r − F (cid:48) (cid:32) − (cid:98) δm u (cid:33) ∂ r u = − r − u (cid:98) δ + r − F (cid:48) (cid:32) − (cid:98) δm u (cid:33) r Υ − w, w ≥ F (cid:48) ≥
0, we deduce ∂ r ( f (cid:98) δ ) ≥ − r − u (cid:98) δ ≥ (cid:98) δ − m r − . For u ≥ − m (cid:98) δ , using Lemma 10.1.15, we have ∂ r ( f (cid:98) δ ) = ∂ r f ≥ m r . Hence, for all r ≥ m , (cid:98) δ > ∂ r ( f (cid:98) δ ) ≥ m r which is (10.1.39).It remains to evaluate W (cid:98) δ . This is done in the following lemma. Lemma 10.1.24.
Let W (cid:98) δ ( r, m ) := 1 r ≤ m | W (cid:98) δ | . (10.1.40) Then, W (cid:98) δ is supported, for δ > small enough, in the region m + e − (cid:98) δ ≤ r ≤ m . Moreover its primitive, (cid:102) W (cid:98) δ ( r, m ) := (cid:90) r m W (cid:98) δ ( r (cid:48) , m ) dr (cid:48) (10.1.41) verifies the pointwise estimate (cid:102) W (cid:98) δ ( r, m ) (cid:46) (cid:98) δ. (10.1.42) Proof.
Recall that we have chosen w = m to be constant in the region r ≤ m . Hence,in that region, w (cid:98) δ = 14 m F (cid:48) (cid:32) − (cid:98) δm u (cid:33) , ∂ r w (cid:98) δ = 14 m ∂ r (cid:32) F (cid:48) (cid:32) − (cid:98) δm u (cid:33)(cid:33) . Hence, W (cid:98) δ = − r − ∂ r (cid:32) m r Υ ∂ r (cid:32) F (cid:48) (cid:32) − (cid:98) δm u (cid:33)(cid:33)(cid:33) = − m r − ∂ r (cid:32) r Υ ∂ r (cid:32) F (cid:48) (cid:32) − (cid:98) δm u (cid:33)(cid:33)(cid:33) . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Now, setting δ = (cid:98) δm for convenience below, r − ∂ r (cid:0) r Υ ∂ r ( F (cid:48) ( − δ u )) (cid:1) = − δ F (cid:48)(cid:48) ( − δ u ) r − ∂ r (cid:0) r Υ ∂ r u (cid:1) + δ F (cid:48)(cid:48)(cid:48) ( − δ u )Υ( ∂ r u ) . Note that, since r − Υ ∂ r u = w and w = (4 m ) − is constant in r in the region of interest r − ∂ r (cid:0) r Υ ∂ r u (cid:1) = r − ∂ r (cid:0) r r − Υ ∂ r u (cid:1) = r − ∂ r (cid:18) r m (cid:19) = rm . Hence, r − ∂ r (cid:0) r Υ ∂ r ( F (cid:48) ( − δ u )) (cid:1) = − δ F (cid:48)(cid:48) ( − δ u ) r − ∂ r (cid:0) r Υ ∂ r u (cid:1) + δ F (cid:48)(cid:48)(cid:48) ( − δ u )Υ( ∂ r u ) = − δ F (cid:48)(cid:48) ( − δ u ) rm + δ F (cid:48)(cid:48)(cid:48) ( − δ u )Υ( ∂ r u ) . Hence, for r ≤ m , with δ = (cid:98) δm , | W (cid:98) δ | (cid:46) δ | Υ || F (cid:48)(cid:48)(cid:48) ( − δ u ) | ( ∂ r u ) + δ | F (cid:48)(cid:48) ( − δ u ) | or, since | ∂ r u | (cid:46) r − m , in the region of interest, | W (cid:98) δ | (cid:46) δ | r − m | | F (cid:48)(cid:48)(cid:48) ( − δ u ) | + δ | F (cid:48)(cid:48) ( − δ u ) | . Since F (cid:48)(cid:48) ( − δ u ) , F (cid:48)(cid:48)(cid:48) ( − δ u ) are supported in the region 1 ≤ − δ u ≤
3, i.e. − δ ≤ u ≤ − δ ,for (cid:98) δ > e − (cid:98) δ ≤ r − m ≤ e − (cid:98) δ ≤ m . Hence, W (cid:98) δ = 1 r ≤ m | W (cid:98) δ | (cid:46) (cid:98) δ (cid:18) δr − m + 1 (cid:19) κ (cid:98) δ ( r − m )with κ (cid:98) δ ( x ) the characteristic function of the interval [ e − (cid:98) δ , e − (cid:98) δ ]. Note that the primitiveof W (cid:98) δ , i.e. (cid:102) W (cid:98) δ ( r, m ) = (cid:90) r m W (cid:98) δ ( r (cid:48) , m ) dr (cid:48) , is a positive, increasing function. Moreover, (cid:102) W (cid:98) δ ( r ) (cid:46) (cid:90) m m W (cid:98) δ ( r ) dr (cid:46) δ + δ (cid:90) e − (cid:98) δ e − (cid:98) δ x dx (cid:46) δ as desired. E [ f R, w ](Ψ) = ˙ E [ f R, w ](Ψ) + 4Υ r − mr f | Ψ | , ˙ E [ f R, w ](Ψ) = ∂ r f | R (Ψ) | + r − W | Ψ | + r − (cid:18) − mr (cid:19) f (cid:98) Q . Using the functions f (cid:98) δ , w (cid:98) δ and W (cid:98) δ introduced in definition 10.1.22, we have˙ E [ f (cid:98) δ R, w (cid:98) δ ](Ψ) = 1 r f (cid:98) δ (cid:18) − mr (cid:19) (cid:98) Q + ∂ r ( f (cid:98) δ ) | R Ψ | + W (cid:98) δ | Ψ | . Note that in view of the estimates (10.1.38) (10.1.39), and Lemma 10.1.24, we immediatelydeduce the existence of a constant
C > (cid:98) δ such that˙ E [ f (cid:98) δ R, w (cid:98) δ ](Ψ) ≥ C − (cid:2) | R Ψ | + |∇ / Ψ | + Υ | Ψ | (cid:3) − W (cid:98) δ | Ψ | on r ≤ m . (10.1.43)where W (cid:98) δ is a non-negative potential supported in the region 2 m + e − (cid:98) δ ≤ r ≤ m , whoseprimitive (cid:102) W (cid:98) δ ( r ) = (cid:82) r m W (cid:98) δ ( r (cid:48) m ) dr (cid:48) verifies (cid:102) W (cid:98) δ (cid:46) (cid:98) δ . Combining this with estimates of theprevious section we derive the following. Proposition 10.1.25.
There exists a constant
C > , and for any small enough (cid:98) δ > ,there exists functions f (cid:98) δ ∈ C ( r > , w (cid:98) δ ∈ C ( r > and h ∈ C ( r > verifying, forall r > , | f (cid:98) δ ( r ) | (cid:46) (cid:98) δ − , w (cid:98) δ (cid:46) r − , h (cid:46) r − , such that E [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) = ˙ E [ f (cid:98) δ R, w (cid:98) δ , hR ] + E (cid:15) [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) satisfies (cid:90) S ˙ E [ f (cid:98) δ R, w (cid:98) δ , hR ] ≥ C − (cid:90) S (cid:32) m r | R (Ψ) | + r − (cid:18) − mr (cid:19) (cid:18) |∇ / Ψ | + m r | T Ψ | (cid:19) + mr | Ψ | (cid:33) − (cid:90) S W (cid:98) δ | Ψ | , E (cid:15) [ f (cid:98) δ R, w (cid:98) δ , hR ] = (cid:15) Q · ( f (cid:98) δ R ) ¨ π + O ( r − u − − δ dec trap (1 + | f (cid:98) δ | )) | Ψ | , where W (cid:98) δ is non-negative, supported in the region m + e − (cid:98) δ ≤ r ≤ m , and such that itsprimitive (cid:102) W (cid:98) δ ( r ) = (cid:82) r m W (cid:98) δ verifies (cid:102) W (cid:98) δ (cid:46) (cid:98) δ . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Proof.
We choose h to be the function of ( r, m ) introduced in Proposition 10.1.20, f (cid:98) δ tobe the function of ( r, m ) introduced in definition 10.1.22, and W (cid:98) δ , introduced in Lemma10.1.24. Also, by an abuse of notation, we denote by w (cid:98) δ, the function denoted by w (cid:98) δ indefinition 10.1.22. Then, combining Proposition 10.1.20 in the region r ≥ m with theestimate (10.1.43) in the region r ≤ m , we immediately obtain (cid:90) S ˙ E [ f (cid:98) δ R, w (cid:98) δ, , hR ] ≥ C − (cid:90) S (cid:32) m r | R (Ψ) | + r − (cid:18) − mr (cid:19) |∇ / Ψ | + mr | Ψ | (cid:33) − (cid:90) S W (cid:98) δ | Ψ | . (10.1.44)(10.1.44) corresponds to the desired estimate without the presence of the term | T Ψ | onthe right hand side. To get the improved estimate of Proposition 10.1.25, we set w (cid:98) δ := w (cid:98) δ, − δ w , (10.1.45)for a small parameter δ > w (cid:98) δ, is our previous choice intro-duced in definition 10.1.22, and where w ( r, m ) := r − m r Υ (cid:18) − mr (cid:19) . (10.1.46)We evaluate (modulo the same type of error terms as before which we include in E (cid:15) ),˙ E [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) = ˙ E [ X (cid:98) δ , w (cid:98) δ, , hR ] − δ w L (Ψ) + δ | Ψ | r − ∂ r ( r Υ ∂ r w ) . Now, since L (Ψ) = − e Ψ · e Ψ + |∇ / Ψ | + V | Ψ | = Υ − (cid:0) −| T Ψ | + | R Ψ | (cid:1) + |∇ / Ψ | + V | Ψ | , we have, − δ w L (Ψ) + δ | Ψ | r − ∂ r ( r Υ ∂ r w )= − δ r − m r Υ (cid:18) − mr (cid:19) L (Ψ) + δ | Ψ | r − ∂ r (cid:32) r Υ ∂ r (cid:32) r − m r Υ (cid:18) − mr (cid:19) (cid:33)(cid:33) = 12 δ r − (cid:18) − mr (cid:19) m r | T Ψ | + O ( δ ) (cid:32) m r | R (Ψ) | + r − (cid:18) − mr (cid:19) |∇ / Ψ | + mr | Ψ | (cid:33) E [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) = ˙ E [ X (cid:98) δ , w (cid:98) δ, , hR ] + 12 δ r − (cid:18) − mr (cid:19) m r | T Ψ | (10.1.47)+ O ( δ ) (cid:32) m r | R (Ψ) | + r − (cid:18) − mr (cid:19) |∇ / Ψ | + mr | Ψ | (cid:33) . The desired estimate now follows from (10.1.44) and (10.1.47) provided δ > C > O ( δ ) inthe above identity can be absorbed. Note that the vectorfields T and R become both proportional to e for Υ = 0 which meansthat the estimate of Proposition 10.1.25 degenerates along Υ = 0, i.e. it does not control e (Ψ) there. In this section we make use of the Dafermos-Rodnianski red shift vectorfieldto compensate for this degeneracy. The crucial ingredient here is the favorable sign of ω in a small neighborhood of r = 2 m . Lemma 10.1.26.
Let π (3) , π (4) denote the deformation tensors of e , e . In the region r ≤ m all components are O ( (cid:15) ) with the exception of, π (3)44 = − ω = 8 mr + O ( (cid:15) ) , π (3) θθ = κ + ϑ = − r + O ( (cid:15) ) ,π (3) ϕϕ = κ − ϑ = − r + O ( (cid:15) ) ,π (4)34 = 4 ω = − mr + O ( (cid:15) ) , π (4) θθ = κ + ϑ = 2Υ r + O ( (cid:15) ) ,π (4) ϕϕ = κ − ϑ = 2Υ r + O ( (cid:15) ) . Proof.
Immediate verification in view of our assumptions.
Lemma 10.1.27.
Given the vectorfield, Y = a ( r, m ) e + b ( r, m ) e , (10.1.48) and assuming sup r ≤ m (cid:16) | a | + | ∂ r a | + | ∂ m a | + | b | + | ∂ r b | + | ∂ m b | (cid:17) (cid:46) , CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS we have, for r ≤ m , Q αβ ( Y ) π αβ = (cid:18) mr a − Υ ∂ r a (cid:19) Q + ∂ r b Q + (cid:18) ∂ r a − mr b − Υ ∂ r b (cid:19) Q + 2 r ( b Υ − a ) e Ψ · e Ψ + 8 Υ r ( a − Υ b ) | Ψ | + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) . Moreover, with the notation (10.1.13) , E [ Y, Q αβ ( Y ) π αβ + 4 r − mr ( − a + b Υ) | Ψ | + O ( (cid:15) ) r − | Ψ | . (10.1.49) Proof.
In view of | e ( r ) − Υ , e ( r ) + 1 | (cid:46) (cid:15) Lemma 10.1.5, and the assumptions on the derivatives of a and b w.r.t. ( r, m ), we have e ( a ) = Υ ∂ r a + O ( (cid:15) ) , e ( a ) = − ∂ r a + O ( (cid:15) ) ,e ( b ) = Υ ∂ r b + O ( (cid:15) ) , e ( b ) = − ∂ r b + O ( (cid:15) ) , e θ ( a ) = e θ ( b ) = 0 . We infer, Q αβ ( Y ) π αβ = a Q αβ π (3) αβ − ( Q e a + Q e a ) + b Q αβ π (4) αβ − ( Q e b + Q e b ) + O ( (cid:15) ) |Q (Ψ) | = a Q αβ π (3) αβ + b Q αβ π (4) αβ − Q Υ ∂ r a − Q ( − ∂ r a + Υ ∂ r b ) + Q ∂ r b + O ( (cid:15) ) |Q (Ψ) | . Note that, Q θθ + Q ϕϕ = e Ψ · e Ψ − V | Ψ | = e Ψ · e Ψ − r | Ψ | + O ( (cid:15) ) r − | Ψ | . (10.1.50)Hence, Q αβ π (3) αβ = Q π (3)44 + Q θθ π (3) θθ + Q ϕϕ π (3) ϕϕ + O ( (cid:15) ) |Q (Ψ) | = 14 Q mr − r ( Q θθ + Q ϕϕ ) + O ( (cid:15) ) |Q (Ψ) | = 2 mr Q − r e Ψ · e Ψ + 8 Υ r | Ψ | + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) , Q αβ π (4) αβ = 2 Q π (4)34 + Q θθ π (4) θθ + Q ϕϕ π (4) ϕϕ + O ( (cid:15) ) |Q (Ψ) | = 12 Q ( − mr ) + 2Υ r ( Q θθ + Q ϕϕ ) + O ( (cid:15) ) |Q (Ψ) | = − mr Q + 2Υ r e Ψ · e Ψ − r | Ψ | + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) . Q αβ ( Y ) π αβ = a (cid:20) mr Q − r e Ψ e Ψ + 8 Υ r | Ψ | (cid:21) + b (cid:20) − mr Q + 2Υ r e Ψ · e Ψ − r | Ψ | (cid:21) − Q Υ ∂ r a − Q ( − ∂ r a + Υ ∂ r b ) + Q ∂ r b + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) = (cid:18) mr a − Υ ∂ r a (cid:19) Q + Q ∂ r b + 2 r ( b Υ − a ) e Ψ e Ψ + (cid:18) ∂ r a − mr b − Υ ∂ r b (cid:19) Q + 8 Υ r ( a − Υ b ) | Ψ | + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) . To prove the second part of the lemma we recall (see (10.1.13)), E [ Y, Q αβ ( Y ) π αβ − Y ( V ) | Ψ | and, relying on Lemma 10.1.5, we have on r ≤ mY ( V ) = ( − a + b Υ) ∂ r V + O ( (cid:15) ) = ( − a + b Υ) (cid:18) − r − mr (cid:19) + O ( (cid:15) )which concludes the proof of the lemma. Corollary 10.1.28.
If we choose, a (2 m, m ) = 1 , b (2 m, m ) = 0 , ∂ r a (2 m, m ) ≥ m , ∂ r b (2 m, m ) ≥ m , then, at r = 2 m , we have Q αβ ( Y ) π αβ ≥ m ( | e Ψ | + | e Ψ | + Q ) + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) (10.1.51) and, E [ Y, ≥ m (cid:18) | e Ψ | + | e Ψ | + Q + 1 m | Ψ | (cid:19) + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) . (10.1.52) Moreover the estimates remain true if we add to Y a multiple of T = ( e + Υ e ) .Proof. Recall from Lemma 10.1.27 that we have, for r ≤ m , Q αβ ( Y ) π αβ = (cid:18) mr a − Υ ∂ r a (cid:19) Q + ∂ r b Q + (cid:18) ∂ r a − mr b − Υ ∂ r b (cid:19) Q + 2 r ( b Υ − a ) e Ψ · e Ψ + 8 Υ r ( a − Υ b ) | Ψ | + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Hence, at r = 2 m , using Υ = 0, a = 1, b = 0, ∂ r a ≥ (4 m ) − and ∂ r b ≥ m ) − , wededuce Q αβ ( Y ) π αβ = 12 m Q + ∂ r b Q + ∂ r a Q − m e Ψ · e Ψ + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) ≥ m | e (Ψ) | + 54 m | e (Ψ) | + 14 m Q − m e Ψ · e Ψ + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) from which the desired lower bound in (10.1.51) follows.Also, at r = 2 m , using (10.1.49), Υ = 0, a = 1, and b = 0, we have E [ Y, Q αβ ( Y ) π αβ + 4 r − mr ( − a + b Υ) | Ψ | + O ( (cid:15) ) r − | Ψ | = 12 Q αβ ( Y ) π αβ + 14 m | Ψ | ≥ m (cid:18) | e Ψ | + | e Ψ | + Q + 1 m | Ψ | (cid:19) + O ( (cid:15) ) (cid:0) |Q (Ψ) | + r − | Ψ | (cid:1) which yields (10.1.52).We are now ready to prove the following result. Proposition 10.1.29.
Given a small parameter δ H > there exists a smooth vectorfield Y H supported in the region | Υ | ≤ δ H such that the following estimate holds, E [ Y H , ≥ m | Υ |≤ δ H (cid:16) | e Ψ | + | e Ψ | + (cid:98) Q + m − | Ψ | (cid:17) − m δ − H δ H ≤ Υ ≤ δ H (cid:16) | e Ψ | + | e Ψ | + (cid:98) Q + m − | Ψ | (cid:17) + O ( (cid:15) )1 | Υ |≤ δ H (cid:0) |Q (Ψ) | + m − | Ψ | (cid:1) . Moreover, for | Υ | ≤ δ H , we have Y H = e + e + O ( δ H )( e + e ) . Proof.
We introduce the vectorfield Y (0) := ae + be + 2 T, a ( r, m ) := 1 + 54 m ( r − m ) , b ( r, m ) := 54 m ( r − m ) , T = ( e + Υ e ). Also, we pick positive bump function κ = κ ( r ), supported in theregion in [ − ,
2] and equal to 1 for [ − ,
1] and define, for sufficiently small δ H > Y H := κ H Y (0) , κ H := κ (cid:32) Υ δ H (cid:33) . (10.1.53)We have E [ Y H , Q · ( Y H ) π − Y H ( V ) | Ψ | = κ H Q · ( Y ) π + Q ( Y (0) , dκ H ) + κ H Y (0) ( V ) | Ψ | = κ H E [ Y (0) , O ( δ − H )1 δ H ≤ Υ ≤ δ H (cid:16) | e Ψ | + | e Ψ | + (cid:98) Q + m − | Ψ | (cid:17) . Note from the definition of Y (0) and the choice of a and b that Corollary 10.1.28 appliesto Y (0) . In particular, we deduce from (10.1.52) for δ H > E [ Y H , ≥ m | Υ |≤ δ H (cid:16) | e Ψ | + | e Ψ | + (cid:98) Q + m − | Ψ | (cid:17) − m δ − H δ H ≤ Υ ≤ δ H (cid:16) | e Ψ | + | e Ψ | + (cid:98) Q + m − | Ψ | (cid:17) + O ( (cid:15) )1 | Υ |≤ δ H (cid:0) |Q (Ψ) | + m − | Ψ | (cid:1) as desired. We consider the combined Morawetz triplet(
X, w, M ) := ( X (cid:98) δ , w (cid:98) δ , hR ) + (cid:15) H ( Y H , , , (10.1.54)with (cid:15) H > X (cid:98) δ = f (cid:98) δ R, w (cid:98) δ , hR ) is thetriplet given by Proposition 10.1.25 and Y H the vectorfield of Proposition 10.1.29.Recall, see Proposition 10.1.25, that ˙ E (cid:98) δ := ˙ E [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) verifies, (cid:90) S ˙ E (cid:98) δ ≥ C − (cid:90) S (cid:32) m r | R (Ψ) | + (cid:18) − mr (cid:19) r − (cid:18) (cid:98) Q + m r | T Ψ | (cid:19) + Υ mr | Ψ | (cid:33) − (cid:90) S W (cid:98) δ | Ψ | . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
According to Proposition 10.1.29, we write for E H = E ( Y H , , E H = ˙ E H + E H ,(cid:15) , ˙ E H ≥ m | Υ |≤ δ H (cid:16) | e Ψ | + | e Ψ | + (cid:98) Q + m − | Ψ | (cid:17) − m δ − H δ H ≤ Υ ≤ δ H (cid:16) | e Ψ | + | e Ψ | + (cid:98) Q + m − | Ψ | (cid:17) , E H ,(cid:15) = O ( (cid:15) ) (cid:0) |Q (Ψ) | + m − | Ψ | (cid:1) | Υ |≤ δ H . Note that, for | Υ | ≥ δ H we have, | R Ψ | + | T Ψ | = 12 ( | e Ψ | + Υ | e Ψ | ) ≥ δ H ( | e Ψ | + | e Ψ | ) . We now proceed to find a lower bound for the expression ˙ E (cid:98) δ + (cid:15) H ˙ E H . For brevity the S integration is omitted below. Region δ H ≤ | Υ | ≤ δ H . ˙ E (cid:98) δ + (cid:15) H ˙ E H ≥ m − C − (cid:104) δ H ( | e Ψ | + | e Ψ | ) + m − | Ψ | + |∇ / Ψ | (cid:105) − W (cid:98) δ | Ψ | − (cid:15) H m δ − H (cid:0) | e Ψ | + | e Ψ | + |∇ / Ψ | + m − | Ψ | (cid:1) . Therefore, choosing (cid:15) H ≤ (2 C ) − δ H , we deduce,˙ E (cid:98) δ + (cid:15) H ˙ E H ≥ m − δ H (2 C ) − (cid:0) | e Ψ | + | e Ψ | + |∇ / Ψ | + m − | Ψ | (cid:1) − W (cid:98) δ | Ψ | . Region | Υ | ≤ δ H . (cid:15) H ˙ E H + ˙ E (cid:98) δ ≥ (cid:15) H m (cid:16) | e Ψ | + | e Ψ | + (cid:98) Q + m − | Ψ | (cid:17) − W (cid:98) δ | Ψ | . Region Υ ≥ δ H . In this region ˙ E (cid:98) δ + (cid:15) H ˙ E H = ˙ E (cid:98) δ . Hence (ignoring the S -integration),˙ E (cid:98) δ + (cid:15) H ˙ E H ≥ C − (cid:32) m r | R (Ψ) | + (cid:18) − mr (cid:19) r − (cid:18) (cid:98) Q + m r | T Ψ | (cid:19) + mr | Ψ | (cid:33) − W (cid:98) δ | Ψ | . R, T near r = 2 m according to (5.1.11), i.e.˘ R := θ
12 ( e − e ) + (1 − θ )Υ − R = 12 (cid:104) ˘ θe − e (cid:105) , ˘ T := θ
12 ( e + e ) + (1 − θ )Υ − T = 12 (cid:104) ˘ θe + e (cid:105) , where θ a smooth bump function equal 1 on | Υ | ≤ δ H vanishing for | Υ | ≥ δ H , andwhere ˘ θ = θ + Υ − (1 − θ ) = (cid:40) , for | Υ | ≤ δ H , Υ − , for | Υ | ≥ δ H . Note that 2( | ˘ R Ψ | + | ˘ T Ψ | ) = | e Ψ | + ˘ θ | e Ψ | . Thus in the region | Υ | ≤ δ H we have | e Ψ | + | e Ψ | = 2( | ˘ R Ψ | + | ˘ T Ψ | ) and therefore,˙ E (cid:98) δ + (cid:15) H ˙ E H ≥ (cid:15) H m (cid:16) | e Ψ | + | e Ψ | + (cid:98) Q + m − | Ψ | (cid:17) − W (cid:98) δ | Ψ | = (cid:15) H m (cid:16) | ˘ R Ψ | + | ˘ T Ψ | + (cid:98) Q + m − | Ψ | (cid:17) − W (cid:98) δ | Ψ | . In the region δ H ≤ | Υ | ≤ δ H , we have | ˘ R Ψ | + | ˘ T Ψ | (cid:46) | e Ψ | + δ − H | e Ψ | . Hence, for (cid:15) H ≤ (2 C ) − δ H , we deduce,˙ E (cid:98) δ + (cid:15) H ˙ E H ≥ m − δ H (2 C ) − (cid:0) | e Ψ | + | e Ψ | + |∇ / Ψ | + m − | Ψ | (cid:1) − W (cid:98) δ | Ψ | ≥ m − δ H (2 C ) − (cid:16) δ H (cid:16) | ˘ R Ψ | + | ˘ T Ψ | (cid:17) + |∇ / Ψ | + m − | Ψ | (cid:17) − W (cid:98) δ | Ψ | . Finally, for Υ ≥ δ H we have ˘ R = Υ − R, ˘ T = Υ − T . Hence,˙ E (cid:98) δ + (cid:15) H ˙ E H ≥ C − (cid:32) m r | R (Ψ) | + (cid:18) − mr (cid:19) r − (cid:18) (cid:98) Q + m r | T Ψ | (cid:19) + Υ mr | Ψ | (cid:33) − W (cid:98) δ | Ψ | ≥ C − (cid:32) δ H m r | ˘ R (Ψ) | + (cid:18) − mr (cid:19) r − (cid:18) (cid:98) Q + δ H m r | ˘ T Ψ | (cid:19) + Υ mr | Ψ | (cid:33) − W (cid:98) δ | Ψ | . We deduce the following.92
CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Proposition 10.1.30.
Let
C > the constant of Proposition 10.1.25. Consider thecombined Morawetz triplet ( X, w, M ) := ( X (cid:98) δ , w (cid:98) δ , hR ) + (cid:15) H ( Y H , , , (10.1.55) with C − δ H ≤ (cid:15) H ≤ (2 C ) − δ H where, for given fixed (cid:98) δ > , ( X (cid:98) δ , w (cid:98) δ , hR ) is the triplet ofProposition 10.1.25 and Y H the vectorfield of Proposition 10.1.29, supported in | Υ | ≤ δ H with δ H > sufficiently small, independent of (cid:98) δ . Let ˙ E (cid:98) δ , ˙ E H be the principal parts of E [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) and respectively E H [ Y H , , and E (cid:98) δ,(cid:15) , E H ,(cid:15) the corresponding errorterms, i.e., E [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) = ˙ E (cid:98) δ + E (cid:98) δ,(cid:15) , E H [ Y H , , E H + E H ,(cid:15) . Then, provided δ H > is sufficiently small, we have1. In the region − δ H ≤ Υ , r ≤ m , we have with a constant Λ − H := C − δ H > (cid:90) S ( ˙ E (cid:98) δ + (cid:15) H ˙ E H ) ≥ m − Λ − H (cid:90) S (cid:16) | ˘ R (Ψ) | + | ˘ T Ψ | + |∇ / Ψ | + m − | Ψ | (cid:17) − (cid:90) S W (cid:98) δ | Ψ | .
2. In the region r ≥ m , where ˙ E (cid:98) δ + (cid:15) H ˙ E H = ˙ E (cid:98) δ and W (cid:98) δ = 0 , we have the same estimateas in Proposition 10.1.25, i.e. (cid:90) S ( ˙ E (cid:98) δ + (cid:15) H ˙ E H ) ≥ C − (cid:90) S (cid:32) m r | R (Ψ) | + r − (cid:18) − mr (cid:19) (cid:18) |∇ / Ψ | + m r | T Ψ | (cid:19) + mr | Ψ | (cid:33) .
3. The (cid:15) -error terms verify the upper bound estimate, E (cid:98) δ,(cid:15) + (cid:15) H E H ,(cid:15) (cid:46) C(cid:15) (cid:98) δ − u − − δ dec trap (cid:2) r − | e Ψ | + r − ( | e Ψ | + |∇ / Ψ | ) (cid:3) + C(cid:15) (cid:98) δ − u − − δ dec trap r − | e Ψ | ( | e Ψ | + |∇ / Ψ | ) + C(cid:15) (cid:98) δ − u − − δ dec trap r − | Ψ | . Proof.
It only remains to check the last part. In view of Proposition 10.1.20 we have, E (cid:98) δ,(cid:15) = E (cid:15) [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) = (cid:15) Q · ( X (cid:98) δ ) ¨ π + O (cid:0) (cid:15)r − u − − δ dec trap ( | f (cid:98) δ | + 1) (cid:1) | Ψ | and | f (cid:98) δ | (cid:46) (cid:98) δ − . Hence, E (cid:98) δ,(cid:15) = E (cid:15) [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) = (cid:15) (cid:18) Q · X (cid:98) δ ¨ π + O ( (cid:98) δ − r − u − − δ dec trap ) | Ψ | (cid:19) . π = ( X (cid:98) δ ) ¨ π for simplicity, Q · ¨ π = 14 ( Q ¨ π + 2 Q ¨ π + Q ˙ π ) −
12 ( Q A ¨ π A + Q A ¨ π A ) + Q AB ¨ π AB . Thus, recalling part 1 and 2 of Proposition 10.1.9, and Lemma 10.1.8,
Q · ¨ π (cid:46) r − u − − δ dec trap | e Ψ | + r − u − − δ dec trap (cid:0) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) + r − u − − δ dec trap | e Ψ | ( | e Ψ | + |∇ / Ψ | ) . Finally, since r ∼ m and u trap = 1 on | Υ | ≤ δ H , the error terms generated by the redshift vectorfield Y H , E H ,(cid:15) = O ( (cid:15) )1 | Υ |≤ δ H (cid:0) |Q (Ψ) | + m − | Ψ | (cid:1) can easily be absorbed on the right hand side to derive the desired estimate. Elimination of W (cid:98) δ We now proceed to eliminate the potential W (cid:98) δ by a procedure analogous to that used insection 10.1.8. More precisely we set, in view of (10.1.13), E (cid:98) δ = E [ f (cid:98) δ R, w (cid:98) δ , hR ](Ψ) , E (cid:48) (cid:98) δ = E [ f (cid:98) δ R, w (cid:98) δ , hR + h ˘ R )](Ψ) , and, E (cid:48) (cid:98) δ = E (cid:98) δ + h Ψ ˘ R Ψ + 12 D µ ( h ˘ R µ ) | Ψ | , where h is a smooth, compactly supported function supported in the region r ≤ m .Thus, we have in view of Proposition 10.1.30, ignoring the integration on S ,˙ E (cid:48) (cid:98) δ + (cid:15) H ˙ E H = ˙ E (cid:98) δ + (cid:15) H ˙ E H + h Ψ ˘ R Ψ + 12 D µ ( h ˘ R µ ) | Ψ | ≥ I (Ψ) + m − Λ − H (cid:18) | ˘ R (Ψ) | + | ˘ T Ψ | + |∇ / Ψ | + m − | Ψ | (cid:19) (10.1.56)where, I (Ψ) : = 12 Λ − H m − | ˘ R (Ψ) | + Ψ h ˘ R Ψ + 12 D µ ( h ˘ R µ ) | Ψ | − W (cid:98) δ | Ψ | Recall that W (cid:98) δ is supported is supported in the region 2 m < r ≤ m . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS so that we have I (Ψ) ≥ (cid:104) D µ ( h ˘ R µ ) − W (cid:98) δ − m Λ H h (cid:105) | Ψ | . (10.1.57)We focus on the coefficient in front of | Ψ | on the RHS of (10.1.57). Ignoring the errorterms in (cid:15) (which can easily be incorporated in the upper bound for E (cid:98) δ,(cid:15) + (cid:15) H E H ,(cid:15) of theprevious proposition), we have, Div ˘ R = 12 (cid:16) D µ (˘ θ ( e ) µ ) − D µ (( e ) µ (cid:17) = 14 (cid:16) ˘ θ tr π (4) − tr π (3) (cid:17) + 12 e (˘ θ ) = O ( δ − H )and, using in particular Lemma 10.1.5, D µ ( h ˘ R µ ) = ˘ Rh + h Div ˘ R = 12 ∂ r h (˘ θe r − e r ) + h Div ˘ R = 12 ∂ r h (˘ θ Υ + 1) + h O ( δ − H ) ≥ ∂ r h + h O ( δ − H ) . Together with (10.1.57), we infer I (Ψ) ≥ (cid:104) ∂ r h − W (cid:98) δ − m Λ H h + h O ( δ − H ) (cid:105) | Ψ | . (10.1.58)We now consider the choice of the function h = h ( r, m ). Recall (see Lemma 10.1.24) that W (cid:98) δ is supported in the region 2 m + e − (cid:98) δ ≤ r ≤ m and that its primitive (cid:102) W (cid:98) δ ( r ) := (cid:82) r m W (cid:98) δ verifies (cid:102) W (cid:98) δ (cid:46) m − (cid:98) δ . We choose h =: (cid:40) (cid:102) W (cid:98) δ , for r ≤ m , for r ≥ m (10.1.59)and since (cid:102) W (cid:98) δ (cid:46) m − (cid:98) δ , we may extend h in m ≤ r ≤ m such that h is C and we havefor all r > | h | (cid:46) m − (cid:98) δ, | ∂ r h | (cid:46) m − (cid:98) δ. (10.1.60)In view of (10.1.58), this choice of h yields I (Ψ) ≥ − O (cid:16) m − (cid:98) δ + Λ H m − (cid:98) δ + (cid:98) δ ( δ H ) − (cid:17) | Ψ | . (cid:98) δ (cid:28) δ H Λ − H , i.e. (cid:98) δ (cid:28) δ H (recall that Λ − H = C − δ H ) and h defined as above,we infer I (Ψ) ≥ − m − Λ − H m − | Ψ | which together with (10.1.56) finally yields (cid:90) S ( ˙ E (cid:48) (cid:98) δ + (cid:15) H ˙ E H ) ≥ (cid:90) S I (Ψ) + m − Λ − H (cid:90) S (cid:18) | ˘ R (Ψ) | + | ˘ T Ψ | + |∇ / Ψ | + m − | Ψ | (cid:19) ≥ m − Λ − H (cid:90) S (cid:18) | ˘ R (Ψ) | + | ˘ T Ψ | + |∇ / Ψ | + 12 m − | Ψ | (cid:19) . Summary of results so far
We summarize the result in the following,
Proposition 10.1.31.
Consider the combined Morawetz triplet ( X, w, M ) := ( f (cid:98) δ R, w (cid:98) δ , hR ) + (cid:15) H ( Y H , ,
0) + (0 , , h ˇ R ) (10.1.61) with ( f (cid:98) δ R, w (cid:98) δ , hR ) the triplet of Proposition 10.1.25, Y H the red shift vectorfield of Propo-sition 10.1.29 (corresponding to the small parameter δ H ) and h the C function abovesatisfying (10.1.59) (10.1.60) . Let ˙ E [ X, w, M ] the principal part of E [ X, w, M ] (indepen-dent of (cid:15) ) and E (cid:15) [ X, w, M ] the error term in (cid:15) such that E = ˙ E + E (cid:15) .We choose the small strictly positive parameters (cid:15) H , δ H , δ such that , (cid:15) H = δ H , (cid:98) δ = δ H . (10.1.62) Then, there holds (cid:90) S E [ X, w, M ](Ψ) ≥ δ H (cid:90) S (cid:34) m r | ˘ R Ψ | + r − (cid:18) − mr (cid:19) (cid:18) (cid:98) Q + m r | ˘ T Ψ | (cid:19) + mr | Ψ | (cid:35) , E (cid:15) [ X, w, M ](Ψ) ≤ δ − H (cid:15)u − − δ dec trap (cid:2) r − | e Ψ | + r − ( | e Ψ | + |∇ / Ψ | ) (cid:3) + δ − H (cid:15)u − − δ dec trap r − | e Ψ | ( | e Ψ | + |∇ / Ψ | ) + δ − H (cid:15)u − − δ dec trap r − | Ψ | . (10.1.63) Note that (10.1.62) verifies all the restrictions we have encountered so far, i.e. δ H (cid:28) (cid:15) H (cid:28) δ H and0 < (cid:98) δ (cid:28) δ H . Note that δ H (cid:28) Λ − H (recall that Λ − H = C − δ H ) and δ − H (cid:29) (cid:98) δ − in view of (10.1.62). CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS Q In this section we prove lower bounds for for Q ( X + 2Λ T, e ) and Q ( X + 2Λ T, e ) in theregion r H ≤ r , for r H to be determined and Λ sufficiently large. Proposition 10.1.32.
Under the assumptions of Proposition 10.1.31, and with the choice
Λ := 14 δ − H , (10.1.64) the following inequalities hold true for r ≥ m (1 − δ H ) .1. For the region such that r ≥ m (1 − δ H ) and Υ ≤ δ H , we have Q ( X + Λ T, e ) ≥ (cid:15) H Q + 12 Λ Q , Q ( X + Λ T, e ) ≥ (cid:15) H Q + 12 Λ Q .
2. For the region δ H ≤ Υ ≤ , we have Q ( X + Λ T, e ) ≥ δ − H ( Q + Q ) , Q ( X + Λ T, e ) ≥ δ − H ( Q + Q ) .
3. For the region r ≥ m , we have Q ( X + Λ T, e ) ≥
14 Λ ( Q + Q ) , Q ( X + Λ T, e ) ≥
14 Λ ( Q + Q ) .
4. The null components of Q are given by (recall Proposition 10.1.9), Q = | e Ψ | , Q = | e Ψ | , Q = |∇ / Ψ | + 4Υ r (1 + O ( (cid:15) )) | Ψ | . Proof.
Since X = f (cid:98) δ R + (cid:15) H Y H and T = ( e + Υ e ), R = ( e − Υ e ), we write, Q ( X + 2Λ T, e ) = Q ( X, e ) + Λ Q ( e + Υ e , e ) = Q ( X, e ) + Λ ( Q + Υ Q )= (cid:15) H Q ( Y H , e ) + Λ ( Q + Υ Q ) + 12 f (cid:98) δ ( Q − Υ Q ) . m (1 − δ H ) ≤ r ≤ m we have Y H = e + e + O ( δ H )( e + e ), Υ ≥ f (cid:98) δ <
0. Hence, in that region, Q ( X + 2Λ T, e ) ≥ (cid:15) H ( Q + Q ) + (cid:18) Λ − | f (cid:98) δ | (cid:19) Q − | Υ | (cid:18) Λ + 12 | f (cid:98) δ | (cid:19) Q ≥ (cid:18) (cid:15) H − | Υ | (cid:18) Λ + 12 | f (cid:98) δ | (cid:19)(cid:19) Q + (cid:18) (cid:15) H + Λ − | f (cid:98) δ | (cid:19) Q . Thus, we need to choose Λ such that12 | f (cid:98) δ | ≤ Λ ≤ (cid:15) H δ H − | f (cid:98) δ | Now, recall (10.1.62) as well as the fact that | f (cid:98) δ | is of size O (( (cid:98) δ ) − ). Thus it suffices tochoose Λ such that O ( δ − H ) ≤ Λ ≤ δ − H − O ( δ − H ) , i.e. it suffices to choose, for δ H > δ − H , to deduce the inequality, Q ( X + 2Λ T, e ) ≥ (cid:15) H Q + 12 Λ Q . Next, in the region 0 ≤ Υ ≤ δ H , the sign of Υ is more favorable and we have Q ( X + 2Λ T, e ) ≥ (cid:15) H ( Q + Q ) + (cid:18) Λ − | f (cid:98) δ | (cid:19) Q + | Υ | (cid:18) Λ + 12 | f (cid:98) δ | (cid:19) Q ≥ (cid:18) (cid:15) H + | Υ | (cid:18) Λ + 12 | f (cid:98) δ | (cid:19)(cid:19) Q + (cid:18) (cid:15) H + Λ − | f (cid:98) δ | (cid:19) Q . In particular, we simply need Λ (cid:29) (cid:98) δ − , which is in particular satisfied by (10.1.64), todeduce the same inequality, Q ( X + 2Λ T, e ) ≥ (cid:15) H Q + 12 Λ Q . In the region δ H ≤ Υ ≤ , where f (cid:98) δ ≤
0, and using the fact that | f (cid:98) δ | is of size O (( (cid:98) δ ) − ) Q ( X + 2Λ T, e ) = (cid:15) H Q ( Y H , e ) + Λ ( Q + Υ Q ) + 12 f (cid:98) δ ( Q − Υ Q ) ≥ Λ (cid:16) Q + δ H Q (cid:17) − O ( (cid:98) δ − ) Q . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Hence, for the choice (10.1.64), and in view of (10.1.62), we infer Q ( X + 2Λ T, e ) ≥ δ − H ( Q + Q ) . Finally, for r ≥ m where we have 0 ≤ f (cid:98) δ (cid:46) ≤ Υ ≤ Y H = 0, Q ( X + 2Λ T, e ) = Λ ( Q + Υ Q ) + 12 f (cid:98) δ ( Q − Υ Q ) ≥ Λ (cid:18) Q + 13 Q (cid:19) − O (1) Q and hence, (10.1.64) implies Q ( X + 2Λ T, e ) ≥
14 Λ ( Q + Q )as desired. The proof for Q ( X + Λ T, e ) is similar. We are now ready to state our first Morawetz estimate which is simply obtained byintegrating the pointwise inequality in Proposition 10.1.30 on our domain M = ( int ) M ∪ ( ext ) M described at the beginning of the section, with X replaced by X + Λ T for Λ > τ , note that we have N Σ = ae + be , ≤ a, b ≤ , a + b ≥ , (10.1.65)with b = 0 , a = 1 on ( int ) M , a = 0 , b = 1 on M r ≥ m , a, b ≥
14 on ( trap ) M . We recall the following quantities for Ψ in regions M ( τ , τ ) ⊂ M in the past of Σ( τ )and future of Σ( τ ).1. Morawetz bulk quantityMor[Ψ]( τ , τ ) = (cid:90) M ( τ ,τ ) m r | ˘ Rψ | + mr | Ψ | + (cid:18) − mr (cid:19) r (cid:18) |∇ / ψ | + m r | ˘ T ψ | (cid:19) .
2. Basic energy quantity E [Ψ]( τ ) = (cid:90) Σ( τ ) (cid:18)
12 ( N Σ , e ) | e Ψ | + 12 ( N Σ , e ) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:19) . A and Σ ∗ F [Ψ]( τ , τ ) = (cid:90) A ( τ ,τ ) (cid:0) δ − H | e Ψ | + δ H | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) + (cid:90) Σ ∗ ( τ ,τ ) (cid:0) | e Ψ | + | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) , with A ( τ , τ ) = A ∩ M ( τ , τ ) and Σ ∗ ( τ , τ ) = Σ ∗ ∩ M ( τ , τ ).The following theorem is our first Morawetz estimate. Theorem 10.1.33.
Consider the equation (10.1.9) , i.e. ˙ (cid:3) Ψ = V Ψ + N , with V = − κκ and a domain M ( τ , τ ) ⊂ M . Then, we have E [Ψ]( τ ) + Mor [Ψ]( τ , τ ) + F [Ψ]( τ , τ ) (cid:46) ( E [Ψ]( τ ) + J [ N, Ψ]( τ , τ ) + Err (cid:15) ( τ , τ )[Ψ]) ,J [ N, Ψ]( τ , τ ) : = (cid:90) M ( τ ,τ ) ( | ˘ R Ψ | + | ˘ T Ψ | + r − | Ψ | ) | N | , Err (cid:15) [Ψ]( τ , τ ) = (cid:90) M ( τ ,τ ) E (cid:15) [Ψ] , (10.1.66) where, E (cid:15) [Ψ] (cid:46) (cid:15)u − − δ dec trap (cid:2) r − | e Ψ | + r − ( | e Ψ | + |∇ / Ψ | + r − | Ψ | + | e Ψ | ( | e Ψ | + |∇ / Ψ | )) (cid:3) . Proof.
Recall that, see (10.1.12) E [ X, w, M ](Ψ) := D µ P µ [ X, w, M ] − (cid:18) X (Ψ) + 12 w Ψ (cid:19) · N [Ψ]where, P µ = P µ [ X, w, M ] = Q µν X ν + 12 w Ψ ˙ D µ Ψ − | Ψ | ∂ µ w + 14 | Ψ | M µ with triplet, ( X, w, M ) := ( f (cid:98) δ R, w (cid:98) δ , hR ) + (cid:15) H ( Y H , ,
0) + (0 , , h ˘ R )given in Proposition 10.1.30. Replacing X by ˇ X = X + Λ T in the calculation above wededuce, ˇ P µ = P µ [ ˇ X, w, M ] = Q µν ˇ X ν + 12 w Ψ ˙ D µ Ψ − | Ψ | ∂ µ w + 14 | Ψ | M µ . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
By the divergence theorem we have, (cid:90) A ˇ P · N A + (cid:90) Σ ˇ P · N Σ + (cid:90) M ( τ ,τ ) E + (cid:90) Σ ∗ ˇ P · N Σ ∗ = (cid:90) Σ ˇ P · N Σ − (cid:90) M ( τ ,τ ) ( ˇ X (Ψ) + 12 w Ψ) N [Ψ] (10.1.67)where E = E [ ˇ X, w, M ](Ψ). Now, E [ ˇ X, w, M ](Ψ) = E [ X, w, M ](Ψ) + 12 Λ
Q · ( T ) π − T ( V ) | Ψ | . According to Lemma 10.1.7 T ( V ) = O ( (cid:15) ) r − u − − δ dec trap , and all components of ( T ) π are O ( (cid:15)r − u − − δ dec trap ) except for ( T ) π which is O ( (cid:15)r − u − − δ dec trap ). We easily deduce,Λ |Q · ( T ) π | + | T ( V ) || Ψ | (cid:46) Λ E (cid:15) . Thus in view of to Proposition 10.1.31, we have , (cid:90) M ( τ ,τ ) E ≥ δ H (cid:90) M ( τ ,τ ) (cid:34) m r | ˘ R Ψ | + r − (cid:18) − mr (cid:19) (cid:18) |∇ / Ψ | + m r | ˘ T Ψ | (cid:19) + mr | Ψ | (cid:35) − O (cid:0) δ − H (cid:1) (cid:90) M ( τ ,τ ) E (cid:15) i.e., (cid:90) M ( τ ,τ ) E ≥ δ H Mor[Ψ]( τ , τ ) − O (cid:0) δ − H (cid:1) Err (cid:15) ( τ , τ ) . (10.1.68)We now analyze the boundary terms in (10.1.67). Boundary term along A Along the spacelike hypersurface A , i.e. r = 2 m (1 − δ H ), the unit normal N A is given by N A = 12 (cid:113) e ( r ) e ( r ) (cid:18) e + e ( r ) e ( r ) e (cid:19) = 12 (cid:112) δ H + O ( (cid:15) ) (cid:16) e + ( δ H + O ( (cid:15) )) e (cid:17) , Recall from (10.1.64) that we have Λ = δ − H (cid:28) δ − H . h, h = 0 as well as w = − δ w where δ > w isgiven by (10.1.46) w ( r, m ) = r − m r Υ (cid:18) − mr (cid:19) . In particular, we have on A in view of the formula for w and for N A | w | (cid:46) δ H , | N A ( w ) | (cid:46) (cid:112) δ H . Hence,
P · N A = Q ( X + Λ T, N A ) − δ w Ψ N A (Ψ) + δ | Ψ | N A ( w )= 2 (cid:112) δ H + O ( (cid:15) ) Q ( X + Λ T, e ) + 2 (cid:112) δ H + O ( (cid:15) ) Q ( X + Λ T, e ) − O ( (cid:112) δ H )Ψ e (Ψ) − O ( δ H )Ψ e (Ψ) − O ( (cid:112) δ H ) | Ψ | . Thus, in view of Proposition 10.1.32, we infer
P · N A ≥ (cid:112) δ H + O ( (cid:15) ) (cid:18) (cid:15) H Q + 12 Λ Q (cid:19) + 2 (cid:112) δ H + O ( (cid:15) ) (cid:18) (cid:15) H Q + 12 Λ Q (cid:19) − O ( (cid:112) δ H )Ψ e (Ψ) − O ( δ H )Ψ e (Ψ) − O ( (cid:112) δ H ) | Ψ | Using in particular (10.1.62) and (10.1.64), we deduce
P · N A ≥ (cid:112) δ H + O ( (cid:15) ) (cid:18) δ H (cid:16) |∇ / Ψ | + O ( δ H ) | Ψ | (cid:17) + 18 δ − H | e Ψ | (cid:19) +2 (cid:112) δ H + O ( (cid:15) ) (cid:18) δ H | e Ψ | + 18 δ − H (cid:16) |∇ / Ψ | + O ( δ H ) | Ψ | (cid:17)(cid:19) − O ( (cid:112) δ H )Ψ e (Ψ) − O ( δ H )Ψ e (Ψ) − O ( (cid:112) δ H ) | Ψ | ≥ δ − H |∇ / Ψ | + 18 δ − H | e Ψ | + 14 δ H | e Ψ | − O ( (cid:112) δ H )Ψ e (Ψ) − O ( δ H )Ψ e (Ψ) − O ( (cid:112) δ H ) | Ψ | . Recalling the Poincar´e inequality (10.1.26), (cid:90) S |∇ / Ψ | ≥ r − (cid:0) − O ( (cid:15) ) (cid:1) (cid:90) S Ψ da S , we deduce, in this region, (cid:90) A ( τ ,τ ) P · N A ≥ (cid:90) A ( τ ,τ ) (cid:16) δ − H | e Ψ | + δ H | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) as desired in view of the definition of the flux along A .02 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Boundary terms along Σ( τ ) , Σ( τ )Along a hypersurface Σ( τ ) with timelike unit future normal N Σ( τ ) = ae + be , we have, P · N Σ = Q ( X + Λ T, N Σ ) + 12 w Ψ N Σ (Ψ) − N Σ ( w ) | Ψ | + 12 N Σ · ( hR + h ˘ R ) | Ψ | and E [Ψ]( τ ) = (cid:90) Σ( τ ) (cid:16) b | e Ψ | + 2 a | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) .
1. In the region r ≥ m (1 − δ H ), Υ ≤ δ H we have h = 0, h = O ( (cid:98) δ ) and N Σ = e (i.e. a = 1, b = 0). Also, we have w = − δ w , where δ > w is given by (10.1.46) w ( r, m ) = r − m r Υ (cid:18) − mr (cid:19) . In particular, we have in the region of interest, in view of the formula for w andfor N Σ | w | (cid:46) δ H , | N Σ ( w ) | = | e ( w ) | (cid:46) . We infer
P · N Σ = Q ( X + Λ T, e ) − δ w Ψ e (Ψ) + δ e ( w ) | Ψ | + 12 h e · ˘ R | Ψ | = Q ( X + Λ T, e ) − O ( δ H ) w Ψ e (Ψ) − O (1) | Ψ | . where we used the fact that ˘ R = ( e − e ) in the region of interest. Thus, accordingto Proposition 10.1.32, P · N Σ ≥ (cid:15) H Q + 12 Λ Q − O ( δ H ) | Ψ || e (Ψ) | − O (1) | Ψ | . Using in particular (10.1.62) and (10.1.64), we deduce
P · N Σ ≥ δ H | e Ψ | + 18 δ − H ( |∇ / Ψ | + O ( (cid:15) ) | Ψ | ) − O ( δ H ) | Ψ || e Ψ | − O (1) | Ψ | Together with the Poincar´e inequality (10.1.26), we deduce (cid:90) Σ r ≥ m − δ H ) , Υ ≤ δ H ( τ ) P · N Σ ≥ δ H (cid:90) Σ r ≥ m − δ H ) , Υ ≤ δ H ( τ ) (cid:16) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) ≥ δ H E r ≥ m (1 − δ H ) , Υ ≤ δ H [Ψ]( τ ) . ≥ δ H , we have w = O ( r − ), N Σ ( w ) = O ( r − ), h = O ( r − ) and h = O ( r − ). We infer P · N Σ = a Q ( X + Λ T, e ) + b Q ( X + Λ T, e ) − O ( r − ) | Ψ | ( a | e Ψ | + b | e Ψ | ) − O ( r − ) | Ψ | . Thus, according to Proposition 10.1.32,
P · N Σ ≥ δ − H ( a Q + b Q + ( a + b ) Q ) − O (1)( a | e Ψ | + b | e Ψ | ) − O ( r − ) | Ψ | = δ − H (cid:18) a | e Ψ | + b | e Ψ | + ( a + b ) (cid:18) |∇ / Ψ | + 4Υ r | Ψ | (cid:19)(cid:19) − O (1) (cid:16) a | e Ψ | + b | e Ψ | + r − | Ψ | (cid:17) ≥ δ − H (cid:32) a | e Ψ | + b | e Ψ | + ( a + b ) (cid:32) |∇ / Ψ | + 4 δ H r | Ψ | (cid:33)(cid:33) − O (1) (cid:16) a | e Ψ | + b | e Ψ | + r − | Ψ | (cid:17) . Hence, for δ H > a ≤ a , b ≤ b and a + b ≥
1, we inferin this region
P · N Σ ≥ δ − H (cid:90) Σ Υ ≥ δ H ( τ ) (cid:16) b | e Ψ | + 2 a | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) = δ − H E [Ψ] Υ ≥ δ H ( τ )In view of the above estimates in r ≥ m (1 − δ H ), Υ ≤ δ H and in Υ ≥ δ H , we deduce,everywhere, (cid:90) Σ( τ ) P · N Σ ≥ δ H E [Ψ]( τ ) . (10.1.69) Boundary terms along Σ ∗ On Σ ∗ , we have N Σ ∗ = T + O (cid:16) (cid:15) + mr (cid:17) ( e + e ) ,w = O ( r − ), N Σ ∗ ( w ) = O ( (cid:15)r − ), h = O ( r − ) and h = 0.04 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Proceeding as before, we have along Σ ∗ , P · N Σ = (cid:18)
12 + O (cid:16) (cid:15) + mr (cid:17)(cid:19) Q ( X + Λ T, e ) + (cid:18)
12 + O (cid:16) (cid:15) + mr (cid:17)(cid:19) Q ( X + Λ T, e ) − O ( r − ) | Ψ | ( | e Ψ | + | e Ψ | ) − O ( r − ) | Ψ | ≥ Q ( X + Λ T, e ) + 14 Q ( X + Λ T, e ) − O (cid:16) | e Ψ | + | e Ψ | + r − | Ψ | (cid:17) . Thus, according to Proposition 10.1.32, we have
P · N Σ ≥
116 Λ ( Q + Q + 2 Q ) − O (cid:16) | e Ψ | + | e Ψ | + r − | Ψ | (cid:17) = 116 Λ (cid:18) | e Ψ | + | e Ψ | + 2 (cid:18) |∇ / Ψ | + 4Υ r | Ψ | (cid:19)(cid:19) − O (cid:16) | e Ψ | + | e Ψ | + r − | Ψ | (cid:17) ≥ δ − H (cid:16) | e Ψ | + | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) − O (cid:16) | e Ψ | + | e Ψ | + r − | Ψ | (cid:17) ≥ δ − H (cid:16) | e Ψ | + | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) and hence (cid:90) Σ ∗ ( τ ,τ ) P · N Σ ∗ ≥ δ − H (cid:90) Σ ∗ ( τ ,τ ) (cid:16) | e Ψ | + | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) . (10.1.70) The inhomogeneous term (cid:82) M ( τ ,τ ) ( ˇ X (Ψ) + w Ψ) N [Ψ]Recall that, ˇ X = X + Λ T = f (cid:98) δ R + Y H + Λ T . We easily check, recalling the properties of f (cid:98) δ , w and Λ and the definition of J [ N, Ψ], (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) M ( τ ,τ ) (cid:18) ˇ X (Ψ) + 12 w Ψ (cid:19) N [Ψ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ − H (cid:90) M ( τ ,τ ) (cid:16) | ˘ R Ψ | + | ˘ T Ψ | + r − | Ψ | (cid:17) | N (Ψ) | = δ − H J [ N, Ψ]( τ , τ ) . (10.1.71)Going back to (10.1.67) we deduce, E [Ψ]( τ ) + (cid:90) M ( τ ,τ ) E + F [Ψ]( τ , τ ) ≤ δ − H ( E [Ψ]( τ ) + J [ N, Ψ]( τ , τ )) . In view of (10.1.68) we obtain, E [Ψ]( τ ) + Mor[Ψ]( τ , τ ) + F [Ψ]( τ , τ ) ≤ δ − H (cid:0) E [Ψ]( τ ) + J [ N, Ψ]( τ , τ ) (cid:1) + O (cid:16) δ − H (cid:17) Err (cid:15) ( τ , τ ) . This concludes the proof of Theorem 10.1.33. E (cid:15) Recall that Err (cid:15) ( τ , τ ) = (cid:82) M ( τ ,τ ) E (cid:15) where, E (cid:15) (cid:46) (cid:15)u − − δ dec trap (cid:104) r − | e Ψ | + r − (cid:0) | e Ψ | + |∇ / Ψ | + r − | Ψ | + | e Ψ | ( | e Ψ | + |∇ / Ψ | ) (cid:1)(cid:105) . • In the trapping region M trap , i.e. m ≤ r ≤ m , where u trap = 1 + τ and Σ( τ ) isstrictly spacelike, we have (cid:90) Σ trap ( τ ) E (cid:15) (cid:46) (cid:15)τ − − δ dec trap (cid:90) Σ trap ( τ ) (cid:0) | e Ψ | + | e Ψ | + |∇ / Ψ | + | Ψ | (cid:1) (cid:46) (cid:15)τ − − δ dec trap E [Ψ]( τ ) . Thus, (cid:90) M trap ( τ ,τ ) E (cid:15) (cid:46) (cid:15) (cid:90) τ τ τ − − δ dec trap E [Ψ]( τ ) (cid:46) (cid:15) (cid:18)(cid:90) τ τ (1 + τ ) − − δ (cid:19) sup τ ∈ [ τ ,τ ] E [Ψ]( τ ) (cid:46) (cid:15) sup τ ∈ [ τ ,τ ] E [Ψ]( τ )and therefore, for small (cid:15) >
0, the integral (cid:82) M trap ( τ ,τ ) E (cid:15) can be absorbed on theleft hand side of (10.1.66). • In the non trapping region M trap (cid:14) we write, with a fixed δ > E (cid:15) (cid:46) (cid:15)r − − δ | e Ψ | + r − δ (cid:0) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) . Hence, (cid:90) M trap (cid:14) ( τ ,τ ) E (cid:15) (cid:46) (cid:15) (cid:90) M r ≥ m ( τ ,τ ) r − − δ | e Ψ | + (cid:15) (cid:90) M r ≥ m ( τ ,τ ) r − δ (cid:0) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) + (cid:15) (cid:90) ( trap (cid:14) ) M r ≤ m ( τ ,τ ) (cid:0) | e Ψ | + | e Ψ | + |∇ / Ψ | + | Ψ | (cid:1) . Note that for (cid:15) > ( trap (cid:14) ) M r ≤ m , can beabsorbed by the left hand side of (10.1.66).06 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
As a consequence we deduce the following.
Corollary 10.1.34.
The statement of Theorem 10.1.33 remains true if we replace Err (cid:15) in the statement of the theorem withErr (cid:15) = (cid:90) M r ≥ m ( τ ,τ ) E (cid:15) , E (cid:15) (cid:46) (cid:15)r − − δ | e Ψ | + (cid:15)r − δ (cid:0) | e Ψ | + |∇ / Ψ | + | r − | Ψ | (cid:1) , for a fixed δ > . Remark 10.1.35.
Note that the error terms Err (cid:15) cannot yet be absorbed to the let handside of (10.1.66) . In fact we need additional estimates. The Morawetz bulk quantity (5.1.13) ,Mor [Ψ]( τ , τ ) := (cid:90) M ( τ ,τ ) m r | ˘ R Ψ | + mr | Ψ | + (cid:18) − mr (cid:19) r (cid:18) |∇ / Ψ | + m r | ˘ T Ψ | (cid:19) is quite weak for r large with regard to the terms | ˘ R Ψ | and | ˘ T Ψ | , while, using thePoincar´e inequality, Mor [Ψ] controls the term (cid:82) M r ≥ m ( τ ,τ ) r − ( |∇ / Ψ | + r − | Ψ | ) . In thenext section we show how we can estimate (cid:82) M ≥ R ( τ ,τ ) r − − δ | e Ψ | by (cid:82) M ≥ R ( τ ,τ ) r − − δ | e Ψ | and then, we provide estimates for the remaining terms. Note also that the weight r − − δ is optimal in estimating e Ψ in the wave zone region. We are now ready to prove Theorem 10.1.1. Note that it suffices to improve the previousMorawetz estimate of Theorem 10.1.33 by replacing the quantity Mor[Ψ]( τ , τ ) withMorr[Ψ]( τ , τ ) := Mor[Ψ]( τ , τ ) + (cid:90) M far ( τ ,τ ) r − − δ | e (Ψ) | . In view of the Morawetz estimate (10.1.66) and corollary 10.1.34 we have E [Ψ]( τ ) + Mor[Ψ]( τ , τ ) + F [Ψ]( τ , τ ) (cid:46) E [Ψ]( τ ) + J [ N, Ψ]( τ , τ ) + Err (cid:15) ( τ , τ ) ,J [ N, Ψ]( τ , τ ) : = (cid:90) M ( τ ,τ ) ( | ˘ R Ψ | + | ˘ T Ψ | + r − | Ψ | ) | N | , (10.1.72) (cid:15) (cid:46) (cid:15) (cid:90) M ≥ m ( τ ,τ ) r − − δ | e Ψ | + r − δ (cid:0) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) . We divide J [ N ] = J [ N, Ψ] as follows: J [ N ] = J [ N ] trap + J [ N ] trap (cid:14) where, J [ N ] trap : = (cid:90) M trap ( | ˘ R Ψ | + | ˘ T Ψ | + r − | Ψ | ) | N | ,J [ N ] trap (cid:14) : = (cid:90) ( trap (cid:14) ) M ( | ˘ R Ψ | + | ˘ T Ψ | + r − | Ψ | ) | N | . For the trapping region, where the hypersurfaces Σ( τ ) are strictly spacelike, we write, J [ N ] trap ( τ , τ ) = (cid:90) τ τ dτ (cid:90) Σ trap ( τ ) ( | ˘ R Ψ | + | ˘ T Ψ | + r − | Ψ | ) | N |≤ (cid:90) τ τ E [Ψ]( τ ) / (cid:32)(cid:90) Σ trap ( τ ) | N | (cid:33) / ≤ sup τ ∈ [ τ ,τ ] E [Ψ]( τ ) / (cid:90) τ τ (cid:32)(cid:90) Σ trap ( τ ) | N | (cid:33) / (cid:46) λ sup τ ∈ [ τ ,τ ] E [Ψ]( τ ) + λ − (cid:18) (cid:90) τ τ (cid:107) N (cid:107) L (Σ trap ( τ )) (cid:19) . Hence, for λ > E [Ψ]( τ ) + Mor[Ψ]( τ , τ ) + F [Ψ]( τ , τ ) (cid:46) E [Ψ]( τ ) + Err (cid:15) ( τ , τ )+ J trap (cid:14) [ N, Ψ]( τ , τ ) + (cid:18) (cid:90) τ τ (cid:107) N (cid:107) L (Σ trap ( τ )) (cid:19) . On the other hand we have, J [ N ] trap (cid:14) ( τ , τ ) = (cid:90) M trap (cid:14) ( τ ,τ ) ( | ˘ R Ψ | + | ˘ T Ψ | + r − | Ψ | ) | N |≤ λ (cid:90) M trap (cid:14) r − − δ ( | ˘ R Ψ | + | ˘ T Ψ | + r − | Ψ | ) + λ − (cid:90) M trap (cid:14) r δ | N | . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
The first integral on the right can be divided further into integrals for r ≤ m and r ≥ m . The first integral can the be easily absorbed by the term M or [Ψ]( τ , τ ), if λ > E [Ψ]( τ ) + Mor[Ψ]( τ , τ ) + F [Ψ]( τ , τ ) (cid:46) E [Ψ]( τ ) + Err (cid:15) ( τ , τ ) + I δ [ N ]( τ , τ )+ (cid:90) M r ≥ m r − − δ ( | e Ψ | + | e Ψ | + r − | Ψ | )where, I δ [ N ]( τ , τ ) : = (cid:90) M trap (cid:14) ( τ ,τ ) r δ | N | + (cid:18) (cid:90) τ τ dτ (cid:107) N (cid:107) L (Σ trap ( τ )) (cid:19) . Recalling the definition of Err (cid:15) in Corollary 10.1.34, we deduce, E [Ψ]( τ ) + Mor[Ψ]( τ , τ ) + F [Ψ]( τ , τ ) (cid:46) E [Ψ]( τ ) + Err (cid:15) ( τ , τ ) + I δ [ N ]( τ , τ ) . (10.1.73)To eliminate the term in e Ψ from the error term we appeal to the following proposition.
Proposition 10.1.36.
Assume (cid:3)
Ψ = V Ψ + N and consider the vectorfield X = f − δ T with f − δ := r − δ for r ≥ m and compactly supported in r ≥ m . With the notation ofProposition 10.1.9, let P µ [ f − δ T, ,
0] = f − δ Q αµ T µ , E [ f − δ T, ,
0] = D µ P µ [ f − δ T, , − f − δ T (Ψ) N. Then,1. We have, for r ≥ m E [ f − δ T, ,
0] = Υ δr − − δ | e Ψ | − δr − − δ | e Ψ | + O (cid:0) (cid:15)r − − δ (cid:0) | D Ψ | + r − | Ψ | (cid:1)(cid:1) .
2. We have, P [ f − δ T, , · e = f − δ Q ( T, e ) ≥ , P [ f − δ T, , · e = f − δ Q ( T, e ) ≥ . We postponed the proof of Proof of Proposition 10.1.36 and continue the proof of Theorem10.1.1. By integration, the proposition provides a bound for (cid:90) M ≥ m ( τ ,τ ) r − − δ | e Ψ | Note that Υ ≥ in r ≥ m . E [Ψ]( τ ), the integrals (cid:82) M ≥ m ( τ ,τ ) r − − δ | e Ψ | and (cid:82) M ≥ m ( τ ,τ ) r − δ T (Ψ) N ,as well as the error terms. The second bulk integral involving the inhomogeneous term N can be estimates exactly like before. Thus combining the new estimate with that in thecorollary 10.1.34 we derive the desired estimate hence concluding the proof of Theorem10.1.1. Proof of Proposition 10.1.36
We consider vectorfields of the form X = f ( r ) T with T = (Υ e + e ). Recall, see Lemma10.1.7, that all components of the deformation tensor ( T ) π of T = ( e + Υ e ) can bebounded by O ( (cid:15)r − ). Since f = O ( r − δ ), we deduce, ( X ) π αβ = f ( T ) π αβ + D α f T β + D β f T α = D α f T β + D β f T α + O ( (cid:15)r − − δ ) . Also, e ( f ) = f (cid:48) e ( r ) = − f (cid:48) + O ( (cid:15)r − − δ ) , e ( f ) = f (cid:48) e ( r ) = Υ f (cid:48) + O ( (cid:15)r − − δ ) . Thus, modulo error terms of the form O ( (cid:15) ) r − − δ (cid:0) | e Ψ | + | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) , wehave Q · ( X ) π = 2 Q αβ T α D β f = 2 (cid:0) Q β T β e f + Q β T β e f (cid:1) = −Q ( e , T ) e f − Q ( e , T ) e f = 12 Q ( e , e + Υ e ) f (cid:48) − f (cid:48) Υ Q ( e , e + Υ e )= 12 f (cid:48) (cid:0) | e Ψ | − Υ | e Ψ | (cid:1) . We now apply Proposition 10.1.9, as well as (10.1.12) (10.1.13), with X = f − δ ( r ) T , w = 0, M = 0 so that P µ [ f − δ T, ,
0] = f − δ Q αµ T µ , E [ f − δ T, ,
0] := D µ P µ [ f − δ T, , − f − δ T (Ψ) N and E [ f − δ T, ,
0] = 12
Q · ( X ) π − f − δ T ( V ) | Ψ | = 14 f (cid:48)− δ ( r ) (cid:0) | e Ψ | − Υ | e Ψ | (cid:1) + O (cid:0) (cid:15)r − − δ (cid:0) | D Ψ | + r − | Ψ | (cid:1)(cid:1) with | D Ψ | = | e Ψ | + | e Ψ | + |∇ / Ψ | + r − | Ψ | . Since f − δ ( r ) = r − δ for r ≥ m , wededuce, for r ≥ m , E [ f − δ T, ,
0] = Υ δr − − δ | e Ψ | − δr − − δ | e Ψ | + O (cid:0) (cid:15)r − − δ (cid:0) | D Ψ | + r − | Ψ | (cid:1)(cid:1) . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
On the other hand, P [ f − δ T, , · e = f − δ Q ( T, e ) ≥ , P [ f − δ T, , · e = f − δ Q ( T, e ) ≥ , as desired. This concludes the proof of Proposition 10.1.36. r p - weighted estimates For convenience, we work in this section with the renormalized frame ( e (cid:48) , e (cid:48) ) defined in(10.2.7) instead of the original frame ( e , e ). To simplify the exposition, we still denoteit as ( e , e ). Recall that the two are frames are equivalent up to lower terms in m/r .In this section we rely on the Morawetz estimates proved in the previous section toestablish r p -weighted estimates in the spirit of Dafermos-Rodnianski [20]. The followingtheorem claims r p -weighted estimates for the solution ψ of the wave equation (5.3.5). Theorem 10.2.1 ( r p -weighted estimates) . Consider a fixed δ > and let R (cid:29) m δ , (cid:15) (cid:28) δ .The following estimates hold true and for all δ ≤ p ≤ − δ , ˙ E p ; R [ ψ ]( τ ) + ˙ B p ; R [ ψ ]( τ , τ ) + ˙ F p [ ψ ]( τ , τ ) (cid:46) E p [ ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) . (10.2.1) Remark 10.2.2.
Note that Theorem 10.1.1 on Morawetz estimates and Theorem 10.2.1on r p -weighted estimates immediately yield for all δ ≤ p ≤ − δ , sup τ ∈ [ τ ,τ ] E p [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) + F p [ ψ ]( τ , τ ) (cid:46) E p [ ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) , (10.2.2) which corresponds to Theorem 5.3.4 in the case s = 0 . Theorem 10.2.1 will be proved in section 5.3.1. We will need in this section strongerassumptions in the region r ≥ m , away from trapping, than those in (10.1.4)–(10.1.6)of the previous section. For convenience we express our conditions with respect to theweights , w p,q ( u, r ) = r − p (1 + τ ) − q − δ dec +2 δ . The assumptions are consistent with the global frame used in Theorem M1, see Lemma 5.1.1. Inparticular, δ > δ dec − δ > δ dec − δ to derivethe r p weighted estimates. R P - WEIGHTED ESTIMATES RP0.
The assumptions
Mor1 – Mor4 made in the previous section hold true.
RP1.
The Ricci coefficients verify, for r ≥ m (cid:12)(cid:12) ξ, ϑ, ϑ, η, η, ζ, ω (cid:12)(cid:12) (cid:46) (cid:15)w , , (cid:12)(cid:12)(cid:12) κ + 2 r (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) χ + 1 r (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) e Φ − χ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)w , , (cid:12)(cid:12)(cid:12) κ − r (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) χ − Υ r | , (cid:12)(cid:12)(cid:12) e Φ − χ (cid:12)(cid:12)(cid:12) (cid:46) (cid:15) min { w , , w , / } , (cid:12)(cid:12)(cid:12) ω + mr (cid:12)(cid:12)(cid:12) , | ξ | (cid:46) (cid:15) min { w , , w , / } . (10.2.3) RP2.
The derivatives of r verify, for r ≥ m , (cid:12)(cid:12) e ( r ) + 1 (cid:12)(cid:12) (cid:46) (cid:15)w , , (cid:12)(cid:12) e ( r ) − Υ (cid:12)(cid:12) (cid:46) (cid:15) min { w , , w , / } , (cid:12)(cid:12)(cid:12) e e ( r ) + 2 mr , e e ( r ) (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)w , . (10.2.4) RP3.
For r ≥ m , (cid:12)(cid:12)(cid:12) ρ + 2 mr (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)w , , (cid:12)(cid:12)(cid:12) K − r (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − , (cid:12)(cid:12) e θ (Φ) (cid:12)(cid:12) (cid:46) r − . (10.2.5) RP4.
We also assume, for r ≥ m , | m − m | (cid:46) (cid:15), | e m, re m | (cid:46) (cid:15)w , , | e e ( m ) , e e ( m ) | (cid:46) (cid:15)w , . (10.2.6)Since the estimates we are establishing are restricted to the far region r > R it is conve-nient, in this section, to work with the with renormalized frame e (cid:48) = Υ e , e (cid:48) = Υ − e , e (cid:48) θ = e θ . (10.2.7)Relative to the new frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) we have, ξ (cid:48) = Υ − ξ, ξ (cid:48) = Υ ξ, ζ (cid:48) = ζ, η (cid:48) = η, χ (cid:48) = Υ − χ, χ (cid:48) = Υ χ CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS and, ω (cid:48) = Υ − (cid:18) ω + 12 e (log Υ) (cid:19) = Υ − (cid:18) ω + 12 Υ − Υ (cid:48) e ( r ) (cid:19) = Υ − (cid:16) − mr + O ( (cid:15)r − (1 + | u | ) − / − δ dec ) + Υ − mr (Υ + O ( (cid:15)r − (1 + | u | ) − / − δ dec ) (cid:17) = O ( (cid:15)r − ) ,ω (cid:48) = Υ (cid:18) ω − e (log Υ) (cid:19) = Υ (cid:18) ω −
12 Υ − Υ (cid:48) e ( r ) (cid:19) = Υ (cid:18) ω −
12 Υ − Υ (cid:48) ( − O ( (cid:15) (1 + | u | ) − − δ dec ) (cid:19) = mr + O ( (cid:15)r − (1 + | u | ) − − δ dec ) . Thus in the new frame we have, for r ≥ m , RP1’.
The Ricci coefficients with respect to the null frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ) verify, for r ≥ m : (cid:12)(cid:12) ξ (cid:48) , ϑ (cid:48) , ϑ, η (cid:48) , η (cid:48) , ζ (cid:48) (cid:12)(cid:12) , | ω (cid:48) − mr | (cid:46) (cid:15)w , , (cid:12)(cid:12)(cid:12) κ (cid:48) + 2Υ r (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) χ (cid:48) + Υ r (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12) e (cid:48) Φ − χ (cid:48) (cid:12)(cid:12) (cid:46) (cid:15)w , , (cid:12)(cid:12)(cid:12) κ (cid:48) − r (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) χ (cid:48) − r (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12) e (cid:48) Φ − χ (cid:48) (cid:12)(cid:12) (cid:46) (cid:15) min { w , , w , / } , (cid:12)(cid:12) ω (cid:48) (cid:12)(cid:12) , | ξ (cid:48) | (cid:46) (cid:15) min { w , , w , / } . (10.2.8) RP2’.
The derivatives of r verify, (cid:12)(cid:12) e (cid:48) ( r ) + Υ (cid:12)(cid:12) (cid:46) (cid:15)w , , (cid:12)(cid:12) e (cid:48) ( r ) − (cid:12)(cid:12) (cid:46) (cid:15)w , , (cid:12)(cid:12)(cid:12) e (cid:48) e (cid:48) ( r ) , e (cid:48) e (cid:48) ( r ) + 2 mr (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)w , . (10.2.9) RP3’.
The Gauss curvature K of S and ρ verify, (cid:12)(cid:12)(cid:12) ρ + 2 mr (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − , (cid:12)(cid:12)(cid:12) K − r (cid:12)(cid:12)(cid:12) (cid:46) (cid:15)r − . (10.2.10) RP4’.
We also assume | m − m | (cid:46) (cid:15), | e (cid:48) m, re (cid:48) m | (cid:46) (cid:15)w , , | e (cid:48) e (cid:48) ( m ) , e (cid:48) e (cid:48) ( m ) | (cid:46) (cid:15)w , . (10.2.11) R P - WEIGHTED ESTIMATES Remark 10.2.3.
In the far region r ≥ m all norms we are using in our estimates areequivalent when expressed relative to the null frame ( e , e , e θ ) or ( e (cid:48) , e (cid:48) , e (cid:48) θ ) . Convention.
For the remaining of this section we shall do all calculations with respect tothe renormalized frame ( e (cid:48) , e (cid:48) , e (cid:48) θ ). For convenience we shall drop the primes, throughoutthis section, since there is no danger of confusion. Note however that the main results,which include the interior region r ≤ R , are always expressed with respect the originalframe. X = f ( r ) e Lemma 10.2.4.
Consider the vectorfield X = f ( r ) e .1. We have the decomposition, ( X ) π = ( X ) Λ g + ( X ) (cid:101) π, ( X ) Λ = 2 r f with symmetric tensor ( X ) (cid:101) π which verifies ( X ) (cid:101) π = − f (cid:48) + 4 fr + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) ( X ) (cid:101) π = 4 f (cid:48) Υ − (cid:48) + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) ( X ) (cid:101) π θ = O ( (cid:15) ) w , / | f | ( X ) (cid:101) π AB = O ( (cid:15) ) w , / | f | ( X ) (cid:101) π θ = O ( (cid:15) ) w , | f | (10.2.12)
2. We have, (cid:3) ( X ) Λ = 2 r f (cid:48)(cid:48) + O (cid:16) mr + (cid:15)w , (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) (10.2.13) Proof.
See Lemma D.3.1 in appendix. X = f ( r ) e We start with the following proposition.14
CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Proposition 10.2.5.
Assume Ψ verifies the equation ˙ (cid:3) g Ψ = V Ψ + N and let X = f e and w = ( X ) Λ = fr and let E := E [ X, w ] = E [ X = f e , w = fr ] .1. We have, E = 12 f (cid:48) | e Ψ | + 12 (cid:18) − f (cid:48) + 2 fr (cid:19) (cid:0) |∇ / Ψ | + V | Ψ | (cid:1) − r f (cid:48)(cid:48) | Ψ | + Err (cid:16) (cid:15), mr , f (cid:17) (Ψ) where,Err (cid:16) (cid:15), mr , f (cid:17) (Ψ) = O (cid:16) mr (cid:17) ( | f | + r | f (cid:48) | ) | e Ψ | + O (cid:16) mr + (cid:15)w , (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) | Ψ | + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) (cid:0) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) + O ( (cid:15) ) w , / | f | (cid:16) | e Ψ | ( | e Ψ | + r − |∇ / Ψ | ) + |∇ / Ψ | + r − | Ψ | (cid:17) .
2. The current, P µ = P µ [ X, w ] = Q µν X ν + 12 w Ψ · D µ Ψ − | Ψ | ∂ µ w verifies, P · e = f (cid:12)(cid:12)(cid:12)(cid:12) e Ψ + 1 r Ψ (cid:12)(cid:12)(cid:12)(cid:12) − r − e ( rf | Ψ | ) + O ( (cid:15)r − ) f | Ψ | , P · e = f Q + 12 r − e (cid:0) rf ψ ) + r − f (cid:48) ψ + O ( mr − + (cid:15)r − ) | rf (cid:48) | | Ψ | .
3. Let θ = θ ( r ) supported for r ≥ R/ with θ = 1 for r ≥ R such that f p = θ ( r ) r p . Let ( p ) P := P [ f p e , w p ] . Then, for all r ≥ R , ( p ) P · e + p r − e ( θr p +1 | Ψ | ) ≥ r p − ( p − | Ψ | . Before proceeding with the proof of Proposition 10.2.5, we first establish the followinglemma.
Lemma 10.2.6.
We have,
Q · ( X ) (cid:101) π = (cid:16) f (cid:48) + O (cid:16) mr (cid:17) ( | f | + r | f (cid:48) | ) (cid:17) | e Ψ | + (cid:18) − f (cid:48) + 2 fr (cid:19) (cid:0) |∇ / Ψ | + V | Ψ | (cid:1) + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) (cid:16) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) + O ( (cid:15) ) w , / | f | (cid:16) | e Ψ | ( | e Ψ | + r − |∇ / Ψ | ) + |∇ / Ψ | + r − | Ψ | (cid:17) . R P - WEIGHTED ESTIMATES Proof.
Recall from Proposition 10.1.9 that we have Q = | e Ψ | , Q = | e Ψ | , Q = |∇ / Ψ | + V | Ψ | , and, |Q AB | ≤ | e Ψ || e Ψ | + |∇ / Ψ | + | V || Ψ | , |Q A | ≤ | e Ψ ||∇ / Ψ | , |Q A | ≤ | e Ψ ||∇ / Ψ | . Hence, in view of Lemma D.3.1 for ( X ) (cid:101) π , we have Q · ( X ) (cid:101) π = 14 Q
44 ( X ) (cid:101) π + 12 Q
34 ( X ) (cid:101) π − Q A ( X ) (cid:101) π A − Q A ( X ) (cid:101) π A + Q AB ( X ) (cid:101) π AB = (cid:16) f (cid:48) Υ − Υ (cid:48) f + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) (cid:17) Q + (cid:18) − f (cid:48) + 2 fr + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) (cid:19) Q + O ( (cid:15) ) w , | f | Q A + O ( (cid:15) ) w , / | f | ( Q A + Q AB )= (cid:16) f (cid:48) + O (cid:16) mr (cid:17) ( | f | + r | f (cid:48) | ) (cid:17) Q + (cid:18) − f (cid:48) + 2 fr (cid:19) Q + O ( (cid:15) )( | f | + r | f (cid:48) | ) w , ( Q + Q A ) + O ( (cid:15) ) w , / | f | ( Q AB + Q )+ O ( (cid:15) ) w , / | f |Q A from which we deduce, Q · ( X ) (cid:101) π = (cid:16) f (cid:48) + O (cid:16) mr (cid:17) ( | f | + r | f (cid:48) | ) (cid:17) | e Ψ | + (cid:18) − f (cid:48) + 2 fr (cid:19) (cid:0) |∇ / Ψ | + V | Ψ | (cid:1) + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) (cid:16) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) + O ( (cid:15) ) w , / | f | (cid:16) | e Ψ | ( | e Ψ | + r − |∇ / Ψ | ) + |∇ / Ψ | + r − | Ψ | (cid:17) as desired.We are now ready to prove Proposition 10.2.5. Proof of Proposition 10.2.5. If Q = Q [Ψ] is the energy momentum tensor of Ψ (recall˙ (cid:3) Ψ = V Ψ + N ) and ( X ) π = ( X ) Λ g + ( X ) (cid:101) π we deduce, Q · ( X ) π = ( X ) Λtr Q + Q · ( X ) (cid:101) π = ( X ) Λ ( −L (Ψ) − V | Ψ | ) + Q · ( X ) (cid:101) π. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Hence, for X = f e and w = ( X ) Λ = fr ,12 Q · ( X ) π + 12 w L [Ψ] = − ( X ) Λ V | Ψ | + 12 Q · ( X ) (cid:101) π. In view of (10.1.13), we infer E : = E [ X, w = ( X ) Λ , M = 0]= 12 Q · ( X ) (cid:101) π − | Ψ | (cid:3) g ( X ) Λ −
12 ( X ( V ) + ( X ) Λ V ) | Ψ | . Recall that V = − κκ . Hence, X ( V ) + ( X ) Λ V = f e ( V ) + 2 fr V = − f (cid:18) e ( κκ ) + 2 r κκ (cid:19) = f (cid:18) κ κ − κρ − r κκ + O ( (cid:15) ) w , (cid:19) = O (cid:16) mr + (cid:15)w , (cid:17) f. Hence, in view of the computation (10.2.13) of (cid:3) ( X ) Λ − | Ψ | (cid:3) g ( X ) Λ −
12 ( X ( V ) + ( X ) Λ V ) | Ψ | = − r f (cid:48)(cid:48) | Ψ | + O (cid:16) mr + (cid:15)w , (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) | Ψ | . We deduce, E = 12 Q · ( X ) (cid:101) π − r f (cid:48)(cid:48) | Ψ | + O (cid:16) mr + (cid:15)w , (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) | Ψ | Using Lemma 10.2.6, we deduce, E = 12 f (cid:48) | e Ψ | + 12 (cid:18) − f (cid:48) + 2 fr (cid:19) (cid:0) |∇ / Ψ | + V | Ψ | (cid:1) − r f (cid:48)(cid:48) | Ψ | + Err (cid:16) (cid:15), mr , f (cid:17) (Ψ)where,Err (cid:16) (cid:15), mr , f (cid:17) (Ψ) = O (cid:16) mr (cid:17) ( | f | + r | f (cid:48) | ) | e Ψ | + O (cid:16) mr + (cid:15)w , (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) | Ψ | + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) (cid:16) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:17) + O ( (cid:15) ) w , / | f | (cid:16) | e Ψ | ( | e Ψ | + r − |∇ / Ψ | ) + |∇ / Ψ | + r − | Ψ | (cid:17) which is the first part of Proposition 10.2.5. R P - WEIGHTED ESTIMATES P · e = f Q + 1 r f Ψ · e Ψ − e ( r − f ) | Ψ | = f (cid:18) | e Ψ | + 1 r Ψ · e Ψ (cid:19) − e ( r − f ) | Ψ | = f (cid:12)(cid:12)(cid:12)(cid:12) e Ψ + 1 r Ψ (cid:12)(cid:12)(cid:12)(cid:12) − r f Ψ · e Ψ − r − f | Ψ | − e ( r − f ) | Ψ | = f (cid:12)(cid:12)(cid:12)(cid:12) e Ψ + 1 r Ψ (cid:12)(cid:12)(cid:12)(cid:12) − r − e ( rf | Ψ | ) + 12 r − e ( rf ) | Ψ | − r − f | Ψ | − e ( r − f ) | Ψ | = f (cid:12)(cid:12)(cid:12)(cid:12) e Ψ + 1 r Ψ (cid:12)(cid:12)(cid:12)(cid:12) − r − e ( rf | Ψ | ) + r − ( e ( r ) − f | Ψ | . Since e ( r ) = r κ + A ) , we have e ( r ) − O ( (cid:15)r − ) . Thus, as desired,
P · e = f (cid:12)(cid:12)(cid:12)(cid:12) e Ψ + 1 r Ψ (cid:12)(cid:12)(cid:12)(cid:12) − r − e ( rf | Ψ | ) + O ( (cid:15)r − ) f | Ψ | . Also,
P · e = f Q + r − f Ψ · e Ψ − e ( r − f ) | Ψ | = f Q + 12 r − f e ( | Ψ | ) − e ( r − f ) | Ψ | = f Q + 12 (cid:2) r − e (cid:0) rf | Ψ | ) − r − e ( rf ) | Ψ | (cid:3) − e ( r − f ) | Ψ | = f Q + 12 r − e (cid:0) rf | Ψ | ) + r − f (cid:48) Υ | Ψ | − r − f (cid:48) ( e ( r ) + Υ) | Ψ | = f Q + 12 r − e (cid:0) rf | Ψ | ) + r − f (cid:48) | Ψ | + O (cid:0) mr − + (cid:15)r − (cid:1) ( r | f (cid:48) | ) | Ψ | as desired. Note that so far we have only used the weaker version e ( r ) − O ( (cid:15) ). This is the first time weneed the stronger version of the estimate in this chapter. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
It remains to prove the last part of Proposition 10.2.5. We have, for r ≥ R ( p ) P · e = r p | e Ψ | + r p − Ψ e Ψ − e ( r p − ) | Ψ | and, ( p ) P · e + p r − e ( θr p +1 | Ψ | ) = r p | e Ψ | + r p − Ψ · e Ψ − e ( r p − ) | Ψ | + pr p − Ψ · e Ψ + p ( p + 1)2 r p − e ( r ) | Ψ | = r p | e Ψ | + ( p + 1) r p − Ψ · e Ψ + p + 12 e ( r ) r p − | Ψ | = r p (cid:34)(cid:12)(cid:12)(cid:12)(cid:12) e Ψ + p + 12 r Ψ (cid:12)(cid:12)(cid:12)(cid:12) + ( p − r | Ψ | (cid:35) + p + 12 ( e ( r ) − r p − | Ψ | ≥ r p − ( p − | Ψ | − O ( (cid:15) ) p + 12 r p − | Ψ | . This concludes the proof of Proposition 10.2.5.In applications we would like to apply Proposition 10.2.5 to f = r p , < p <
2. Wenote however that the presence of the term − r − f (cid:48)(cid:48) | Ψ | on the right hand side of the E identity requires an additional correction if p >
1. This additional correction is takeninto account by the following proposition.
Proposition 10.2.7.
Assume Ψ verifies the equation (cid:3) g Ψ = V Ψ+ N and let X = f ( r ) e , w = ( X ) Λ = fr and M = 2 r − f (cid:48) e . Then,1. We have, with ˇ e = r − e ( r · ) , E [ X, w, M ] = 12 f (cid:48) | ˇ e (Ψ) | + 12 (cid:18) fr − f (cid:48) (cid:19) Q + Err (cid:16) (cid:15), mr ; f (cid:17) [Ψ](10.2.14) with error term,Err (cid:16) (cid:15), mr , f (cid:17) (Ψ) = O (cid:16) mr (cid:17) ( | f | + r | f (cid:48) | ) | e Ψ | + O (cid:16) mr + (cid:15)w , (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) | Ψ | + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) (cid:0) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) + O ( (cid:15) ) w , / | f | (cid:16) | e Ψ | ( | e Ψ | + r − |∇ / Ψ | ) + |∇ / Ψ | + r − | Ψ | (cid:17) . R P - WEIGHTED ESTIMATES
2. The current, P µ = P µ [ X, w, M ] = Q µν X ν + 12 w Ψ D µ Ψ − | Ψ | ∂ µ w + 14 M µ | Ψ | verifies, P · e = f (ˇ e Ψ) − r − e ( rf | Ψ | ) + O ( (cid:15)r − ) f ( | e Ψ | + r − | Ψ | ) , P · e = f Q + 12 r − e (cid:0) rf | Ψ | ) + O ( mr − + (cid:15)r − )( | f | + r | f (cid:48) | ) | Ψ | .
3. Let θ = θ ( r ) supported for r ≥ R/ with θ = 1 for r ≥ R such that f p = θ ( r ) r p . Let ( p ) P := P [ f p e , w p , M p ] . Then, for all r ≥ R , ( p ) P · e + p r − e ( θr p +1 | Ψ | ) ≥ r p − ( p − | Ψ | . Proof.
We start with the first part of Proposition 10.2.7. To this end, we use ( X ) π = − e f + 4 f ω, ( X ) π AB = 2 f (1+3) χ AB so that tr ( X ) π = − ( X ) π + g AB ( X ) π AB = 2 e f − f ω + 2 f κ, and we compute D µ M µ = D µ (2 r − f (cid:48) e ) µ = D µ (cid:18) f (cid:48) rf X µ (cid:19) = 2 f (cid:48) rf Div X + X (cid:18) f (cid:48) rf (cid:19) = f (cid:48) rf tr ( X ) π + X (cid:18) f (cid:48) rf (cid:19) = f (cid:48) rf (2 e f − f ω + 2 f κ ) + 2 f e (cid:18) f (cid:48) rf (cid:19) = f (cid:48) rf (cid:18) e ( r ) f (cid:48) + 4 fr + O (cid:16) mr + (cid:15)w , (cid:17)(cid:19) + 2 f (cid:18) f (cid:48)(cid:48) rf − f (cid:48) ( f + rf (cid:48) ) r f (cid:19) e ( r )= 4 f (cid:48) r + 2( f (cid:48) ) rf + 2 f (cid:48)(cid:48) r − f (cid:48) r − f (cid:48) ) rf + O (cid:16) mr + (cid:15)w , (cid:17) ( | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | )= 2 f (cid:48) r + 2 f (cid:48)(cid:48) r + O (cid:16) mr + (cid:15)w , (cid:17) ( | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | ) . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
We also have 12 Ψ · D µ Ψ M µ = r − f (cid:48) Ψ · D Ψ . Since we have E [ X, w, M ] = E [ X, w ] + 14 ( D µ M µ ) | Ψ | + 12 Ψ · D µ Ψ M µ , we infer E [ X, w, M ] = E [ X, w ] + (cid:18) f (cid:48) r + f (cid:48)(cid:48) r + O (cid:16) mr + (cid:15)w , (cid:17) ( | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | ) (cid:19) | Ψ | + r − f (cid:48) Ψ · D Ψ . Together with Proposition 10.2.5, this yields E [ X, w, M ] = 12 f (cid:48) (cid:16) | e Ψ | + 2 r − Ψ · D Ψ + r − | Ψ | (cid:17) + 12 (cid:18) − f (cid:48) + 2 fr (cid:19) (cid:0) |∇ / Ψ | + V | Ψ | (cid:1) +Err (cid:16) (cid:15), mr , f (cid:17) (Ψ) + O (cid:16) mr + (cid:15)w , (cid:17) ( | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | ) | Ψ | = 12 f (cid:48) | e Ψ + r − Ψ | + 12 (cid:18) − f (cid:48) + 2 fr (cid:19) (cid:0) |∇ / Ψ | + V | Ψ | (cid:1) +Err (cid:16) (cid:15), mr , f (cid:17) (Ψ) + O (cid:16) mr + (cid:15)w , (cid:17) ( | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | ) | Ψ | = 12 f (cid:48) | ˇ e Ψ + r − (1 − e ( r ))Ψ | + 12 (cid:18) − f (cid:48) + 2 fr (cid:19) (cid:0) |∇ / Ψ | + V | Ψ | (cid:1) +Err (cid:16) (cid:15), mr , f (cid:17) (Ψ) + O (cid:16) mr + (cid:15)w , (cid:17) ( | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | ) | Ψ | and hence E [ X, w, M ] = 12 f (cid:48) | ˇ e Ψ | + 12 (cid:18) − f (cid:48) + 2 fr (cid:19) (cid:0) |∇ / Ψ | + V | Ψ | (cid:1) + Err (cid:16) (cid:15), mr , f (cid:17) (Ψ)whereErr (cid:16) (cid:15), mr , f (cid:17) (Ψ) = O (cid:16) mr (cid:17) ( | f | + r | f (cid:48) | ) | e Ψ | + O (cid:16) mr + (cid:15)w , (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) | Ψ | + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) (cid:0) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) + O ( (cid:15) ) w , / | f | (cid:16) | e Ψ | ( | e Ψ | + r − |∇ / Ψ | ) + |∇ / Ψ | + r − | Ψ | (cid:17) . This is the desired estimate (10.2.14).Next, we consider the second part of Proposition 10.2.7. P µ [ X, w, M ] = P µ [ X, w ] + 14 | Ψ | M µ = P µ [ X, w ] + 12 r − f (cid:48) | Ψ | e . R P - WEIGHTED ESTIMATES P [ X, w, M ] = P [ X, w ] = f (cid:12)(cid:12)(cid:12) ˇ e Ψ + (1 − e ( r ))Ψ (cid:12)(cid:12)(cid:12) − r − e ( rf | Ψ | ) + O ( (cid:15)r − ) f | Ψ | = f | ˇ e Ψ | − r − e ( rf | Ψ | ) + O ( (cid:15)r − ) f ( | e Ψ | + r − | Ψ | ) , P [ X, w, M ] = P [ X, w ] − r − f (cid:48) | Ψ | = f Q + 12 r − e (cid:0) rf | Ψ | ) + r − f (cid:48) | Ψ | − r − f (cid:48) | Ψ | + O ( (cid:15)r − )( | f | + r | f (cid:48) | ) | Ψ | = f Q + 12 r − e (cid:0) rf | Ψ | ) + O ( (cid:15)r − )( | f | + r | f (cid:48) | ) | Ψ | as desired. The last part follows from the third part of Proposition 10.2.5. Lemma 10.2.8. On Σ ∗ , we have P · N Σ ∗ = 12 f Q + 12 f (ˇ e Ψ) + 12 div Σ ∗ (cid:16) r − f | Ψ | ν Σ ∗ (cid:17) + O ( mr − + (cid:15) )( | f | + r | f (cid:48) | )( | e Ψ | + |∇ / Ψ | + r − | Ψ | ) Proof.
Recall that there exists a constant c ∗ such that u + r = c ∗ on Σ ∗ . In particular,the unit normal N Σ ∗ is collinear to − g αβ ∂ α ( u + r ) ∂ β = e ( u + r ) e + e ( u + r ) e = e ( r ) e + ( e ( u ) + e ( r )) e and since g (cid:16) e ( r ) e + ( e ( u ) + e ( r )) e , e ( r ) e + ( e ( u ) + e ( r )) e (cid:17) = − e ( r )( e ( u ) + e ( r )) , we infer N Σ ∗ = (cid:112) e ( r )2 (cid:112) e ( u ) + e ( r ) e + (cid:112) e ( u ) + e ( r )2 (cid:112) e ( r ) e . In particular, we have
P · N Σ ∗ = (cid:112) e ( r )2 (cid:112) e ( u ) + e ( r ) P · e + (cid:112) e ( u ) + e ( r )2 (cid:112) e ( r ) P · e . Now, recall from Proposition 10.2.7 that we have
P · e = f (ˇ e Ψ) − r − e ( rf | Ψ | ) + O ( (cid:15)r − ) f ( | e Ψ | + r − | Ψ | ) , P · e = f Q + 12 r − e (cid:0) rf | Ψ | ) + O ( mr − + (cid:15)r − )( | f | + r | f (cid:48) | ) | Ψ | . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
We deduce
P · N Σ ∗ = (cid:112) e ( r )2 (cid:112) e ( u ) + e ( r ) (cid:18) f Q + 12 r − e (cid:0) rf | Ψ | ) + O ( mr − + (cid:15)r − )( | f | + r | f (cid:48) | ) | Ψ | (cid:19) + (cid:112) e ( u ) + e ( r )2 (cid:112) e ( r ) (cid:18) f (ˇ e Ψ) − r − e ( rf | Ψ | ) + O ( (cid:15)r − ) f ( | e Ψ | + r − | Ψ | ) (cid:19) = (cid:112) e ( r )2 (cid:112) e ( u ) + e ( r ) f Q + (cid:112) e ( u ) + e ( r )2 (cid:112) e ( r ) f (ˇ e Ψ) + (cid:112) e ( r )2 (cid:112) e ( u ) + e ( r ) 12 r − e (cid:0) rf | Ψ | ) − (cid:112) e ( u ) + e ( r )2 (cid:112) e ( r ) 12 r − e ( rf | Ψ | )+ O ( mr − + (cid:15)r − )( | f | + r | f (cid:48) | ) | Ψ | + O ( (cid:15)r − ) f ( | e Ψ | + r − | Ψ | )= 12 √ − Υ (1 + O ( (cid:15) )) f Q + √ − Υ2 (1 + O ( (cid:15) )) f (ˇ e Ψ) + 12 r − ν Σ ∗ (cid:0) rf | Ψ | )+ O ( mr − + (cid:15)r − )( | f | + r | f (cid:48) | ) | Ψ | + O ( (cid:15)r − ) f ( | e Ψ | + r − | Ψ | )= 12 f Q + 12 f (ˇ e Ψ) + 12 r − ν Σ ∗ (cid:0) rf | Ψ | )+ O ( mr − + (cid:15) )( | f | + r | f (cid:48) | )( | e Ψ | + |∇ / Ψ | + r − | Ψ | )where we used e ( r ) = 1 + O ( (cid:15) ) , e ( r ) = − Υ + O ( (cid:15) ) , e ( u ) = 2 + O ( (cid:15) ) , and where ν Σ ∗ denotes the vectorfield ν Σ ∗ = (cid:112) e ( r )2 (cid:112) e ( u ) + e ( r ) e − (cid:112) e ( u ) + e ( r )2 (cid:112) e ( r ) e . Next, note from the formula that ν Σ ∗ is unitary and orthogonal to N Σ ∗ so that ν Σ ∗ is aunit vectorfield, tangent to Σ ∗ and normal to e θ . Furthermore, since ( ν Σ ∗ , e θ , e ϕ ) is anorthonormal frame of Σ ∗ , we havediv Σ ∗ ( ν Σ ∗ ) = g ( D ν Σ ∗ ν Σ ∗ , ν Σ ∗ ) + g ( D e θ ν Σ ∗ , e θ ) + g ( D e ϕ ν Σ ∗ , e ϕ ) . Since ν Σ ∗ is a unit vector, the first term vanishes, and hencediv Σ ∗ ( ν Σ ∗ ) = g ( D e θ ν Σ ∗ , e θ ) + g ( D e ϕ ν Σ ∗ , e ϕ )= (cid:112) e ( r )2 (cid:112) e ( u ) + e ( r ) g ( D θ e , e θ ) − (cid:112) e ( u ) + e ( r )2 (cid:112) e ( r ) g ( D θ e , e θ ) + ν Σ ∗ (Φ)= (cid:112) e ( r )2 (cid:112) e ( u ) + e ( r ) κ − (cid:112) e ( u ) + e ( r )2 (cid:112) e ( r ) κ. R P - WEIGHTED ESTIMATES Σ ∗ (cid:16) r − f | Ψ | ν Σ ∗ (cid:17) = r − ν Σ ∗ (cid:0) rf | Ψ | ) + ν Σ ∗ ( r − ) rf | Ψ | + div Σ ∗ ( ν Σ ∗ ) r − f | Ψ | = r − ν Σ ∗ (cid:0) rf | Ψ | ) + (cid:18) div Σ ∗ ( ν Σ ∗ ) − ν Σ ∗ ( r ) r (cid:19) r − f | Ψ | = r − ν Σ ∗ (cid:0) rf | Ψ | ) + (cid:32) (cid:112) e ( r )2 (cid:112) e ( u ) + e ( r ) (cid:18) κ − e ( r ) r (cid:19) − (cid:112) e ( u ) + e ( r )2 (cid:112) e ( r ) (cid:18) κ − e ( r ) r (cid:19) (cid:33) r − f | Ψ | and hence r − ν Σ ∗ (cid:0) rf | Ψ | ) = div Σ ∗ (cid:16) r − f | Ψ | ν Σ ∗ (cid:17) + O ( (cid:15)r − ) f | Ψ | We finally obtain
P · N Σ ∗ = 12 f Q + 12 f (ˇ e Ψ) + 12 r − ν Σ ∗ (cid:0) rf | Ψ | )+ O ( mr − + (cid:15) )( | f | + r | f (cid:48) | )( | e Ψ | + |∇ / Ψ | + r − | Ψ | )= 12 f Q + 12 f (ˇ e Ψ) + 12 div Σ ∗ (cid:16) r − f | Ψ | ν Σ ∗ (cid:17) + O ( mr − + (cid:15) )( | f | + r | f (cid:48) | )( | e Ψ | + |∇ / Ψ | + r − | Ψ | )which concludes the proof of the lemma. Consider the function f p = f p,R defined by, f p = (cid:40) r p , if r ≥ R, , if r ≤ R , (10.2.15)where R is a fixed, sufficiently large constant which will be chosen in the proof. We alsoconsider X p = f p e , w p = 2 f p r , M p = 2 f (cid:48) p r e . The proof relies on Proposition 10.2.7.24
CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Step 0. (Reduction to the region r ≥ R ) In view of the definition of E [ X p , w p , M p ], see(10.1.12), and in view of the choice of X p and w p , we have D µ P µ [ X p , w p , M p ] = E [ X p , w p , M p ] + f p ( r )ˇ e Ψ · N. We integrate this identity on the domain M ( τ , τ ) to derive, (cid:90) Σ( τ ) P · N Σ + (cid:90) Σ ∗ ( τ ,τ ) P · N Σ ∗ + (cid:90) M ( τ ,τ ) E = (cid:90) Σ( τ ) P · N Σ − (cid:90) M ( τ ,τ ) f p ˇ e Ψ · N. Denoting the boundary terms, K ≥ R ( τ , τ ) : = (cid:90) Σ ≥ R ( τ ) P · e + (cid:90) Σ ∗ ( τ ,τ ) P · N Σ ∗ − (cid:90) Σ ≥ R ( τ ) P · e ,K ≤ R ( τ , τ ) : = (cid:90) Σ ≤ R ( τ ) P · N Σ − (cid:90) Σ ≤ R ( τ ) P · N Σ , we write, K ≥ R ( τ , τ ) + (cid:90) M ≥ R ( τ ,τ ) E = K ≤ R ( τ , τ ) − (cid:90) M ≤ R ( τ ,τ ) E − (cid:90) M ( τ ,τ ) f p ˇ e Ψ · N. (10.2.16)We have the following lemma. Lemma 10.2.9.
For p ≥ δ , we have K ≥ R ( τ , τ ) + (cid:90) M ≥ R ( τ ,τ ) E (cid:46) R p +2 (cid:16) E [Ψ]( τ ) + J p [ N, ψ ]( τ , τ ) + O ( (cid:15) ) ˙ B sδ ; 4 m [ ψ ]( τ , τ ) (cid:17) . Proof of Lemma 10.2.9.
The terms (cid:82) Σ ≤ R ( τ ) P · N Σ and (cid:82) M ≤ R ( τ ,τ ) E on the right can beestimated as follows (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) Σ ≤ R ( τ ) P · N Σ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) R p E [Ψ]( τ ) , (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) Σ ≤ R ( τ ) P · N Σ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) R p E [Ψ]( τ ) , (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) M ≤ R ( τ ,τ ) E (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) R p +2 Mor[Ψ]( τ , τ ) . Hence, K ≤ R ( τ , τ ) − (cid:90) M ≤ R ( τ ,τ ) E (cid:46) R p +2 ( E [Ψ]( τ ) + E [Ψ]( τ ) + Mor[Ψ]( τ , τ )) . R P - WEIGHTED ESTIMATES δ > E [Ψ]( τ ) + Morr[Ψ]( τ , τ ) + F [Ψ]( τ , τ ) (cid:46) E [Ψ]( τ ) + J δ [ N, ψ ]( τ , τ )+ O ( (cid:15) ) ˙ B sδ ; 4 m [ ψ ]( τ , τ )which implies K ≥ R ( τ , τ ) + (cid:90) M ≥ R ( τ ,τ ) E (cid:46) R p +2 (cid:16) E [Ψ]( τ ) + J δ [ N, ψ ]( τ , τ ) + O ( (cid:15) ) ˙ B sδ ; 4 m [ ψ ]( τ , τ ) (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ( τ ,τ ) f p ˇ e Ψ · N (cid:12)(cid:12)(cid:12)(cid:12) . Together with the definition (5.3.7) of J p and the fact that p ≥ δ , we infer K ≥ R ( τ , τ ) + (cid:90) M ≥ R ( τ ,τ ) E (cid:46) R p +2 (cid:16) E [Ψ]( τ ) + J p [ N, ψ ]( τ , τ ) + O ( (cid:15) ) ˙ B sδ ; 4 m [ ψ ]( τ , τ ) (cid:17) which concludes the proof of Lemma 10.2.9.The proof of Theorem 10.2.1 now proceeds according to the following steps. Step 1. (Bulk terms for r ≥ R ) We prove the following lower bound for (cid:82) M ≥ R ( τ ,τ ) E . Lemma 10.2.10.
Given a fixed δ > we have for all δ ≤ p ≤ − δ and R (cid:29) mδ , (cid:15) (cid:28) δ , (cid:90) M ≥ R ( τ ,τ ) E ≥ (cid:90) M ≥ R ( τ ,τ ) r p − (cid:16) p | ˇ e (Ψ) | + (2 − p )( |∇ / Ψ | + r − | Ψ | ) (cid:17) − O ( (cid:15) ) Morr [Ψ]( τ , τ ) . (10.2.17) Proof of Lemma 10.2.10.
We make use of Proposition 10.2.7 according to which, E [ X, w, M ] = 12 f (cid:48) p | ˇ e (Ψ) | + 12 (cid:18) f p r − f (cid:48) p (cid:19) Q + Err (cid:16) (cid:15), mr ; f p (cid:17) [Ψ]= r p − (cid:20) p | ˇ e (Ψ) | + 12 (2 − p )( |∇ / Ψ | + V | Ψ | ) (cid:21) + Err (cid:16) (cid:15), mr ; f p (cid:17) [Ψ] ≥ r p − (cid:20) p | ˇ e (Ψ) | + 12 (2 − p )( |∇ / Ψ | + r − | Ψ | ) (cid:21) + Err (cid:16) (cid:15), mr ; f p (cid:17) [Ψ]26 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS where,Err (cid:16) (cid:15), mr , f p (cid:17) (Ψ) = r p O (cid:16) mr (cid:17) (cid:2) | e Ψ | + r − Ψ | (cid:3) + r p O ( (cid:15) ) w , (cid:0) | e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) + r p O ( (cid:15) ) w , / (cid:16) | e Ψ | ( | e Ψ | + r − |∇ / Ψ | ) + |∇ / Ψ | + r − | Ψ | (cid:17) (cid:46) Err (cid:16) mr (cid:17) + Err( (cid:15) ) , Err (cid:16) mr (cid:17) = O (cid:16) mr (cid:17) r p − (cid:2) | ˇ e Ψ | + r − | Ψ | (cid:3) , Err( (cid:15) ) = O ( (cid:15) ) r p − (cid:0) | ˇ e Ψ | + |∇ / Ψ | + r − | Ψ | + r − | e Ψ | (cid:1) . Thus, (cid:90) M ≥ R ( τ ,τ ) E ≥ (cid:90) M ≥ R ( τ ,τ ) r p − (cid:16) p | ˇ e (Ψ) | + 12 (2 − p )( |∇ / Ψ | + r − | Ψ | ) (cid:17) − O (cid:16) mR (cid:17) (cid:90) M ≥ R ( τ ,τ ) r p − (cid:2) | ˇ e Ψ | + r − | Ψ | (cid:3) − O ( (cid:15) ) (cid:90) M ≥ R ( τ ,τ ) r p − (cid:0) | ˇ e Ψ | + |∇ / Ψ | + r − | Ψ | (cid:1) − O ( (cid:15) ) (cid:90) M ≥ R ( τ ,τ ) r p − | e Ψ | . For δ ≤ p ≤ − δ , R (cid:29) mδ and (cid:15) (cid:28) δ we can absorb all error terms except the last, i.e. (cid:90) M ≥ R ( τ ,τ ) E ≥ (cid:90) M ≥ R ( τ ,τ ) r p − (cid:16) p | ˇ e (Ψ) | + (2 − p )( |∇ / Ψ | + r − | Ψ | ) (cid:17) − O ( (cid:15) ) (cid:90) M ≥ R ( τ ,τ ) r p − | e Ψ | . Note also that for all δ ≤ p ≤ − δ we have, (cid:90) M ≥ R ( τ ,τ ) r p − | e Ψ | (cid:46) Morr( τ , τ ) . Hence, for all δ ≤ p ≤ − δ and R (cid:29) mδ , (cid:15) (cid:28) δ , (cid:90) M ≥ R ( τ ,τ ) E ≥ (cid:90) M ≥ R ( τ ,τ ) r p − (cid:16) p | ˇ e (Ψ) | + (2 − p )( |∇ / Ψ | + r − | Ψ | ) (cid:17) − O ( (cid:15) )Morr[Ψ]( τ , τ )as desired. R P - WEIGHTED ESTIMATES K ≥ R ( τ , τ ) + ˙ B p,R [Ψ]( τ , τ ) (cid:46) R p +2 (cid:16) E [Ψ]( τ ) + J p [ N, ψ ]( τ , τ ) (10.2.18)+ O ( (cid:15) ) ˙ B sδ ; 4 m [ ψ ]( τ , τ ) (cid:17) . Step 2. (Boundary terms for r ≥ R .) Recall that according to Proposition 10.2.7, P · e = f p | ˇ e Ψ | − r − e ( rf p | Ψ | ) + O ( (cid:15)r − ) f p ( | e Ψ | + r − | Ψ | ) , and according to Lemma 10.2.8 P · N Σ ∗ = 12 f Q + 12 f (ˇ e Ψ) + 12 div Σ ∗ (cid:16) r − f | Ψ | ν Σ ∗ (cid:17) + O ( mr − + (cid:15) )( | f | + r | f (cid:48) | )( | e Ψ | + |∇ / Ψ | + r − | Ψ | )Recalling the definition of K ≥ R = (cid:90) Σ ≥ R ( τ ) P · e + (cid:90) Σ ∗ ( τ ,τ ) P · N Σ ∗ − (cid:90) Σ ≥ R ( τ ) P · e we write, K ≥ R = (cid:90) Σ ≥ R ( τ ) f p | ˇ e Ψ | + 12 (cid:90) Σ ∗ ( τ ,τ ) r p (cid:16) Q + (ˇ e Ψ) (cid:17) − (cid:90) Σ ≥ R ( τ ) f p (ˇ e Ψ) − (cid:90) Σ ≥ R ( τ ) r − e ( rf p | Ψ | ) + 12 (cid:90) Σ ∗ ( τ ,τ ) div Σ ∗ (cid:16) r − f | Ψ | ν Σ ∗ (cid:17) + 12 (cid:90) Σ ≥ R ( τ ) r − e ( rf p | Ψ | ) + (cid:90) Σ ∗ ( τ ,τ ) O ( mr − + (cid:15) ) r p − ( | e Ψ | + |∇ / Ψ | + r − | Ψ | )+ O ( (cid:15) ) (cid:32)(cid:90) Σ ≥ R ( τ ) r p − ( | e Ψ | + r − | Ψ | ) − (cid:90) Σ ≥ R ( τ ) r p − ( | e Ψ | + r − | Ψ | ) (cid:33) . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Now, the following integrations by parts hold true (cid:90) Σ ≥ R ( τ ) r − e ( rf p | Ψ | )= (cid:90) r ≥ R (cid:18)(cid:90) S r r − e ( rf p | Ψ | ) (cid:19) e ( r )= (cid:90) r ≥ R e ( r ) e (cid:18)(cid:90) S r r − f p | Ψ | (cid:19) − (cid:90) Σ ≥ R ( τ ) (cid:0) e (cid:0) r − (cid:1) + κr − (cid:1) rf p | Ψ | = (cid:90) S ∗ ( τ ) r p − | Ψ | − (cid:90) S R ( τ ) r p − | Ψ | − (cid:90) Σ ≥ R ( τ ) (cid:18) κ − e ( r ) r (cid:19) r − f p | Ψ | = (cid:90) S ∗ ( τ ) r p − | Ψ | − (cid:90) S R ( τ ) r p − | Ψ | + O ( (cid:15) ) (cid:90) Σ ≥ R ( τ ) r p − | Ψ | and (cid:90) Σ ∗ ( τ ,τ ) div Σ ∗ (cid:16) r − f | Ψ | ν Σ ∗ (cid:17) = (cid:90) S ∗ ( τ ) r p − | Ψ | − (cid:90) S ∗ ( τ ) r p − | Ψ | where S ∗ ( τ ) denotes the 2-sphere Σ ∗ ∩ Σ( τ ). Note that the boundary terms cancel, exceptthe one on r = R , and hence K ≥ R = (cid:90) Σ ≥ R ( τ ) f p | ˇ e Ψ | + 12 (cid:90) Σ ∗ ( τ ,τ ) r p (cid:16) Q + (ˇ e Ψ) (cid:17) − (cid:90) Σ ≥ R ( τ ) f p | ˇ e Ψ | + (cid:90) Σ ∗ ( τ ,τ ) O ( mr − + (cid:15) ) r p − ( | e Ψ | + |∇ / Ψ | + r − | Ψ | )+ O ( (cid:15) ) (cid:32)(cid:90) Σ ≥ R ( τ ) r p − ( | e Ψ | + r − | Ψ | ) − (cid:90) Σ ≥ R ( τ ) r p − ( | e Ψ | + r − | Ψ | ) (cid:33) + 12 (cid:90) S R ( τ ) r p − | Ψ | − (cid:90) S R ( τ ) r p − | Ψ | . Using Q + | ˇ e Ψ | = |∇ / Ψ | + 4Υ r | Ψ | + (cid:12)(cid:12)(cid:12)(cid:12) e Ψ + 1 r Ψ (cid:12)(cid:12)(cid:12)(cid:12) = |∇ / Ψ | + 4Υ r | Ψ | + ( e Ψ) + 1 r | Ψ | + 2 r Ψ · e (Ψ) ≥ |∇ / Ψ | + 4Υ − r | Ψ | + 23 | e Ψ | R P - WEIGHTED ESTIMATES ≥ / r ≥ R and R large enough, we infer K ≥ R ≥ (cid:32)(cid:90) Σ ≥ R ( τ ) r p | ˇ e Ψ | − (cid:90) Σ ≥ R ( τ ) r p | ˇ e Ψ | + ˙ F p [Ψ]( τ , τ ) (cid:33) + O ( (cid:15) ) (cid:32)(cid:90) Σ ≥ R ( τ ) r p − | Ψ | − (cid:90) Σ ≥ R ( τ ) r p − | Ψ | (cid:33) + 12 (cid:90) S R ( τ ) r p − | Ψ | − (cid:90) S R ( τ ) r p − | Ψ | (10.2.19)Next, recall that according to Proposition 10.2.7, we have P · e ≥ r p − ( p − | Ψ | − p r − e ( rf p | Ψ | ) . We infer (cid:90) Σ ≥ R ( τ ) P · e ≥ (cid:90) Σ ≥ R ( τ ) r p − ( p − | Ψ | − p (cid:90) Σ ≥ R ( τ ) r − e ( rf p | Ψ | ) . Integrating by parts similarly as before, we infer (cid:90) Σ ≥ R ( τ ) P · e ≥ (cid:90) Σ ≥ R ( τ ) r p − ( p − | Ψ | − p (cid:90) S ∗ ( τ ) r p − | Ψ | + p (cid:90) S R ( τ ) r p − | Ψ | + O ( (cid:15) ) (cid:90) Σ ≥ R ( τ ) r p − | Ψ | . Arguing as for the proof of (10.2.19) except for the boundary term on Σ ≥ R ( τ ) for whichwe use the above estimate, we deduce K ≥ R ≥ (cid:90) Σ ≥ R ( τ ) r p − ( p − | Ψ | + 12 (cid:32) − (cid:90) Σ ≥ R ( τ ) r p | ˇ e Ψ | + ˙ F p [Ψ]( τ , τ ) (cid:33) + O ( (cid:15) ) (cid:32)(cid:90) Σ ≥ R ( τ ) r p − ( | e Ψ | + r − | Ψ | ) − (cid:90) Σ ≥ R ( τ ) r p − ( | e Ψ | + r − | Ψ | ) (cid:33) + 1 − p (cid:90) S ∗ ( τ ) r p − | Ψ | + p (cid:90) S R ( τ ) r p − | Ψ | − (cid:90) S R ( τ ) r p − | Ψ | . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
We first focus on the case δ ≤ p ≤ − δ , in which case the previous estimate yields K ≥ R ≥ δ (cid:90) Σ ≥ R ( τ ) r p − | Ψ | + 12 (cid:32) − (cid:90) Σ ≥ R ( τ ) r p | ˇ e Ψ | + ˙ F p [Ψ]( τ , τ ) (cid:33) + O ( (cid:15) ) (cid:32)(cid:90) Σ ≥ R ( τ ) r p − ( | e Ψ | + r − | Ψ | ) − (cid:90) Σ ≥ R ( τ ) r p − | Ψ | (cid:33) + 12 (cid:90) S R ( τ ) r p − | Ψ | − (cid:90) S R ( τ ) r p − | Ψ | . Together with (10.2.19) and the fact that (cid:15) (cid:28) δ by assumption, we infer in view of thedefinition of ˙ E p,R [Ψ] for δ ≤ p ≤ − δ ,˙ E p,R [Ψ]( τ ) + ˙ F p [Ψ]( τ , τ ) (cid:46) K ≥ R + ˙ E p,R [Ψ]( τ ) + (cid:90) S R ( τ ) r p − | Ψ | . Together with (10.2.18), we deduce for δ ≤ p ≤ − δ ˙ E p,R [Ψ]( τ ) + ˙ F p [Ψ]( τ , τ ) + ˙ B p,R [Ψ]( τ , τ ) (cid:46) ˙ E p,R [Ψ]( τ ) + (cid:90) S R ( τ ) r p − | Ψ | + R p +2 (cid:16) E [Ψ]( τ ) + J p [ N, ψ ]( τ , τ )+ O ( (cid:15) ) ˙ B sδ ; 4 m [ ψ ]( τ , τ ) (cid:17) . In view of the improved Morawetz Theorem 10.1.1, and thanks also to the term ˙ B p,R [Ψ]( τ , τ )on the left hand side, we may absorb the term O ( (cid:15) ) ˙ B sδ ; 4 m [ ψ ]( τ , τ ) (cid:17) and obtain˙ E p,R [Ψ]( τ ) + ˙ F p [Ψ]( τ , τ ) + ˙ B p,R [Ψ]( τ , τ ) (cid:46) R p +2 (cid:16) E p [Ψ]( τ ) + J p [ N, ψ ]( τ , τ ) (cid:17) (10.2.20)which is the desired estimate in the case δ ≤ p ≤ − δ .Finally, we focus on the remaining case, i.e. 1 − δ ≤ p ≤ − δ . Combining (10.2.19) and(10.2.18), arguing as in the proof of (10.2.20), and in view of the definition of ˙ E p,R [Ψ] for1 − δ ≤ p ≤ − δ , we obtain˙ E p,R [Ψ]( τ ) + ˙ F p [Ψ]( τ , τ ) + ˙ B p,R [Ψ]( τ , τ ) (cid:46) R p +2 ( E p [Ψ]( τ ) + J p [ N, ψ ]( τ , τ ))+ O ( (cid:15) ) (cid:90) Σ ≥ R ( τ ) r p − | Ψ | + (cid:90) Σ ≥ R ( τ ) r − − δ | Ψ | (cid:46) R p +2 ( E p [Ψ]( τ ) + J p [ N, ψ ]( τ , τ )) + E − δ [Ψ]( τ ) p ≤ − δ so that p − ≤ − − δ . Together with the factthat ˙ E p,R [Ψ]( τ ) ≥ ˙ E − δ,R [Ψ]( τ ) for p ≥ − δ and (10.2.20), we infer˙ E p,R [Ψ]( τ ) + ˙ F p [Ψ]( τ , τ ) + ˙ B p,R [Ψ]( τ , τ ) (cid:46) R p +2 ( E p [Ψ]( τ ) + J p [ N, ψ ]( τ , τ ))for all δ ≤ p ≤ − δ as desired. This concludes the proof of Theorem 10.2.1. We use a variation of the method of [4] to derive slightly stronger weighted estimates.This allows us to prove Theorem 5.3.5 for s = 0 in section 10.4.6. The proof for higherorder derivatives s ≤ k small + 29 will be provided in section 10.4.6.As in the previous section we rely on the assumptions (10.2.8)–(10.2.11) to which we add, RP5.
The assumptions
RP0 – RP4 hold true for one extra derivative with respect to d . RP6. e ( m ) satisfies the following improvement of RP4 | d ≤ e ( m ) | (cid:46) (cid:15)w , . (10.3.1) ˇ ψ Proposition 10.3.1.
Assume ψ verifies (cid:3) ψ = − κκψ + N . Then ˇ ψ = f ˇ e ψ verifies:1. In the region r ≥ m , ( (cid:3) + κκ ) ˇ ψ = r (cid:18) e ( N ) + 3 r N (cid:19) + 2 r (cid:18) − mr (cid:19) e ˇ ψ + O ( r − ) d ≤ ψ + r Γ b e d ψ + d ≤ (Γ b ) d ≤ ψ + r d ≤ (Γ g ) e ψ + d ≤ (Γ g ) d ψ.
2. In the region m ≤ r ≤ m , ( (cid:3) + κκ ) ˇ ψ = f (cid:18) e ( N ) + 3 r N (cid:19) + O (1) d ≤ ψ. The proof of Proposition 10.3.1 is postponed to Appendix D.4.32
CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS r p weighted estimates for ˇ ψ The goal of this section is to prove Theorem 5.3.5 in the case s = 0. The proof for higherorder derivatives s ≤ k small + 29 will be provided in section 10.4.6. Proof of Theorem 5.3.5 in the case s = 0 . We write, in accordance to Proposition 10.3.1 (cid:3) ˇ ψ − V ˇ ψ = ˇ N + f (cid:18) e + 3 r (cid:19) N where,ˇ N = r (cid:0) − mr (cid:1) e ˇ ψ + O ( r − ) d ≤ ψ + r Γ b e d ψ + d ≤ (Γ b ) d ≤ ψ + r d ≤ (Γ g ) e ψ + d ≤ (Γ g ) d ψ, r ≥ m ,O (1) d ≤ ψ, m ≤ r ≤ m . (10.3.2)We apply the first part of Proposition 10.2.7 to ψ replaced by ˇ ψ . This yields, using also(10.1.12), Div P q ( ˇ ψ ) = E q ( ˇ ψ ) + f q ˇ e ˇ ψ · ˇ N + f q ˇ e ˇ ψ · f (cid:18) e + 3 r (cid:19) N, where, with f = f q , X q = f q e , w q = f q r , M q = 2 r − f (cid:48) q e , E q ( ˇ ψ ) = E [ X q , w q , M q ] = 12 f (cid:48) q | ˇ e ( ˇ ψ ) | + 12 (cid:18) f q r − f (cid:48) q (cid:19) Q ( ˇ ψ ) + Err q ( ˇ ψ ) , Err q ( ˇ ψ ) : = Err (cid:16) (cid:15), mr ; f q (cid:17) [ ˇ ψ ]= O (cid:16) mr (cid:17) r q | e ˇ ψ | + O (cid:16) mr + (cid:15)w , (cid:17) r q | ˇ ψ | + O ( (cid:15) ) w , r q (cid:0) | e ˇ ψ | + |∇ / ˇ ψ | + r − | ˇ ψ | (cid:1) + O ( (cid:15) ) w , / r q (cid:16) (cid:0) | e ˇ ψ | + r − |∇ / ˇ ψ | (cid:1) | e ˇ ψ | + |∇ / ˇ ψ | + r − | ˇ ψ | (cid:17) , P k ( ˇ ψ ) = P [ X q , w q , M q ]( ˇ ψ ) . We then integrate on the domain M ( τ , τ ) to derive, exactly as in the proof of Theorem10.2.1 (see section 10.2.3), (cid:90) Σ( τ ) P q · e + (cid:90) Σ ∗ ( τ ,τ ) P q · N Σ ∗ + (cid:90) M ( τ ,τ ) (cid:0) E q + f q ˇ e ˇ ψ ˇ N (cid:1) = (cid:90) Σ( τ ) P q · e − (cid:90) M ( τ ,τ ) f q ˇ e ˇ ψ · f (cid:18) e + 3 r (cid:19) N. (10.3.3) E q ; R [ ˇ ψ ]( τ ) + (cid:90) M ( τ ,τ ) ( E q + r q ˇ e ( ˇ ψ ) ˇ N ) + ˙ F q [ ˇ ψ ]( τ , τ ) (cid:46) E q [ ˇ ψ ]( τ ) + J q (cid:20) ˇ ψ, f (cid:18) e + 3 r (cid:19) N (cid:21) ( τ , τ ) . Since all terms for r ≤ R can be controlled by one derivative of ψ , we infer˙ E q [ ˇ ψ ]( τ ) + Morr[ ˇ ψ ]( τ , τ ) + (cid:90) M ≥ R ( τ ,τ ) ( E q + r q ˇ e ( ˇ ψ ) ˇ N ) + ˙ F q [ ˇ ψ ]( τ , τ ) (cid:46) E q [ ˇ ψ ]( τ ) + ˇ J q [ ˇ ψ, N ]( τ , τ ) + R q +3 ( E [ ψ ]( τ ) + Morr [ ψ ]( τ , τ )) . (10.3.4)Also, since δ ≤ max( q, δ ) ≤ − δ , we have in view of Theorem 5.3.4 in the case s = 1 sup τ ∈ [ τ ,τ ] E q,δ ) [ ψ ]( τ ) + B q,δ ) [ ψ ]( τ , τ ) (cid:46) E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ]( τ , τ ) , (10.3.5)In view of (10.3.4) and (10.3.5), it thus only remains to estimate the integral (cid:90) M ≥ R ( τ ,τ ) ( E q + r q ˇ e ( ˇ ψ ) ˇ N ) , i.e. we need to derive the analog of (10.2.17) used in the proof of Theorem 10.2.1.This is achieved in Proposition 10.3.2 below, which together with (10.3.4) and (10.3.5)immediately yields the proof of Theorem 5.3.5 in the case s = 0. Proposition 10.3.2.
The following estimate holds true, (cid:90) M ≥ R ( τ ,τ ) ( E q + r q ˇ e ( ˇ ψ ) ˇ N ) ≥ (cid:90) M ≥ R ( τ ,τ ) r q − (cid:0) (2 + q ) | ˇ e ˇ ψ | + (2 − q ) |∇ / ˇ ψ | + 2 r − | ˇ ψ | (cid:1) − O ( (cid:15) ) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) − O (1) (cid:0) E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ] (cid:1) . (10.3.6)We now focus on the proof of Proposition 10.3.2. In view of the definition of ˇ N , we have The proof of Theorem 5.3.4 for higher derivatives s ≥
1, even though proved later in section 10.4.5,is in fact independent of the proof of Theorem 5.3.5 and can thus be invoked here. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS for r ≥ R , ˇ N = A + A + A ,A = 2 r e ˇ ψ = 2 r (ˇ e ˇ ψ − r − ˇ ψ ) ,A = − mr e ˇ ψ + O ( r − ) d ≤ ψ,A = Err[ (cid:3) g ˇ ψ ] , Err[ (cid:3) g ˇ ψ ] = r Γ g e d ψ + r d ≤ (Γ g ) d ≤ ψ + d ≤ (Γ g ) d ψ. Also, recall that we have for r ≥ R E q ( ˇ ψ ) = E [ X q , w q , M q ] = q r q − | ˇ e ( ˇ ψ ) | + 2 − q r q − Q ( ˇ ψ ) + Err q ( ˇ ψ ) . Consequently, we write,( E q + r q ˇ e ( ˇ ψ ) ˇ N ) = I + I + I ,I : = 12 r q − (cid:0) q | ˇ e ˇ ψ | + (2 − q ) |∇ / ˇ ψ | + 4(2 − q ) r − ˇ ψ (cid:1) + 2 r q − ˇ e ˇ ψ (ˇ e ˇ ψ − r − ˇ ψ )= 12 r q − (cid:0) ( q + 4) | ˇ e ˇ ψ | + (2 − q ) |∇ / ˇ ψ | + 4(2 − q ) r − ˇ ψ − r − ˇ e ˇ ψ ˇ ψ (cid:1) ,I : = r q − ˇ e ( ˇ ψ ) (cid:2) − me ˇ ψ + O (1) d ≤ ψ (cid:3) + O (cid:16) mr (cid:17) r q − ( ˇ ψ ) ,I : = Err q ( ˇ ψ ) + r q ˇ e ( ˇ ψ ) A . (10.3.7)We will rely on the following two lemmas. Lemma 10.3.3.
The following lower bound estimate holds true for q ≤ − δ and r ≥ R ,where R is sufficiently large, I + I ≥ r q − (cid:0) (2 + q ) | ˇ e ˇ ψ | + (2 − q ) |∇ / ˇ ψ | + 2 r − | ˇ ψ | (cid:1) − O (1) r q − ( d ≤ ψ ) . (10.3.8) Lemma 10.3.4.
The following estimate holds true for the error term I (cid:90) M ≥ R ( τ ,τ ) | I | (cid:46) (cid:15) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) + (cid:16) m R + (cid:15) (cid:17) ˙ B q,R [ ˇ ψ ]( τ , τ )+ (cid:15) (cid:18) sup τ ≤ τ ≤ τ E q [ ψ ]( τ ) + B q [ ψ ]( τ , τ ) + J q [ ψ, N ]( τ , τ ) (cid:19) . We postpone the proof of Lemma 10.3.3 and Lemma 10.3.4 to finish the proof of Propo-sition 10.3.2.
Proof of Proposition 10.3.2.
In view of Lemma 10.3.3 and Lemma 10.3.4, we have (cid:90) M ≥ R ( τ ,τ ) ( E q + r q ˇ e ( ˇ ψ ) ˇ N ) = (cid:90) M ≥ R ( τ ,τ ) ( I + I ) + (cid:90) M ≥ R ( τ ,τ ) I ≥ (cid:90) M ≥ R ( τ ,τ ) r q − (cid:0) (2 + q ) | ˇ e ˇ ψ | + (2 − q ) |∇ / ˇ ψ | + 2 r − | ˇ ψ | (cid:1) − O (1) (cid:90) M ≥ R ( τ ,τ ) r q − ( ψ ) − O ( (cid:15) ) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) + O (cid:16) m R + (cid:15) (cid:17) ˙ B q,R [ ˇ ψ ]( τ , τ ) − O ( (cid:15) ) (cid:18) sup τ ≤ τ ≤ τ E q [ ψ ]( τ ) + B q [ ψ ]( τ , τ ) + J q [ ψ, N ]( τ , τ ) (cid:19) so that, since 1 − δ < q ≤ − δ , and for R sufficiently large and small (cid:15) , (cid:90) M ≥ R ( τ ,τ ) ( E q + r q ˇ e ( ˇ ψ ) ˇ N ) ≥ (cid:90) M ≥ R ( τ ,τ ) r q − (cid:0) (2 + q ) | ˇ e ˇ ψ | + (2 − q ) |∇ / ˇ ψ | + 2 r − | ˇ ψ | (cid:1) − O ( (cid:15) ) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) − O ( (cid:15) ) (cid:18) sup τ ≤ τ ≤ τ E q [ ψ ]( τ ) + B q [ ψ ]( τ , τ ) + J q [ ψ, N ]( τ , τ ) (cid:19) . In view of (10.3.5), we infer (cid:90) M ≥ R ( τ ,τ ) ( E q + r q ˇ e ( ˇ ψ ) ˇ N ) ≥ (cid:90) M ≥ R ( τ ,τ ) r q − (cid:0) (2 + q ) | ˇ e ˇ ψ | + (2 − q ) |∇ / ˇ ψ | + 2 r − | ˇ ψ | (cid:1) − O ( (cid:15) ) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) − O (1) (cid:0) E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ] (cid:1) which concludes the proof.It finally remains to prove Lemma 10.3.3 and Lemma 10.3.4. Proof of Lemma 10.3.3.
Note that,( q + 4) | ˇ e ˇ ψ | − r − (ˇ e ˇ ψ ) ˇ ψ + 4(2 − q ) r − | ˇ ψ | = ( q + 2) | ˇ e ˇ ψ | + (6 − q ) r − | ˇ ψ | + 2 (cid:0) ˇ e ˇ ψ − r − ˇ ψ (cid:1) ≥ ( q + 2) | ˇ e ˇ ψ | + (6 − q ) r − | ˇ ψ | ≥ ( q + 2) | ˇ e ˇ ψ | + 2 r − | ˇ ψ | , CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS where we used the fact that q ≤ − δ . Hence, I ≥ r q − (cid:0) (2 + q ) | ˇ e ˇ ψ | + (2 − q ) |∇ / ˇ ψ | + 2 r − | ˇ ψ | (cid:1) . We also have, I ≤ O (cid:16) mr (cid:17) r q − (cid:0) | ˇ e ˇ ψ | + r − | ˇ ψ | (cid:1) + O (1) (cid:0) r q − (ˇ e ˇ ψ ) (cid:1) (cid:0) r q − ( d ≤ ψ ) (cid:1) . Thus if m /R is sufficiently small, and since q ≤ − δ , we deduce, for r ≥ R , I + I ≥ r q − (cid:0) (2 + q ) | ˇ e ˇ ψ | + (2 − q ) |∇ / ˇ ψ | + 2 r − | ˇ ψ | (cid:1) − O (1) r q − ( d ≤ ψ ) as desired. Proof of Lemma 10.3.4.
Recall that I = Err q ( ˇ ψ ) + r q ˇ e ( ˇ ψ ) A ,A = Err[ (cid:3) g ˇ ψ ] , Err[ (cid:3) g ˇ ψ ] = r Γ b e d ψ + d ≤ (Γ b ) d ≤ ψ + r d ≤ (Γ g ) e ψ + d ≤ (Γ g ) d ψ, Err q ( ˇ ψ ) = O (cid:16) mr (cid:17) r q | e ˇ ψ | + O (cid:16) mr + (cid:15)w , (cid:17) r q | ˇ ψ | + O ( (cid:15) ) w , r q (cid:0) | e ˇ ψ | + |∇ / ˇ ψ | + r − | ˇ ψ | (cid:1) + O ( (cid:15) ) w , / r q (cid:16) (cid:0) | e ˇ ψ | + r − |∇ / ˇ ψ | (cid:1) | e ˇ ψ | + |∇ / ˇ ψ | + r − | ˇ ψ | (cid:17) . Hence, | I | (cid:46) (cid:15)r q | ˇ e ( ˇ ψ ) | (cid:16) τ − − δ dec (cid:0) | ˇ e d ≤ ψ | + | d ≤ ψ | (cid:1) + r − τ − − δ dec (cid:0) | e ψ | + r − | d ψ | (cid:1)(cid:17) + (cid:16) mr + (cid:15) (cid:17) r q − (cid:0) | ˇ e ˇ ψ | + |∇ / ˇ ψ | + r − | ˇ ψ | (cid:1) + (cid:15)r q − τ − − δ dec | e ˇ ψ || e ˇ ψ | + (cid:15)r q − |∇ / ˇ ψ || e ˇ ψ | . This yields, using q ≤ − δ , (cid:90) M ≥ R ( τ ,τ ) | I | (cid:46) (cid:15) (cid:18) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) (cid:19) (cid:18) sup τ ≤ τ ≤ τ E q [ ψ ]( τ ) + B q [ ψ ]( τ , τ ) (cid:19) + (cid:16) m R + (cid:15) (cid:17) ˙ B q,R [ ˇ ψ ]( τ , τ )+ (cid:15) (cid:18) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) + ˙ B q [ ˇ ψ ]( τ , τ ) (cid:19) (cid:32)(cid:90) M ≥ R ( τ ,τ ) r q − | e ˇ ψ | (cid:33) (cid:46) (cid:15) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) + (cid:16) m R + (cid:15) (cid:17) ˙ B q,R [ ˇ ψ ]( τ , τ )+ (cid:15) (cid:18) sup τ ≤ τ ≤ τ E q [ ψ ]( τ ) + B q [ ψ ]( τ , τ ) (cid:19) + (cid:15) (cid:90) M ≥ R ( τ ,τ ) r q − | e ˇ ψ | . e ˇ ψ . For this we need to appeal to the formula inLemma 5.3.7 which we recall below, (cid:3) ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ We have for r ≥ m e ˇ ψ = e ( re ( rψ )) = re ( re ψ ) + e ( r ) e ( rψ ) = r e e ψ + 2 re ( r ) e ψ + e ( r ) e ( r ) ψ = r (cid:18) − (cid:3) ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ (cid:19) + 2 re ( r ) e ψ + e ( r ) e ( r ) ψ so that | e ˇ ψ | (cid:46) r | N | + r | e ψ | + | d ≤ ψ | and hence (cid:90) M ≥ R ( τ ,τ ) r q − | e ˇ ψ | (cid:46) (cid:90) M ≥ R ( τ ,τ ) r q − (cid:16) r | N | + r | e ψ | + | d ≤ ψ | (cid:17) . Since q ≤ − δ , we infer (cid:90) M ≥ R ( τ ,τ ) r q − | e ˇ ψ | (cid:46) (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r − δ | N | + B q [ ψ ]( τ , τ ) (cid:46) J q [ ψ, N ]( τ , τ ) + B q [ ψ ]( τ , τ )and thus (cid:90) M ≥ R ( τ ,τ ) | I | (cid:46) (cid:15) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) + (cid:16) m R + (cid:15) (cid:17) ˙ B q,R [ ˇ ψ ]( τ , τ )+ (cid:15) (cid:18) sup τ ≤ τ ≤ τ E q [ ψ ]( τ ) + B q [ ψ ]( τ , τ ) (cid:19) + (cid:15) (cid:90) M ≥ R ( τ ,τ ) r q − | e ˇ ψ | (cid:46) (cid:15) sup τ ≤ τ ≤ τ ˙ E q,R [ ˇ ψ ]( τ ) + (cid:16) m R + (cid:15) (cid:17) ˙ B q,R [ ˇ ψ ]( τ , τ )+ (cid:15) (cid:18) sup τ ≤ τ ≤ τ E q [ ψ ]( τ ) + B q [ ψ ]( τ , τ ) + J q [ ψ, N ]( τ , τ ) (cid:19) . which concludes the proof of Lemma 10.3.4.38 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
We have proved, respectively in section 10.2 and section 10.3.2, Theorem 5.3.4 on basicweighted estimates (see Remark 10.2.2) and Theorem 5.3.5 on higher weighted estimatesonly in the case s = 0. In this section, we conclude the proof of these theorems byrecovering higher order derivatives one by one. Recall that any Ricci coefficient either belongs to Γ g or Γ b , where Γ g and Γ b are definedin section 5.1.2. We make use of the following non sharp consequence of the estimates ofLemma 5.1.1. We assume, concerning the Ricci coefficients | d k (Γ g ) | (cid:46) (cid:15)r u δ dec − δ trap for k ≤ k small + 30 , | d k (Γ b ) | (cid:46) (cid:15)ru δ dec − δ trap for k ≤ k small + 30 , | d k ( α, β, ˇ ρ ) | (cid:46) (cid:15)r u δ dec − δ trap for k ≤ k small + 30 , | d k α | + r | d k β | (cid:46) (cid:15)ru δ dec − δ trap for k ≤ k small + 30 , where we recall that δ dec and δ are such that we have in particular 0 < δ < δ dec . So far, we have proved Theorem 5.3.4 in the case s = 0 in section 10.2, and Theorem5.3.5 on higher weighted estimates in the case s = 0 in section 10.3. We now conclude theproof of these theorems by recovering higher order derivatives one by one. Since goingfrom s = 0 to s = 1 is analogous to going from s to s + 1, we will in fact consider only theformer. More precisely, we assume the following bounds proved respectively in sectionsection 10.2 and section 10.3,sup τ ∈ [ τ ,τ ] E p [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) + F p [ ψ ]( τ , τ ) (cid:46) E p [ ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) , (10.4.1) Respectively s ≤ k small + 30 in the case of Theorem 5.3.4, and s ≤ k small + 29 in the case of Theorem5.3.5. Recall that Theorem 5.3.4 in the case s = 0 is obtained as a consequence of Theorem 10.1.1 onMorawetz and energy estimates, and Theorem 10.2.1 on r p -weighted estimates, see Remark 10.2.2. τ ∈ [ τ ,τ ] E q [ ˇ ψ ]( τ ) + B q [ ˇ ψ ]( τ , τ ) (cid:46) E q [ ˇ ψ ]( τ ) + ˇ J q [ ˇ ψ, N ]( τ , τ )+ E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ] , (10.4.2)and our goal is to prove the corresponding estimates for s = 1. We will proceed as follows1. we first commute the wave equation for ψ and ˇ ψ with T and derive (10.4.1) for T ψ instead of ψ , and (10.4.2) for T ˇ ψ instead of ˇ ψ ,2. we then commute the wave equation for ψ and ˇ ψ with r d/ and derive (10.4.1) for r d/ ψ instead of ψ , and (10.4.2) for r d/ ˇ ψ instead of ˇ ψ ,3. we then use the wave equation satisfied by ψ to derive an estimate for R ψ in r ≤ m with a degeneracy at r = 3 m ,4. we then commute the wave equation for ψ with R and remove the degeneracy at r = 3 m for R ψ ,5. we then commute the wave equation for ψ with the redshift vectorfield Y H and derive(10.4.1) for Y H ψ instead of ψ ,6. we then commute the wave equation for ψ and ˇ ψ with f e and derive (10.4.1) for re ψ instead of ψ , and (10.4.2) for f e ˇ ψ instead of ˇ ψ , where f = r for r ≥ m and f = 0 for r ≤ m ,7. we finally gather all estimates and conclude.We will follow the above strategy in section 10.4.5 to prove Theorem 5.3.4, and in section10.4.6 to prove Theorem 5.3.5. To this end, we first derive several commutator identitiesand estimates. Commutation with T Lemma 10.4.1.
We have, schematically, the following commutator formulae [ T, e ] = Γ g d , [ T, e ] = Γ b d , [ T, d/ k ] = Γ b d + Γ b , [ T, d (cid:63) / k ] = Γ b d + Γ b . Note that any finite region in r strictly containing the trapping region would suffice. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Proof.
Recall that we have[ e , e ] = 2 ωe − ωe + 2( η − η ) e θ . We infer 2[
T, e ] = [ e + Υ e , e ]= Υ[ e , e ] − e (Υ) e = Υ (cid:0) ωe − ωe + 2( η − η ) e θ (cid:1) − e (Υ) e = Υ (cid:18) ωe − (cid:18) ω + 12 Υ − e (Υ) (cid:19) e + 2( η − η ) e θ (cid:19) = ( r − Γ b + Γ g ) d = Γ g d , and 2[ T, e ] = [ e + Υ e , e ]= [ e , e ] − e (Υ) e = − ωe + 2 ωe − η − η ) e θ − e (Υ) e = − ωe + 2 (cid:18) ω − e (Υ) (cid:19) e − η − η ) e θ = − ωe + 2 (cid:18) ω + 12 Υ − e (Υ) − T (Υ) (cid:19) e − η − η ) e θ = (Γ g + Γ b ) d = Γ b d . Next, recall in view of Lemma 2.1.51, the following commutation formulae for reducedscalars.1. If f ∈ s k ,[ d/ k , e ] f = 12 κ d/ k f + Com k ( f ) ,Com k ( f ) = − ϑ d (cid:63) / k +1 f + ( ζ − η ) e f − kηe Φ f − ξ ( e f + ke (Φ) f ) − kβf, [ d/ k , e ] = 12 κ d/ k f + Com k ( f ) , Com k ( f ) = − ϑ d (cid:63) / k +1 f − ( ζ + η ) e f − kηe Φ f − ξ ( e f + ke (Φ) f ) − kβf. f ∈ s k − [ d (cid:63) / k , e ] f = 12 κ d (cid:63) / k f + Com ∗ k ( f ) ,Com ∗ k ( f ) = − ϑ d/ k − f − ( ζ − η ) e f − ( k − ηe Φ f + ξ ( e f − ( k − e (Φ) f ) − ( k − βf, [ d (cid:63) / k , e ] f = 12 κ d/ k f + Com ∗ k ( f ) , Com ∗ k ( f ) = − ϑ d/ k − f + ( ζ + η ) e f − ( k − ηe Φ f + ξ ( e f − ( k − e (Φ) f ) − ( k − βf. We infer, schematically,2[
T, d/ k ] = [ e + Υ e , d/ k ]= [ e , d/ k ] + Υ[ e , d/ k ] − e θ (Υ) e = −
12 ( κ + Υ κ ) d/ k + Γ b d + r − Γ b + 2 e θ (cid:16) mr (cid:17) e = Γ b d + Γ b . The estimate for [
T, d (cid:63) / k ] is similar and left to the reader. This concludes the proof of thelemma. Lemma 10.4.2.
We have T ( κ ) = d ≤ Γ g , T (cid:18) ω − κ (cid:19) = d ≤ Γ b , T ( K ) = d ≤ Γ g . Proof.
We have2 T ( κ ) = ( e + Υ e ) κ = − κ − ωκ + 2 d/ ξ + 2( η + η + 2 ζ ) ξ − ϑ +Υ (cid:18) − κκ + 2 ωκ + 2 d/ η + 2 ρ − ϑϑ + 2( ξξ + η ) (cid:19) = − κ − ωκ − κκ Υ + 2 ωκ Υ + 2Υ ρ +2 d/ ξ + 2Υ d/ η + 2( η + η + 2 ζ ) ξ − ϑ + Υ (cid:18) − ϑϑ + 2( ξξ + η ) (cid:19) = r − d Γ b + r − Γ b = d ≤ Γ g . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Also, we have T ( r ) = e ( r ) + Υ e ( r ) = e ( r ) − Υ + Υ( e ( r ) − ∈ r Γ b . We infer T (cid:18) ω − κ (cid:19) = T (cid:18) r (cid:19) + Γ b − T (cid:18) κ + 2 r (cid:19) = − T ( r ) r + d Γ b = d ≤ Γ b and T ( K ) = T (cid:18) r (cid:19) + T (cid:18) K − r (cid:19) = − T ( r ) r + r − Γ b = r − d (Γ b ) + r − Γ b = d ≤ Γ g . This concludes the proof of the lemma.
Corollary 10.4.3.
We have [ T, (cid:3) ] ψ = d ≤ (Γ g ) d ≤ ψ. Proof.
Recall that we have (cid:3) ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ. We infer[ T, (cid:3) ] ψ = − [ T, e ] e ψ − e [ T, e ] ψ + [ T, (cid:52) / ] ψ + (cid:18) ω − κ (cid:19) [ T, e ] ψ + T (cid:18) ω − κ (cid:19) e ψ − κ [ T, e ] ψ − T ( κ ) e ψ + 2 η [ T, e θ ] ψ + 2 T ( η ) e θ ψ. and hence, using also (cid:52) / = − d (cid:63) / d/ + 2 K ,[ T, (cid:3) ] ψ = − [ T, e ]( r − d ψ ) − d [ T, e ] ψ − r − d [ T, d/ ] ψ − [ T, d (cid:63) / ] r − d ψ + 2 T ( K ) ψ + (cid:18) ω − κ (cid:19) [ T, e ] ψ + T (cid:18) ω − κ (cid:19) r − d ψ − κ [ T, e ] ψ − T ( κ ) d ψ +2 η [ T, e θ ] ψ + 2 T ( η ) r − d ψ. T, e ] = Γ g d , [ T, e ] = Γ b d , [ T, d/ k ] = Γ b d + Γ b , [ T, d (cid:63) / k ] = Γ b d + Γ b and T ( κ ) = d ≤ Γ g , T (cid:18) ω − κ (cid:19) = d ≤ Γ b , T ( K ) = d ≤ Γ g , we deduce, schematically,[ T, (cid:3) ] ψ = d ≤ (Γ g ) d ≤ ψ + r − d ≤ (Γ b ) d ≤ ψ = d ≤ (Γ g ) d ≤ ψ. This concludes the proof of the corollary.
Commutation with angular derivativesLemma 10.4.4.
We have, schematically, [ r d/ k , e ] f, [ r d (cid:63) / k , e ] f = Γ g d ≤ f, [ r (cid:52) / k , e ] f = d ≤ (Γ g ) d ≤ f [ r d/ k , e ] f = − rηe ( f ) + Γ b d ≤ f, [ r d (cid:63) / k , e ] f = rηe ( f ) + Γ b d ≤ f. Proof.
Recall from Lemma 2.2.13 that the following commutation formulae holds true,1. If f ∈ s k , [ r d/ k , e ] = r (cid:20) Com k ( f ) − A d/ k f (cid:21) , [ r d/ k , e ] f = r (cid:20) Com k ( f ) − A d/ k f (cid:21) .
2. If f ∈ s k − [ r d (cid:63) / k , e ] f = r (cid:20) Com ∗ k ( f ) − A d (cid:63) / k f (cid:21) , [ r d (cid:63) / k , e ] f = r (cid:20) Com ∗ k ( f ) − A d (cid:63) / k f (cid:21) , CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS where A = 2 /re ( r ) − κ and A = 2 /re ( r ) − κ . Now, we haveCom k ( f ) = r − Γ g d ≤ f, Com ∗ k ( f ) = r − Γ g d ≤ f,Com k ( f ) = − ηe ( f ) + r − Γ b d ≤ f, Com ∗ k ( f ) = ηe ( f ) + r − Γ b d ≤ f, which together with the fact that A ∈ Γ g and A ∈ Γ b implies, schematically,[ r d/ k , e ] f, [ r d (cid:63) / k , e ] f = Γ g d ≤ f, [ r d/ k , e ] f = − rηe ( f ) + Γ b d ≤ f, [ r d (cid:63) / k , e ] f = rηe ( f ) + Γ b d ≤ f. Since (cid:52) / k = − d (cid:63) / k d/ k + kK , We infer[ r (cid:52) / k , e ] = [ − r d (cid:63) / k d/ k + kr K, e ]= − [ r d (cid:63) / k , e ] r d/ − r d (cid:63) / k [ r d/ k , e ]+This concludes the proof of the lemma. Corollary 10.4.5.
We have r d/ ( (cid:3) ψ ) − ( (cid:3) − K )( r d/ ψ ) = − rη (cid:3) ψ + d ≤ (Γ g ) d ≤ ψ and r d (cid:63) / ( (cid:3) φ ) − ( (cid:3) − K )( r d (cid:63) / φ ) = rη (cid:3) φ + d ≤ (Γ g ) d ≤ φ. Proof.
Recall that we have (cid:3) ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ and (cid:3) φ = − e e φ + (cid:52) / φ + (cid:18) ω − κ (cid:19) e φ − κe φ + 2 ηe θ φ We infer r d/ ( (cid:3) ψ ) − (cid:3) ( r d/ ψ ) = − [ r d/ , e ] e ψ − e [ r d/ , e ] ψ + r ( d/ (cid:52) / − (cid:52) / d/ ) ψ + [ r, (cid:52) / ] d/ ψ + re θ (cid:18) ω − κ (cid:19) e ψ + (cid:18) ω − κ (cid:19) [ r d/ , e ] ψ − re θ ( κ ) e ψ − κ [ r d/ , e ] ψ + 2 r d/ ( ηe θ ψ ) − ηe θ ( r d/ ψ ) , r d (cid:63) / ( (cid:3) φ ) − (cid:3) ( r d (cid:63) / φ ) = − [ r d (cid:63) / , e ] e φ − e [ r d (cid:63) / , e ] φ + r ( d (cid:63) / (cid:52) / − (cid:52) / d (cid:63) / ) φ + [ r, (cid:52) / ] d (cid:63) / φ − re θ (cid:18) ω − κ (cid:19) e φ + (cid:18) ω − κ (cid:19) [ r d (cid:63) / , e ] φ + 12 re θ ( κ ) e φ − κ [ r d (cid:63) / , e ] φ + 2 r d/ ( ηe θ φ ) − ηe θ ( r d/ φ ) , and hence, using also in particular the following identities from Proposition 2.1.25 d/ (cid:52) / − (cid:52) / d/ = − K d/ + 2 e θ ( K ) ,d (cid:63) / (cid:52) / − (cid:52) / d (cid:63) / = − K d (cid:63) / − e θ ( K ) , we infer, r d/ ( (cid:3) ψ ) − ( (cid:3) − K )( r d/ ψ ) = − [ r d/ , e ] e ψ − d [ r d/ , e ] ψ + 2 re θ ( K ) ψ + [ r, (cid:52) / ]( r − d ψ )+ re θ (cid:18) ω − κ (cid:19) r − d ψ + (cid:18) ω − κ (cid:19) [ r d/ , e ] ψ − re θ ( κ ) d ψ − κ [ r d/ , e ] ψ + 2 d ( r − η d ψ ) − r − η d ( d ψ ) , and r d (cid:63) / ( (cid:3) φ ) − ( (cid:3) − K )( r d (cid:63) / φ ) = − [ r d (cid:63) / , e ] e φ − d [ r d (cid:63) / , e ] φ − re θ ( K ) φ + [ r, (cid:52) / ]( r − d φ ) − re θ (cid:18) ω − κ (cid:19) r − d φ + (cid:18) ω − κ (cid:19) [ r d (cid:63) / , e ] φ + 12 re θ ( κ ) d φ − κ [ r d (cid:63) / , e ] φ + 2 d ( r − η d φ ) − r − η d ( d φ ) . This yields, schematically, r d/ ( (cid:3) ψ ) − ( (cid:3) − K )( r d/ ψ )= − [ r d/ , e ] e ψ − d [ r d/ , e ] ψ + r − [ r d/ , e ] ψ − κ [ r d/ , e ] ψ + d ≤ (Γ g ) d ≤ ψ and r d (cid:63) / ( (cid:3) φ ) − ( (cid:3) − K )( r d (cid:63) / φ )= − [ r d (cid:63) / , e ] e φ − d [ r d (cid:63) / , e ] φ + r − [ r d (cid:63) / , e ] φ − κ [ r d (cid:63) / , e ] φ + d ≤ (Γ g ) d ≤ φ where we used the fact that r − d ≤ Γ b is at least as good as d ≤ Γ g and the fact that r − e θ ( r ) is Γ g .46 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Next, we rely on[ r d/ k , e ] f, [ r d (cid:63) / k , e ] f = Γ g d ≤ f, [ r d/ k , e ] f = − rηe ( f ) + Γ b d ≤ f, [ r d (cid:63) / k , e ] f = rηe ( f ) + Γ b d ≤ f to infer r d/ ( (cid:3) ψ ) − ( (cid:3) − K )( r d/ ψ ) = rηe e ψ + 12 rκηe ψ + d ≤ (Γ g ) d ≤ ψ and r d (cid:63) / ( (cid:3) φ ) − ( (cid:3) − K )( r d (cid:63) / φ ) = − rηe e φ − rκηe φ + d ≤ (Γ g ) d ≤ φ. This yields r d/ ( (cid:3) ψ ) − ( (cid:3) − K )( r d/ ψ ) = rη (cid:18) − (cid:3) ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ + 2 ηe θ ψ (cid:19) + d ≤ (Γ g ) d ≤ ψ = − rη (cid:3) ψ + d ≤ (Γ g ) d ≤ ψ and r d (cid:63) / ( (cid:3) φ ) − ( (cid:3) − K )( r d (cid:63) / φ ) = − rη (cid:18) − (cid:3) φ + (cid:52) / φ + (cid:18) ω − κ (cid:19) e φ + 2 ηe θ φ (cid:19) + d ≤ (Γ g ) d ≤ φ = rη (cid:3) φ + d ≤ (Γ g ) d ≤ φ. where we used the fact that r − d ≤ Γ b is at least as good as d ≤ Γ g . This concludes theproof of the corollary. Commutation with R in the region r ≤ r We derive in the following lemma commutator identities that are non sharp as far asdecay in r is concerned. This is sufficient for our needs since we will commute the waveequation with R only in the region r ≤ r for a fixed r ≥ m large enough. We will usein particular the following estimate Also, recall thatmax k ≤ k small +30 | d (Γ g ) | (cid:46) (cid:15)r u δ dec − δ trap . (10.4.3) Lemma 10.4.6.
We have [ R, e ] = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d , [ R, e ] = − mr e + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d , [ r d/ k , R ] f, [ r d (cid:63) / k , R ] f = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ≤ f, [ r (cid:52) / k , R ] f = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ≤ f. Proof.
Recall that R is defined by R = 12 ( e − Υ e )and that we have [ e , e ] = 2 ωe − ωe + 2( η − η ) e θ . We infer[
R, e ] = 12 [ − Υ e , e ] = − Υ2 [ e , e ] + 12 e (Υ) e = (cid:16) Υ ωe + mr e ( r ) (cid:17) e + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d , and[ R, e ] = 12 [ e − Υ e , e ] = 12 [ e , e ] + 12 e (Υ) e = (cid:16) ωe + mr e ( r ) (cid:17) e + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d , and hence,[ R, e ] = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d , [ R, e ] = − mr e + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d . Also, recall that we have[ r d/ k , e ] f, [ r d (cid:63) / k , e ] f = Γ g d ≤ f, [ r (cid:52) / k , e ] f = d ≤ (Γ g ) d ≤ f [ r d/ k , e ] f = − rηe ( f ) + Γ b d ≤ f, [ r d (cid:63) / k , e ] f = rηe ( f ) + Γ b d ≤ f. We infer[ r d/ k , e ] f, [ r d (cid:63) / k , e ] f = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ≤ f, [ r (cid:52) / k , e ] f = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ≤ f [ r d/ k , e ] f = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ≤ f, [ r d (cid:63) / k , e ] f = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ≤ f. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Together with the definition for R , we deduce[ r d/ k , R ] f, [ r d (cid:63) / k , R ] f = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ≤ f, [ r (cid:52) / k , R ] f = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ≤ f. This concludes the proof of the lemma.
Corollary 10.4.7.
We have in the region r ≤ r (cid:3) ( Rψ ) = (cid:18) − mr (cid:19) d ψ + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ψ + O (1) d ≤ ψ + O (1) d ≤ N. Proof.
Recall that we have (cid:3) ψ = − e ( e ( ψ )) + (cid:52) / ψ − κe ψ + (cid:18) − κ + 2 ω (cid:19) e ψ + 2 ηe θ ψ. Multiplying by r , we infer r (cid:3) ψ = − r e ( e ( ψ )) + r (cid:52) / ψ − r κe ψ + r (cid:18) − κ + 2 ω (cid:19) e ψ + 2 rηre θ ψ and hence R ( r (cid:3) ψ ) = r (cid:3) ( Rψ ) − [ R, r e e ] ψ + [ R, r (cid:52) / ] ψ − R ( r κ ) e ψ − r κ [ R, e ] ψ + R (cid:18) r (cid:18) − κ + 2 ω (cid:19)(cid:19) e ψ + r (cid:18) − κ + 2 ω (cid:19) [ R, e ] ψ + 2 R ( rη ) re θ ψ +2 rη [ R, re θ ] ψ. Using the commutation identities of the previous lemma, we infer in the region r ≤ r R ( r (cid:3) ψ ) = r (cid:3) ( Rψ ) − [ R, r e e ] ψ + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ψ + O (1) d ψ. Also, since ψ satisfies (cid:3) ψ = V ψ + N , we infer in the region r ≤ r r (cid:3) ( Rψ ) = [ R, r e e ] ψ + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ψ + O (1) d ≤ ψ + O (1) d ≤ N. Next, recall that we have[
R, e ] = O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d , [ R, e ] = − mr e + O ( (cid:15) ) d . R, r e e ] ψ = R ( r ) e e + r [ R, e ] e + r e [ R, e ]= 12 (cid:16) e ( r ) − Υ e ( r ) (cid:17) e e ψ + r e (cid:18) − mr e ψ (cid:19) + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ψ = 2 (cid:16) r − m (cid:17) e e ψ + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ψ + O (1) d ψ and thus, in the region r ≤ r , (cid:3) ( Rψ ) = (cid:18) − mr (cid:19) d ψ + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ψ + O (1) d ≤ ψ + O (1) d ≤ N as desired. Commutation with the redshift vectorfield
Let a positive bump function κ = κ ( r ), supported in the region in [ − ,
2] and equal to 1for [ − , Y H = κ H Y (0) , κ H := κ (cid:18) Υ δ H (cid:19) where Y (0) is defined by Y (0) = ae + be + 2 T, a = 1 + 54 m ( r − m ) , b = 54 m ( r − m ) . Lemma 10.4.8.
We have [ (cid:3) , e ] ψ = − ωe ( e ψ ) + κe ( e ψ ) + κ (cid:3) ψ + d ≤ (Γ g ) d ψ + r − d ≤ ψ. Proof.
Recall that we have (cid:3) ψ = − e ( e ( ψ )) + (cid:52) / ψ − κe ψ + (cid:18) − κ + 2 ω (cid:19) e ψ + 2 ηe θ ψ. Since we have [ e , e ] = 2 ωe + r − Γ b d /, [ d/ k , e ] = 12 κ d/ k + Γ b d + r − Γ b , [ d (cid:63) / k , e ] = 12 κ d (cid:63) / k + Γ b d + r − Γ b CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
We infer[ (cid:3) , e ] ψ = − [ e , e ]( e ( ψ )) + [ (cid:52) / , e ] − κ [ e , e ]( ψ )+ 12 e ( κ ) e ( ψ ) − e (cid:18) − κ + 2 ω (cid:19) e ( ψ ) + 2 η [ e θ , e ] ψ − e ( η ) e θ ( ψ )= − ωe ( e ψ ) + κ (cid:52) / ψ + d ≤ (Γ g ) d ψ + r − d ≤ ψ. Using again (cid:3) ψ = − e ( e ( ψ )) + (cid:52) / ψ − κe ψ + (cid:18) − κ + 2 ω (cid:19) e ψ + 2 ηe θ ψ, we deduce[ (cid:3) , e ] ψ = − ωe ( e ψ ) + κ (cid:16) (cid:3) ψ + e ( e ψ ) (cid:17) + d ≤ (Γ g ) d ψ + r − d ≤ ψ = − ωe ( e ψ ) + κe ( e ψ ) + κ (cid:3) ψ + d ≤ (Γ g ) d ψ + r − d ≤ ψ. This concludes the proof of the lemma.
Lemma 10.4.9.
The exists a scalar function d satisfying the bound d = 12 m + O ( δ H ) on the support of κ H , such that we have, schematically [ (cid:3) , Y H ] ψ = d Y (0) ( Y H ψ ) + 1 Υ ≤ δ H (cid:18) (cid:3) ψ + d T ψ + d ≤ (Γ g ) d ψ + 1 δ H d ≤ ψ (cid:19) + 1 δ H δ H ≤ Υ ≤ δ H d ≤ ψ. Proof.
We have Y (0) = ae + be + 2 T = ae + b (2 T − Υ e ) + 2 T = ( a − Υ b ) e + 2(1 + b ) T. Thus, in view of the commutator identities[ T, (cid:3) ] ψ = d ≤ (Γ g ) d ≤ ψ, [ (cid:3) , e ] ψ = − ωe ( e ψ ) + κe ( e ψ ) + κ (cid:3) ψ + d ≤ (Γ g ) d ψ + r − d ≤ ψ, (cid:3) , Y (0) ] ψ = [ (cid:3) , ( a − Υ b ) e ] ψ + [ (cid:3) , b ) T ] ψ = ( a − Υ b )[ (cid:3) , e ] ψ + g αβ D α ( a ) D β e ψ + 2(1 + b )[ (cid:3) , T ] ψ +2 g αβ D α ( b ) D β T ψ + d ≤ ψ = ( a − Υ b ) (cid:16) − ωe ( e ψ ) + κe ( e ψ ) + κ (cid:3) ψ (cid:17) − e ( a ) e ( e ψ ) − e ( a ) e ( e ψ )+ d T ψ + d ≤ (Γ g ) d ψ + d ≤ ψ. Since e = − Υ e + 2 T , we infer schematically[ (cid:3) , Y (0) ] ψ = (cid:18) ( a − Υ b )( − ω − Υ κ ) + Υ2 e ( a ) − e ( a ) (cid:19) e ( e ψ )+ (cid:3) ψ + d T ψ + d ≤ (Γ g ) d ψ + d ≤ ψ. We deduce,[ (cid:3) , Y H ] ψ = [ (cid:3) , κ H Y (0) ] ψ = κ H [ (cid:3) , Y (0) ] ψ + κ (cid:48)H d ≤ ψ + κ (cid:48)(cid:48)H d ≤ ψ = κ H (cid:18) ( a − Υ b )( − ω − Υ κ ) + Υ2 e ( a ) − e ( a ) (cid:19) e ( e ψ )+1 Υ ≤ δ H (cid:18) (cid:3) ψ + d T ψ + d ≤ (Γ g ) d ψ + 1 δ H d ≤ ψ (cid:19) + 1 δ H δ H ≤ Υ ≤ δ H d ≤ ψ. Now, we have κ H e ( e ψ ) = 1 a − Υ b κ H Y (0) ( e ψ ) + T d ψ = 1( a − Υ b ) κ H Y (0) ( Y (0) ψ ) + d T ψ + d ≤ ψ = 1( a − Υ b ) Y (0) ( Y H ψ ) + d T ψ + 1 δ H d ≤ ψ and hence[ (cid:3) , Y H ] ψ = ( a − Υ b )( − ω − Υ κ ) + Υ2 e ( a ) − e ( a )( a − Υ b ) Y (0) ( Y H ψ )+1 Υ ≤ δ H (cid:18) (cid:3) ψ + d T ψ + d ≤ (Γ g ) d ψ + 1 δ H d ≤ ψ (cid:19) + 1 δ H δ H ≤ Υ ≤ δ H d ≤ ψ. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
Now, we have in view of the definition of a and b ,( a − Υ b )( − ω − Υ κ ) + Υ2 e ( a ) − e ( a )( a − Υ b ) = (1 + O (Υ))( − ω + O (Υ)) + O (Υ)(1 + O (Υ)) = 12 m + O ( (cid:15) ) + O (Υ)= 12 m + O ( (cid:15) ) + O (Υ)where we used also our assumptions on ω and m . Thus, we have on the support of κ H ( a − Υ b )( − ω − Υ κ ) + Υ2 e ( a ) − e ( a )( a − Υ b ) = 12 m + O ( (cid:15) + δ H )= 12 m + O ( δ H )where we used the fact that (cid:15) (cid:28) δ H by assumption. Setting d := ( a − Υ b )( − ω − Υ κ ) + Υ2 e ( a ) − e ( a )( a − Υ b ) , this concludes the proof of the lemma. Commutation with re Lemma 10.4.10.
We have, schematically, [ (cid:3) , re ] ψ = Υ r (cid:18) mr Υ (cid:19) ˇ e ( re ψ ) + (cid:3) ψ + Γ g d ψ + 1Υ r − d T ψ + r − d / ψ + r − d ψ. Proof.
Recall that we have (cid:3) ψ = − e ( e ( ψ )) + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ. Since we have [ re , e ] = 2 rωe − r κe + Γ b d , [ re , e ] = − r κe + Γ g d , [ d/ k , re ] = 12 rκ d/ k + Γ g d + Γ g , [ d (cid:63) / k , re ] = 12 rκ d (cid:63) / k + Γ g d + Γ g , (cid:3) , re ] ψ = − [ e , re ] e ψ − e [ e , re ] ψ + [ (cid:52) / , re ] ψ + (cid:18) ω − κ (cid:19) [ e , re ] ψ − re (cid:18) ω − κ (cid:19) e ψ − κ [ e , re ] ψ + 12 re ( κ ) e ψ + 2 η [ e θ , re ] ψ − re ( η ) e θ ψ = (cid:16) rωe − r κe (cid:17) e ψ − e ( rκe ψ ) + rκ (cid:52) / ψ + 12 re ( κ ) e ψ + Γ g d ψ + r − d ψ = (cid:16) rωe − r κe (cid:17) e ψ − rκe e ψ − rκ e ψ + Γ g d ψ + r − d / ψ + r − d ψ. Using again (cid:3) ψ = − e ( e ( ψ )) + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ. we have − rκe e ψ − rκ e ψ = 12 rκ (cid:3) ψ + r − d / ψ + r − d ψ and hence[ (cid:3) , re ] ψ = (cid:16) rωe − r κe (cid:17) e ψ + (cid:3) ψ + Γ g d ψ + r − d / ψ + r − d ψ = (cid:18) rω
1Υ (2 T − e ) − r κe (cid:19) e ψ + (cid:3) ψ + Γ g d ψ + r − d / ψ + r − d ψ = (cid:18) − rω − r κ (cid:19) e ( e ψ ) + (cid:3) ψ + Γ g d ψ + 1Υ r − d T ψ + r − d / ψ + r − d ψ = (cid:18) Υ + 2 mr Υ (cid:19) e ( e ψ ) + (cid:3) ψ + Γ g d ψ + 1Υ r − d T ψ + r − d / ψ + r − d ψ = Υ r (cid:18) mr Υ (cid:19) ˇ e ( re ψ ) + (cid:3) ψ + Γ g d ψ + 1Υ r − d T ψ + r − d / ψ + r − d ψ. This concludes the proof of the lemma.
Recall from Corollary 10.4.5 that we have the following commutator identity r d/ ( (cid:3) ψ ) − ( (cid:3) − K )( r d/ ψ ) = − rη (cid:3) ψ + d ≤ (Γ g ) d ≤ ψ. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
In particular, to derive weighted estimates for r d/ , we need to derive weighted estimatesfor solutions φ to wave equations of the type( (cid:3) − V ) φ = N, where φ is a reduced 1-scalar and the potential V is given by V = V + K = − κκ + K .This is done in the following theorem. Theorem 10.4.11.
Let φ a reduced 1-scalar solution to ( (cid:3) − V ) φ = N, V = − κκ + K. Then, φ satisfies for all δ ≤ p ≤ − δ , sup τ ∈ [ τ ,τ ] E p [ φ ]( τ ) + B p [ φ ]( τ , τ ) + F p [ φ ]( τ , τ ) (cid:46) E p [ φ ]( τ ) + J p [ φ, N ]( τ , τ ) + (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | φ | ( | φ | + | Rφ | )+ (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | φ | ( | φ | + | d φ | ) , (10.4.4) and ˇ φ = f ˇ e φ satisfies for all − δ < q ≤ − δ , sup τ ∈ [ τ ,τ ] E q [ ˇ φ ]( τ ) + B q [ ˇ φ ]( τ , τ ) (cid:46) E q [ ˇ φ ]( τ ) + ˇ J q [ ˇ φ, N ]( τ , τ ) + E q,δ ) [ φ ]( τ ) + J q,δ ) [ φ, N ]+ (cid:90) M ( τ ,τ ) r q − | ˇ φ | ( | ˇ φ | + | d ˇ φ | ) . (10.4.5) Remark 10.4.12.
Although we will not need it, we expect that the last 2 terms in theright-hand side of (10.4.4) and the last term in the right-hand side of (10.4.5) could beremoved.Proof.
We start with the following observations. • (10.4.5) is the analog of (10.4.1), i.e. of Theorem 5.3.5 in the case s = 0, with V replaced by V , and with the reduced 2-scalar ψ replaced by the reduced 1-scalar φ .The proof is in fact significantly easier in view of the presence of the term (cid:90) M ( τ ,τ ) r q − | ˇ φ | ( | ˇ φ | + | d ˇ φ | )on the right-hand side of (10.4.5). • (10.4.4) is the analog of (10.4.2), i.e. of Theorem 5.3.4 in the case s = 0, with V replaced by V , and with the reduced 2-scalar ψ replaced by the reduced 1-scalar φ .The proof is in fact significantly easier in view of the presence of the terms (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | φ | ( | φ | + | Rφ | ) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | φ | ( | φ | + | d φ | )on the right-hand side of (10.4.4). • The boundary terms can be treated as in the proof of (10.4.1) and (10.4.2) in viewof the fact that V is a positive potential . • The only place where there might a potential difficulty concerns the proof of (10.4.5)in ( trap ) M where the second to last term on the right-hand side is required to havea more precise structure.In view of the above observations, and in particular of the last one, we focus on recoveringthe bulk term leading to (10.4.5) in ( trap ) M . To this end, we choose f ad w as inProposition 10.1.16. This yields ˙ E [ f R, w ](Ψ) ≥ f (cid:48) | R (Ψ) | + r − (cid:0) − mr (cid:1) f |∇ / Ψ | + O (cid:16) − mr r (cid:17) | Ψ | . We infer˙ E [ f R, w, M = 2 hR ](Ψ) ≥ f (cid:48) | R (Ψ) | + r − (cid:18) − mr (cid:19) f |∇ / Ψ | + O (cid:18) − mr r (cid:19) | Ψ | + 12 r − (Υ r h ) (cid:48) | Ψ | + h Ψ R (Ψ) . We now choose a smooth h , compactly supported in [5 / m , / m ], such that h (3 m ) = 0and h (cid:48) (3 m ) = 1 . We infer r − (Υ r h ) (cid:48) (3 m ) = 1 / > E [ f R, w, M ](Ψ) ≥ f (cid:48) | R (Ψ) | + r − (cid:18) − mr (cid:19) f |∇ / Ψ | + Υ r | Ψ | + O (cid:18) − mr r (cid:19) | Ψ | ( | Ψ | + | R (Ψ) | ) . We have V = − κκ + K = 4Υ + 1 + O ( (cid:15) ) r in view of the assumptions so that V is indeed a positive potential. Note that Proposition 10.1.16 does not use the particular form of the potential and the type of thereduced scalar φ and hence holds in our more general case. This differs from the choice of h in the proof of (10.4.1) in order to avoid using a Poincar´e inequality(which depends of the type of the reduced scalar) and the particular form of the potential V . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
In view of the choice of f in Proposition 10.1.16, we have f (cid:48) (cid:38) r , (cid:18) − mr (cid:19) f ≥ (cid:18) − mr (cid:19) , and hence, there exists two constants c > C > E [ f R, w , M ](Ψ) ≥ c (cid:32) r | R (Ψ) | + r − (cid:18) − mr (cid:19) |∇ / Ψ | + Υ r | Ψ | (cid:33) − C (cid:12)(cid:12) − mr (cid:12)(cid:12) r | Ψ | ( | Ψ | + | R (Ψ) | ) . The last term above is responsible for the second to last term on the right-hand side of(10.4.5).Next, we have the following consequence of (10.4.1) and Theorem 10.4.11.
Corollary 10.4.13.
Let φ be a reduced k -scalar for k = 1 , such that φ satisfies ( (cid:3) k − W ) φ = O (cid:32) (cid:15)r u δ dec − δ trap (cid:33) d φ + φ where φ and φ are given reduced scalars, and where W = V in the case k = 2 and W = V in the case k = 1 . Then, φ satisfies for all δ ≤ p ≤ − δ , sup τ ∈ [ τ ,τ ] E p [ φ ]( τ ) + B p [ φ ]( τ , τ ) + F p [ φ ]( τ , τ ) (cid:46) E p [ φ ]( τ ) + (cid:15) (cid:32) sup [ τ ,τ ] E [ φ ]( τ ) + B p [ φ ]( τ , τ ) (cid:33) + J p [ φ, φ ]( τ , τ )+ (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | φ | ( | φ | + | Rφ | ) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | φ | ( | φ | + | d φ | ) . Proof.
The wave equation for φ satisfies the assumptions of (10.4.1) and Theorem 10.4.11with N = O (cid:32) (cid:15)r u δ dec − δ trap (cid:33) d φ + φ . Recall that we have δ (cid:28) δ dec in view of (5.1.1), and hence δ dec − δ > τ ∈ [ τ ,τ ] E p [ φ ]( τ ) + B p [ φ ]( τ , τ ) + F p [ φ ]( τ , τ ) (cid:46) E p [ φ ]( τ ) + J p (cid:34) φ, O (cid:32) (cid:15)r u δ dec − δ trap (cid:33) d φ + φ (cid:35) ( τ , τ )+ (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | φ | ( | φ | + | Rφ | ) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | φ | ( | φ | + | d φ | ) . Now, in view of the definition J p,R [ ψ, N ]( τ , τ ) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) M ≥ R ( τ ,τ ) r p ˇ e ψN (cid:12)(cid:12)(cid:12)(cid:12) ,J p [ ψ, N ]( τ , τ ) = (cid:18) (cid:90) τ τ dτ (cid:107) N (cid:107) L ( ( trap ) Σ( τ )) (cid:19) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ | N | + J p, m [ ψ, N ]( τ , τ ) , we have J p (cid:34) φ, O (cid:32) (cid:15)r u δ dec − δ trap (cid:33) d φ + φ (cid:35) ( τ , τ ) (cid:46) J p (cid:34) φ, O (cid:32) (cid:15)r u δ dec − δ trap (cid:33) d φ (cid:35) ( τ , τ ) + J p [ φ, φ ] ( τ , τ )and, for δ ≤ p ≤ − δ , using also δ dec − δ >
0, we have J p (cid:34) φ, O (cid:32) (cid:15)r u δ dec − δ trap (cid:33) d φ (cid:35) ( τ , τ ) (cid:46) (cid:15) (cid:18) (cid:90) τ τ (cid:107) d φ (cid:107) L ( ( trap ) Σ( τ )) dττ δ dec − δ (cid:19) + (cid:15) (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ − | d φ | + (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) M ≥ m ( τ ,τ ) r p − ˇ e ( ψ ) d φ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:15) sup [ τ ,τ ] (cid:107) d φ (cid:107) L ( ( trap ) Σ( τ )) + (cid:15) (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | d ≤ ψ | + (cid:15) (cid:32)(cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | d φ | (cid:33) (cid:32)(cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | d φ | (cid:33) (cid:46) (cid:15) (cid:32) sup [ τ ,τ ] E [ φ ]( τ ) + B p [ φ ]( τ , τ ) (cid:33) + (cid:15) (cid:0) B p [ φ ]( τ , τ ) (cid:1) (cid:0) B p [ φ ]( τ , τ ) (cid:1) . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
We immediately deducesup τ ∈ [ τ ,τ ] E p [ φ ]( τ ) + B p [ φ ]( τ , τ ) + F p [ φ ]( τ , τ ) (cid:46) E p [ φ ]( τ ) + (cid:15) (cid:32) sup [ τ ,τ ] E [ φ ]( τ ) + B p [ φ ]( τ , τ ) (cid:33) + J p [ φ, φ ]( τ , τ )+ (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | φ | ( | φ | + | Rφ | ) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | φ | ( | φ | + | d φ | ) . This concludes the proof of the corollary.Finally, we end this section with the following lemma.
Lemma 10.4.14.
Let φ be a reduced k -scalar for k = 1 , , and let X a vectorfield. Wehave for all δ ≤ p ≤ − δ , (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | d φ | ( | Xφ | + | R ( Xφ ) | ) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | d φ | ( | Xφ | + | d ( Xφ ) | ) (cid:46) ( B p [ Xφ ]( τ , τ )) ( B p [ φ ]( τ , τ )) . Proof.
The proof follows immediately from the definition of B p [ φ ]( τ , τ ). We now conclude the proof of Theorem 5.3.4 for all 0 ≤ s ≤ k small + 30 by recoveringhigher derivatives s ≥ s = 0 provided by (10.4.1).As explained in section 10.4.2, it suffices to recover the estimates for s = 1 from the onefor s = 0 as the procedure to recover the estimate for s + 1 from the one for s is completelyanalogous. We now follow the strategy outlined in section 10.4.2. Recovering estimates for
T ψ
Recall that ψ satisfies (cid:3) ψ + V ψ = N, V = κκ, and recall also from Corollary 10.4.3 that we have[ T, (cid:3) ] ψ = d ≤ (Γ g ) d ≤ ψ. (cid:3) ( T ψ ) +
V T ( ψ ) = T ( N ) + d ≤ (Γ g ) d ≤ ψ. In view of Corollary 10.4.13 with φ = T ( ψ ), φ = d ≤ ψ and φ = T ( N ), and in view of(10.4.3), we deducesup τ ∈ [ τ ,τ ] E p [ T ψ ]( τ ) + B p [ T ψ ]( τ , τ ) + F p [ T ψ ]( τ , τ ) (cid:46) E p [ T ψ ]( τ ) + (cid:15) (cid:32) sup [ τ ,τ ] E [ d ≤ ψ ]( τ ) + B p [ d ≤ ψ ]( τ , τ ) (cid:33) + J p [ T ( ψ ) , T ( N )]( τ , τ )+ (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | T φ | ( | T φ | + | R ( T φ ) | )+ (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | T φ | ( | T φ | + | d ( T φ ) | ) , and hence, using Lemma 10.4.14 with X = T , we infer for any δ ≤ p ≤ − δ ,sup τ ∈ [ τ ,τ ] E p [ T ψ ]( τ ) + B p [ T ψ ]( τ , τ ) + F p [ T ψ ]( τ , τ ) (10.4.6) (cid:46) E p [ T ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) + (cid:15) (cid:32) sup [ τ ,τ ] E [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) (cid:33) + B p [ ψ ]( τ , τ ) . Recovering estimates for r d/ ψ Recall that ψ satisfies (cid:3) ψ + V ψ = N, V = κκ, and recall also from Corollary 10.4.5 that we have r d/ ( (cid:3) ψ ) − ( (cid:3) − K )( r d/ ψ ) = − rη (cid:3) ψ + d ≤ (Γ g ) d ≤ ψ We infer (cid:3) ( r d/ ψ ) + ( V − K ) r d/ ψ = rη (cid:3) ψ + r d/ ( N ) + d ≤ (Γ g ) d ≤ ψ = − rηN + r d/ ( N ) + d ≤ (Γ g ) d ≤ ψ. and hence (cid:3) ( r d/ ψ ) + ( V − K ) r d/ ψ = − rηN + r d/ ( N ) + d ≤ (Γ g ) d ≤ ψ. CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
In view of Corollary 10.4.13 with φ = r d/ ψ , φ = d ≤ ψ and φ = − rηN + r d/ ( N ), andin view of (10.4.3), we deducesup τ ∈ [ τ ,τ ] E p [ r d/ ψ ]( τ ) + B p [ r d/ ψ ]( τ , τ ) + F p [ r d/ ψ ]( τ , τ ) (cid:46) E p [ r d/ ψ ]( τ ) + (cid:15) (cid:32) sup [ τ ,τ ] E [ d ≤ ψ ]( τ ) + B p [ d ≤ ψ ]( τ , τ ) (cid:33) + J p (cid:104) r d/ ψ, − rηN + r d/ ( N ) (cid:105) ( τ , τ ) + (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | r d/ φ | ( | r d/ φ | + | R ( r d/ φ ) | )+ (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r p − | r d/ φ | ( | r d/ φ | + | d ( r d/ φ ) | ) , and hence, using Lemma 10.4.14 with X = r d/ , we infer for any δ ≤ p ≤ − δ ,sup τ ∈ [ τ ,τ ] E p [ r d/ ψ ]( τ ) + B p [ r d/ ψ ]( τ , τ ) + F p [ r d/ ψ ]( τ , τ ) (10.4.7) (cid:46) E p [ r d/ ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) + (cid:15) (cid:32) sup [ τ ,τ ] E [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) (cid:33) + B p [ ψ ]( τ , τ ) . Recovering estimates for Rψ in r ≤ r We start with the following lemma.
Lemma 10.4.15.
Let ψ satisfy (cid:3) ψ = V ψ + N, V = κκ. Then, R ψ satisfies R ψ = − Υ N + T ψ + O ( r − ) d / ψ + O ( r − ) d ψ + O ( r − ) ψ. Proof.
Recall that we have (cid:3) ψ = − e ( e ( ψ )) + (cid:52) / ψ − κe ψ + (cid:18) − κ + 2 ω (cid:19) e ψ + 2 ηe θ ψ and e = T + R, Υ e = ( T − R ) . (cid:3) ψ = − ( T − R )( T + R ) ψ + Υ (cid:52) / ψ − Υ2 κe ψ + Υ (cid:18) − κ + 2 ω (cid:19) e ψ + 2Υ ηe θ ψ = − T ψ + R ψ − [ T, R ] ψ + O ( r − ) d / ψ + O ( r − ) d ψ and hence R ψ = − Υ (cid:3) ψ + T ψ − [ T, R ] ψ + O ( r − ) d / ψ + O ( r − ) d ψ = − Υ N + T ψ − [ T, R ] ψ + O ( r − ) d / ψ + O ( r − ) d ψ + O ( r − ) ψ where we used the fact that (cid:3) ψ = V ψ + N and V = κκ = O ( r − ). Also, we have[ T, R ] ψ = 14 [ e + Υ e , e − Υ e ] ψ = 12 (cid:16) − e (Υ) e + Υ[ e , e ] (cid:17) ψ = O ( r − ) d ψ and thus R ψ = − Υ N + T ψ + O ( r − ) d / ψ + O ( r − ) d ψ + O ( r − ) ψ. This concludes the proof of the lemma.We now estimate Rψ in r ≤ r for a fixed r ≥ m that will be chosen large enough.First, in view of the identity of the previous lemma, i.e. R ψ = − Υ N + T ψ + O ( r − ) d / ψ + O ( r − ) d ψ + O ( r − ) ψ, we infer sup [ τ ,τ ] (cid:90) Σ r ≤ r ( τ ) | R ψ | + (cid:90) M r ≤ r ( τ ,τ ) (cid:18) − mr (cid:19) ( R ψ ) (10.4.8) (cid:46) sup [ τ ,τ ] (cid:32) E [ T ψ ] + E [ r d/ ψ ] + E [ ψ ] + (cid:90) Σ r ≤ r ( τ ) N (cid:33) + (cid:90) M r ≤ r ( τ ,τ ) N + Morr[ T ψ ]( τ , τ ) + Morr[ r d/ ψ ]( τ , τ ) + Morr[ ψ ]( τ , τ ) . Next, we remove the degeneracy of the above estimate at r = 3 m . Recall from Corollary10.4.7 that we have in the region r ≤ m (cid:3) ( Rψ ) = (cid:18) − mr (cid:19) d ψ + O (cid:32) (cid:15)u δ dec − δ trap (cid:33) d ψ + O (1) d ≤ ψ + O (1) d ≤ N. Then,62
CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
1. multiplying Rψ with a cut-off function equal to one on [5 / m , / m ] and vanishingon [9 / m , m ] and inferring the corresponding wave equation from the above onefor Rψ ,2. relying on the Morawetz estimate of Proposition 10.1.12 with the particular choice f ( r ) = r − m ,3. adding a large multiple of the energy estimate,4. using Proposition 10.1.32 for the boundary terms,we easily infer the following estimate (cid:90) ( trap ) M ( τ ,τ ) ( R ψ ) (cid:46) (cid:90) M r ≤ m ( τ ,τ ) (cid:32)(cid:18) − mr (cid:19) ( d ψ ) + ( d ψ ) + ( d ≤ N ) (cid:33) + E [ Rψ ]( τ ) + (cid:15) sup [ τ ,τ ] E [ ψ ]( τ ) . Together with (10.4.8), we infersup [ τ ,τ ] (cid:90) Σ r ≤ r ( τ ) | R ψ | + (cid:90) M r ≤ r ( τ ,τ ) | R ψ | (10.4.9) (cid:46) E [ Rψ ]( τ ) + sup [ τ ,τ ] (cid:16) (cid:15)E [ ψ ]( τ ) + E [ T ψ ]( τ ) + E [ r d/ ψ ]( τ ) + E [ ψ ]( τ ) (cid:17) + J p [ ψ, N ]( τ , τ ) + Morr[ T ψ ]( τ , τ ) + Morr[ r d/ ψ ]( τ , τ ) + Morr[ ψ ]( τ , τ ) . Recovering estimates for Y H ψ Recall that ψ satisfies (cid:3) ψ + V ψ = N, V = κκ, and recall also from Lemma 10.4.9[ (cid:3) , Y H ] ψ = d Y (0) ( Y H ψ ) + 1 Υ ≤ δ H (cid:18) (cid:3) ψ + d T ψ + d ≤ (Γ g ) d ψ + 1 δ H d ≤ ψ (cid:19) + 1 δ H δ H ≤ Υ ≤ δ H d ≤ ψ where the scalar function d satisfying the bound d = 12 m + O ( δ H ) on the support of κ H . (cid:3) ( Y H ψ ) + V Y H ( ψ ) = d Y (0) ( Y H ψ ) + 1 Υ ≤ δ H (cid:18) N + d T ψ + (cid:15) d ψ + 1 δ H d ≤ ψ (cid:19) + 1 δ H δ H ≤ Υ ≤ δ H d ≤ ψ + Y H ( N ) . Then,1. we use the redshift vectorfield Y H as a multiplier,2. we rely on Proposition 10.1.29,3. we use the fact that d ≥ [ τ ,τ ] E [ Y H ψ ] + Morr[ Y H ψ ]( τ , τ ) (cid:46) E [ Y H ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) + (cid:15) Morr[ d ψ ]( τ , τ )+Morr[ Rψ ]( τ , τ ) + Morr[ T ψ ]( τ , τ )+Morr[ r d/ ψ ]( τ , τ ) + Morr[ ψ ]( τ , τ ) . (10.4.10) Recovering estimates for re ψ in r ≥ r Recall that ψ satisfies (cid:3) ψ + V ψ = N, V = κκ, and recall also from Lemma 10.4.10[ (cid:3) , re ] ψ = Υ r (cid:18) mr Υ (cid:19) ˇ e ( re ψ ) + (cid:3) ψ + Γ g d ψ + 1Υ r − d T ψ + r − d / ψ + r − d ψ. We infer (cid:3) ( re ψ ) + V re ( ψ ) = Υ r (cid:18) mr Υ (cid:19) ˇ e ( re ψ ) + O (cid:16) (cid:15)r (cid:17) d ψ + 1Υ r − d T ψ + r − d / ψ + r − d ≤ ψ + N + re ( N ) . Then,64
CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS
1. as in section 10.2.3, we use the vectorfield f p e as a multiplier, where f p = θ r ( r ) r p e and the cut-off θ r ( r ) is equal to one in the region r ≥ r and vanishes in the region r ≤ r / e ( re ψ ) on the right-hand side ispositive for r ≥ m , i.e.Υ r (cid:18) mr Υ (cid:19) ≥ r ≥ m . We easily infersup τ ∈ [ τ ,τ ] E p,r ≥ r [ re ψ ]( τ ) + B p,r ≥ r [ re ψ ]( τ , τ ) + F p,r ≥ r [ re ψ ]( τ , τ ) (cid:46) E p [ re ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) + B p,r / ≤ r Gathering the estimates (10.4.6), (10.4.7), (10.4.9), (10.4.10) and (10.4.11), we infer forany δ ≤ p ≤ − δ ,sup τ ∈ [ τ ,τ ] E p [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) + F p [ ψ ]( τ , τ ) (cid:46) E p [ ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) + (cid:15) (cid:32) sup [ τ ,τ ] E p [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) (cid:33) + sup [ τ ,τ ] E p [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) , and hence sup τ ∈ [ τ ,τ ] E p [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) + F p [ ψ ]( τ , τ ) (cid:46) E p [ ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) + sup [ τ ,τ ] E p [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) . In view of (10.4.1), we deducesup τ ∈ [ τ ,τ ] E p [ ψ ]( τ ) + B p [ ψ ]( τ , τ ) + F p [ ψ ]( τ , τ ) (cid:46) E p [ ψ ]( τ ) + J p [ ψ, N ]( τ , τ ) s = 1. We have thus deduced Theorem 5.3.4 in thecase s = 1 from the case s = 0, i.e. (10.4.1). Since going from s = 0 to s = 1 is analogousto going from s to s + 1, higher order derivatives k ≤ k small + 30 are recovered in the samefashion. This concludes the proof of Theorem 5.3.4. We now conclude the proof of Theorem 5.3.5 for all 0 ≤ s ≤ k small + 29 by recoveringhigher derivatives s ≥ s = 0 provided by (10.4.2).As explained in section 10.4.2, it suffices to recover the estimates for s = 1 from the onefor s = 0 as the procedure to recover the estimate for s + 1 from the one for s is completelyanalogous. We now follow the strategy outlined in section 10.4.2. Recovering estimates for T ˇ ψ Recall from Proposition 10.3.1 that ˇ ψ = f ˇ e ψ satisfies (cid:3) ˇ ψ − V ˇ ψ = 2 r (cid:18) − mr (cid:19) e ˇ ψ + ˇ N + f (cid:18) e + 3 r (cid:19) N where,ˇ N = O ( r − ) d ≤ ψ + r Γ b e d ψ + d ≤ (Γ b ) d ≤ ψ + r d ≤ (Γ g ) e ψ + d ≤ (Γ g ) d ψ, r ≥ m ,O (1) d ≤ ψ, m ≤ r ≤ m , and recall also from Corollary 10.4.3 that we have[ T, (cid:3) ] ˇ ψ = d ≤ (Γ g ) d ≤ ˇ ψ. We infer (cid:3) ( T ˇ ψ ) − V T ( ˇ ψ ) = 2 r (cid:18) − mr (cid:19) e ( T ˇ ψ ) + N T + T (cid:18) f (cid:18) e + 3 r (cid:19) N (cid:19) , CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS where we have, in view of the estimates of Lemma 5.1.1 for d k Γ g and d k Γ b with k ≤ k small + 30, N T = O (cid:0) τ δdec − δ (cid:1) (cid:16) e d ψ + r − d ≤ ψ (cid:17) + O (cid:16) rτ 12 + δdec − δ (cid:17) e d ≤ ψ + O (cid:0) r (cid:1) (cid:16) d ≤ ψ + (cid:15) d ≤ ˇ ψ (cid:17) , r ≥ m ,O (1) d ≤ ψ, m ≤ r ≤ m . In view of (10.4.2) with T ˇ ψ instead of ˇ ψ and with N T + T (cid:18) f (cid:18) e + 3 r (cid:19) N (cid:19) instead of ˇ N + f (cid:0) e + r (cid:1) N , we deducesup τ ∈ [ τ ,τ ] E q [ T ˇ ψ ]( τ ) + B q [ T ˇ ψ ]( τ , τ ) (cid:46) E q [ T ˇ ψ ]( τ ) + J q (cid:20) T ˇ ψ, N T + T (cid:18) f (cid:18) e + 3 r (cid:19) N (cid:19)(cid:21) ( τ , τ )+ E q,δ ) [ T ψ ]( τ ) + J q,δ ) [ T ψ, T N ] (cid:46) E q [ T ˇ ψ ]( τ ) + ˇ J q (cid:2) ˇ ψ, N (cid:3) ( τ , τ ) + J q (cid:2) T ˇ ψ, N T (cid:3) ( τ , τ )+ E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ] , so that it remains to estimate J q [ T ˇ ψ, N T ]( τ , τ ) = J q, m (cid:2) T ˇ ψ, N T (cid:3) ( τ , τ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ≥ m ( τ ,τ ) r q (ˇ e ( T ˇ ψ )) N T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Here, unlike the proof of Theorem 5.3.4 above, the non sharp estimates of section 10.4.1 are notenough, and we need instead to rely on the stronger estimates provided by Lemma 5.1.1. N T , J q [ T ˇ ψ, N T ]( τ , τ ) (cid:46) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ≥ m ( τ ,τ ) r q (ˇ e ( T ˇ ψ )) 1 τ δ dec − δ (cid:16) e d ψ + r − d ≤ ψ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ≥ m ( τ ,τ ) r q (ˇ e ( T ˇ ψ )) 1 rτ + δ dec − δ e d ≤ ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ≥ m ( τ ,τ ) r q (ˇ e ( T ˇ ψ )) 1 r d ≤ ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ≥ m ( τ ,τ ) r q (ˇ e ( T ˇ ψ )) (cid:15)r d ≤ ˇ ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:32) sup [ τ ,τ ] (cid:90) Σ( τ ) r q (ˇ e ( T ˇ ψ )) (cid:33) (cid:40) (cid:32) sup [ τ ,τ ] (cid:90) Σ( τ ) r q (cid:16) ( e d ψ ) + r − ( d ≤ ψ ) (cid:17)(cid:33) + (cid:32)(cid:90) M r ≥ m ( τ ,τ ) r q − ( e d ≤ ψ ) (cid:33) (cid:41) + (cid:15) (cid:90) M ( τ ,τ ) r q − ( d ≤ ˇ ψ ) + (cid:18)(cid:90) M ( τ ,τ ) r q − (ˇ e ( T ˇ ψ )) (cid:19) (cid:18)(cid:90) M ( τ ,τ ) r q − ( d ≤ ψ ) (cid:19) which yields, using in particular the fact that q ≤ − δ , J q [ T ˇ ψ, N T ]( τ , τ ) (cid:46) (cid:32) sup [ τ ,τ ] E q [ T ˇ ψ ]( τ ) (cid:33) (cid:40) sup [ τ ,τ ] E q,δ ) [ ψ ]( τ ) + B q,δ ) [ ψ ]( τ , τ ) (cid:41) + (cid:15)B q [ ˇ ψ ]( τ , τ )+ (cid:16) B q [ T ˇ ψ ]( τ , τ ) (cid:17) (cid:16) B q,δ ) [ ψ ]( τ , τ ) (cid:17) . We deduce sup τ ∈ [ τ ,τ ] E q [ T ˇ ψ ]( τ ) + B q [ T ˇ ψ ]( τ , τ ) (cid:46) E q [ T ˇ ψ ]( τ ) + ˇ J q (cid:2) ˇ ψ, N (cid:3) ( τ , τ ) + (cid:15)B q [ ˇ ψ ]( τ , τ )+ sup [ τ ,τ ] E q,δ ) [ ψ ]( τ ) + B q,δ ) [ ψ ]( τ , τ ) + J q,δ ) [ ψ, N ] . Together with Theorem 5.3.4, this yieldssup τ ∈ [ τ ,τ ] E q [ T ˇ ψ ]( τ ) + B q [ T ˇ ψ ]( τ , τ ) (cid:46) E q [ T ˇ ψ ]( τ ) + ˇ J q (cid:2) ˇ ψ, N (cid:3) ( τ , τ ) + (cid:15)B q [ ˇ ψ ]( τ , τ )+ E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ] . (10.4.12)68 CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS Recovering estimates for r d/ ˇ ψ Recall from Proposition 10.3.1 that ˇ ψ = f ˇ e ψ satisfies (cid:3) ˇ ψ − V ˇ ψ = 2 r (cid:18) − mr (cid:19) e ˇ ψ + ˇ N + f (cid:18) e + 3 r (cid:19) N. Recall also from Corollary 10.4.5 that we have r d/ ( (cid:3) ˇ ψ ) − ( (cid:3) − K )( r d/ ˇ ψ ) = − rη (cid:3) ˇ ψ + d ≤ (Γ g ) d ≤ ˇ ψ We infer (cid:3) ( r d/ ˇ ψ ) + ( V − K ) r d/ ˇ ψ = 2 r (cid:18) − mr (cid:19) e ( r d/ ˇ ψ ) + N r d/ + ( r d/ − rη ) (cid:18) f (cid:18) e + 3 r (cid:19) N (cid:19) where N r d/ = − rη ˇ N + r d/ ( ˇ N ) + d ≤ (Γ g ) d ≤ ˇ ψ. In view of (10.4.5) with r d/ ˇ ψ instead of ˇ ψ and with N r d/ + ( r d/ − rη ) (cid:18) f (cid:18) e + 3 r (cid:19) N (cid:19) instead of ˇ N + f (cid:0) e + r (cid:1) N , we deducesup τ ∈ [ τ ,τ ] E q [ r d/ ˇ ψ ]( τ ) + B q [ r d/ ˇ ψ ]( τ , τ ) (cid:46) E q [ r d/ ˇ ψ ]( τ ) + ˇ J q (cid:20) r d/ ˇ ψ, N r d/ + ( r d/ − rη ) (cid:18) f (cid:18) e + 3 r (cid:19) N (cid:19)(cid:21) ( τ , τ )+ E q,δ ) [ r d/ ψ ]( τ ) + J q,δ ) [ r d/ ψ, r d/ N ]+ (cid:90) M ( τ ,τ ) r q − | r d/ ˇ ψ | ( | r d/ ˇ ψ | + | d ( r d/ ˇ ψ ) | ) (cid:46) E q [ r d/ ˇ ψ ]( τ ) + ˇ J q (cid:2) r d/ ˇ ψ, N r d/ (cid:3) ( τ , τ )+ E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ] + (cid:16) B q [ r d/ ˇ ψ ]( τ , τ ) (cid:17) (cid:16) B q [ ˇ ψ ]( τ , τ ) (cid:17) so that it remains to estimateˇ J q (cid:2) r d/ ˇ ψ, N r d/ (cid:3) ( τ , τ ) = J q, m (cid:2) r d/ ˇ ψ, N r d/ (cid:3) ( τ , τ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) M ≥ m ( τ ,τ ) r q (ˇ e ( r d/ ˇ ψ )) N r d/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . J q [ T ˇ ψ, N T ]( τ , τ ) so weleave the details to the reader. In the end, we arrive at the following analog of (10.4.12)sup τ ∈ [ τ ,τ ] E q [ r d/ ˇ ψ ]( τ ) + B q [ r d/ ˇ ψ ]( τ , τ ) (cid:46) E q [ r d/ ˇ ψ ]( τ ) + B q [ ˇ ψ ]( τ , τ ) + ˇ J q (cid:2) ˇ ψ, N (cid:3) ( τ , τ ) + (cid:15)B q [ ˇ ψ ]( τ , τ )+ E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ] . (10.4.13) Recovering estimates for re ˇ ψ Recall from Proposition 10.3.1 that ˇ ψ = f ˇ e ψ satisfies (cid:3) ˇ ψ − V ˇ ψ = 2 r (cid:18) − mr (cid:19) e ˇ ψ + ˇ N + f (cid:18) e + 3 r (cid:19) N. Recall also from Lemma 10.4.10 that we have[ (cid:3) , re ] ˇ ψ = Υ r (cid:18) mr Υ (cid:19) ˇ e ( re ˇ ψ ) + (cid:3) ˇ ψ + Γ g d ˇ ψ + 1Υ r − d T ˇ ψ + r − d / ˇ ψ + r − d ˇ ψ. We infer (cid:3) ( re ˇ ψ ) − V re ( ˇ ψ ) = (cid:18) r (cid:18) − mr (cid:19) + Υ r (cid:18) mr Υ (cid:19)(cid:19) e ( re ˇ ψ ) + N re + re (cid:18) f (cid:18) e + 3 r (cid:19) N (cid:19) , where N re = re ( ˇ N ) + ˇ N + Γ g d ˇ ψ + 1Υ r − d T ˇ ψ + r − d / ˇ ψ + r − d ˇ ψ. The rest follows along the same lines as the estimate for T ˇ ψ and we arrive at the followinganalog of (10.4.12)sup τ ∈ [ τ ,τ ] E q [ re ˇ ψ ]( τ ) + B q [ re ˇ ψ ]( τ , τ ) (10.4.14) (cid:46) E q [ re ˇ ψ ]( τ ) + B q [ ˇ ψ ]( τ , τ ) + B q [ T ˇ ψ ]( τ , τ ) + B q [ r d/ ˇ ψ ]( τ , τ )+ ˇ J q (cid:2) ˇ ψ, N (cid:3) ( τ , τ ) + (cid:15)B q [ ˇ ψ ]( τ , τ ) + E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ] . Notice that the coefficient in front of the term e ( re ˇ ψ ) in the RHS of the wave equation for re ˇ ψ differs from the one in front of the term e ˇ ψ in the RHS of the wave equation for ˇ ψ . Nevertheless, wemay apply (10.4.2) with re ˇ ψ instead of ˇ ψ since the only property of this coefficient which is used inthat it is positive on r ≥ m , i.e.2 r (cid:18) − mr (cid:19) + Υ r (cid:18) mr Υ (cid:19) ≥ r ≥ m . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS Conclusion of the proof of Theorem 5.3.5 Gathering the estimates (10.4.12), (10.4.13) and (10.4.14), we infer for any − δ < q ≤ − δ ,sup τ ∈ [ τ ,τ ] E q [ ˇ ψ ]( τ ) + B q [ ˇ ψ ]( τ , τ ) (cid:46) E q [ ˇ ψ ]( τ ) + ˇ J q [ ˇ ψ, N ]( τ , τ ) + B q [ ˇ ψ ]( τ , τ )+ (cid:15)B q [ ˇ ψ ]( τ , τ ) + E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ]and hencesup τ ∈ [ τ ,τ ] E q [ ˇ ψ ]( τ ) + B q [ ˇ ψ ]( τ , τ ) (cid:46) E q [ ˇ ψ ]( τ ) + ˇ J q [ ˇ ψ, N ]( τ , τ ) + B q [ ˇ ψ ]( τ , τ )+ E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ] . In view of (10.4.2), we deducesup τ ∈ [ τ ,τ ] E q [ ˇ ψ ]( τ ) + B q [ ˇ ψ ]( τ , τ ) (cid:46) E q [ ˇ ψ ]( τ ) + ˇ J q [ ˇ ψ, N ]( τ , τ )+ E q,δ ) [ ψ ]( τ ) + J q,δ ) [ ψ, N ]which is Theorem 5.3.5 in the case s = 1. We have thus deduced Theorem 5.3.5 in thecase s = 1 from the case s = 0, i.e. (10.4.2). Since going from s = 0 to s = 1 is analogousto going from s to s + 1, higher order derivatives k ≤ k small + 29 are recovered in the samefashion. This concludes the proof of Theorem 5.3.5. The goal of this section is to derive Theorem 10.5.2 and Proposition 10.5.4, see below,which is needed for the proof of Theorem M8 in Chapter 8. Recall that we have used sofar in Chapter 10 the global frame of Proposition 3.5.5. For this last section of Chapter10, we rely instead on the global frame used in Theorem M8, i.e. the one of Proposition3.5.2, as it is more regular and allows us to derive estimates for up to k large derivatives. Remark 10.5.1. Recall that in the frame of Proposition 3.5.2, we only have η ∈ Γ b .Note that the assumptions on the frame used in Chapter 10 are all consistent with η ∈ Γ b ,so that all results in this chapter apply for the frame of Proposition 3.5.2. Unlike the frame of Proposition 3.5.5 for which η ∈ Γ g . Theorem 10.5.2. Let ψ a reduced 2-scalar, and φ a reduced 0-scalar satisfying respec-tively ( (cid:3) + V ) ψ = N , ( (cid:3) + V ) φ = N , V = − r (cid:18) mr (cid:19) , V = 8 mr . Also, assume that the Ricci coefficients and curvature components associated to the globalnull frame we are using satisfy the estimates of section 10.4.1 for k ≤ k small derivatives.Then, for any ≤ s ≤ k large − , we have sup τ ∈ [ τ ,τ ] E sδ [ ψ ]( τ ) + B sδ [ ψ ]( τ , τ ) + F sδ [ ψ ]( τ , τ ) (cid:46) E sδ [ ψ ]( τ ) + sup τ ∈ [ τ ,τ ] E s − δ [ ψ ]( τ ) + B s − δ [ ψ ]( τ , τ ) + F s − δ [ ψ ]( τ , τ )+ D s [Γ] (cid:32) sup M ( τ ,τ ) ru + δ dec trap | d ≤ k small ψ | (cid:33) + (cid:90) M ( τ ,τ ) r δ | d ≤ s N | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M ( τ ,τ ) T ( d s φ ) d s N (cid:12)(cid:12)(cid:12)(cid:12) (10.5.1) and sup τ ∈ [ τ ,τ ] E sδ [ φ ]( τ ) + B sδ [ φ ]( τ , τ ) + F sδ [ φ ]( τ , τ ) (cid:46) E sδ [ φ ]( τ ) + sup τ ∈ [ τ ,τ ] E s − δ [ φ ]( τ ) + B s − δ [ φ ]( τ , τ ) + F s − δ [ φ ]( τ , τ )+ D s [Γ] (cid:32) sup M ( τ ,τ ) ru + δ dec trap | d ≤ k small φ | (cid:33) + (cid:90) Σ( τ ) ( d ≤ s φ ) r + (cid:90) M ( τ ,τ ) r δ | d ≤ s N | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M ( τ ,τ ) T ( d s φ ) d s N (cid:12)(cid:12)(cid:12)(cid:12) , (10.5.2) where D s [Γ] is defined by D s [Γ] := (cid:90) ( int ) M∪ ( ext ) M ( r ≤ m ) ( d ≤ s ˇΓ) + sup r ≥ m (cid:18) r (cid:90) { r = r } | d ≤ s Γ g | + r − (cid:90) { r = r } | d ≤ s Γ b | (cid:19) . The proof of Theorem 10.5.2 relies on the following theorem. Theorem 10.5.3. Let ψ a reduced scalar, and φ a reduced 0-scalar satisfying respectively ( (cid:3) + V ) ψ = N , ( (cid:3) + V ) φ = N , V = − r (cid:18) mr (cid:19) , V = 8 mr . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS Then, we have sup τ ∈ [ τ ,τ ] E δ [ ψ ]( τ ) + B δ [ ψ ]( τ , τ ) + F δ [ ψ ]( τ , τ ) (cid:46) E δ [ ψ ]( τ ) + (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | ψ | ( | ψ | + | Rψ | ) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ − | ψ | ( | ψ | + | d ψ | )+ (cid:90) M ( τ ,τ ) r δ | N | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M ( τ ,τ ) T ( ψ ) N (cid:12)(cid:12)(cid:12)(cid:12) , (10.5.3) and sup τ ∈ [ τ ,τ ] E δ [ φ ]( τ ) + B δ [ φ ]( τ , τ ) + F δ [ φ ]( τ , τ ) (cid:46) E δ [ φ ]( τ ) + (cid:90) A ( τ ,τ ) ∪ Σ( τ ) ∪ Σ ∗ ( τ ,τ ) φ r (10.5.4)+ (cid:90) ( trap ) M ( τ ,τ ) (cid:12)(cid:12)(cid:12)(cid:12) − mr (cid:12)(cid:12)(cid:12)(cid:12) | φ | ( | φ | + | Rφ | ) + (cid:90) ( trap (cid:14) ) M ( τ ,τ ) r δ − | φ | ( | φ | + | d φ | )+ (cid:90) M ( τ ,τ ) r δ | N | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( trap ) M ( τ ,τ ) T ( φ ) N (cid:12)(cid:12)(cid:12)(cid:12) . Proof. The proof of Theorem 10.5.3 is analogous to the one of Theorem 10.4.11. The onlydifferences are • The treatment of the right-hand sides N and N in the spacetime region ( trap ) M . • The boundary term on A ( τ , τ ) ∪ Σ( τ ) ∪ Σ ∗ ( τ , τ ) appearing in the right-hand sideof (10.5.4).The treatment of N and N is similar, so we focus on the one of N . The only estimatein which N appear in the trapping region is the Morawetz estimate. More precisely, itappear under the form, see (10.1.71), (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ( trap ) M ( τ ,τ ) (cid:18) f (cid:98) δ R (Ψ) + Λ T (Ψ) + 12 w Ψ (cid:19) N (cid:12)(cid:12)(cid:12)(cid:12) , This boundary term, as discussed below, is due to the fact that V is positive, which explains why nosuch term is present in (10.5.3) due to the negativity of the potential V for the wave equation satisfiedby ψ . f (cid:98) δ , w are functions which are in particularbounded on ( trap ) M . We infer (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ( trap ) M ( τ ,τ ) (cid:18) f (cid:98) δ R (Ψ) + Λ T (Ψ) + 12 w Ψ (cid:19) N (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:90) ( trap ) M ( τ ,τ ) ( | R (Ψ) | + | Ψ | ) | N | + (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ( trap ) M ( τ ,τ ) T (Ψ) N (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) λB δ [ ψ ]( τ , τ ) + λ − (cid:90) ( trap ) M ( τ ,τ ) | N | + (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ( trap ) M ( τ ,τ ) T (Ψ) N (cid:12)(cid:12)(cid:12)(cid:12) which yields the desired control provided λ > λB δ [ ψ ]( τ , τ ) can be absorbed by the LHS in (10.5.3).Concerning the boundary terms on A ( τ , τ ) ∪ Σ( τ ) ∪ Σ ∗ ( τ , τ ) appearing in the right-hand side of (10.5.4), the potential V does not appear in the boundary term of the r p weighted estimates, but it does appear in the boundary term of the energy estimates .More precisely, it appears in (cid:90) A ( τ ,τ ) ∪ Σ( τ ) ∪ Σ ∗ ( τ ,τ ) Q = (cid:90) A ( τ ,τ ) ∪ Σ( τ ) ∪ Σ ∗ ( τ ,τ ) (cid:16) |∇ / φ | + V φ (cid:17) . Now, we have in view of the definition of V (cid:90) A ( τ ,τ ) ∪ Σ( τ ) ∪ Σ ∗ ( τ ,τ ) Q ≥ (cid:90) A ( τ ,τ ) ∪ Σ( τ ) ∪ Σ ∗ ( τ ,τ ) |∇ / φ | − O (1) (cid:90) A ( τ ,τ ) ∪ Σ( τ ) ∪ Σ ∗ ( τ ,τ ) φ r and the control of the boundary terms follows. This concludes the proof of 10.5.3.We are now in position to prove Theorem 10.5.2. Note first that we have (cid:90) A ( τ ,τ ) ∪ Σ ∗ ( τ ,τ ) ( d ≤ s φ ) r (cid:46) F s − δ [ φ ]( τ , τ )which explains why the term (cid:82) A ( τ ,τ ) ∪ Σ ∗ ( τ ,τ ) ( d ≤ s φ ) r , that one would a priori would expectin view of (10.5.4), is not present on the right-hand side of (10.5.2). Also, the estimatesfor ψ and φ are similar, so we focus on the estimate for ψ . Proof of Theorem 10.5.2. The proof of Theorem 10.5.2 follows along the same lines as theone of Theorem 5.3.4. More precisely, following the strategy in section 10.4.2, we recover The boundary term of the r p weighted estimates involves only Q = ( e φ ) , while the one of theenergy estimate involves also Q = |∇ / φ | + V φ . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS derivatives one by one starting from Theorem 10.5.3 and use it iteratively in conjonctionwith the commutator estimates of section 10.4.3. The only difference is the treatmentof the derivatives for s ≥ k small + 1 as we assume that the estimates of section 10.4.1for the Ricci coefficients and curvature components only hold for k ≤ k small derivatives.Thus, to conclude, we need to consider the terms for which at least k small + 1 derivativesfall on the Ricci coefficients and curvature components. Since on the other hand we have s ≤ k large − 1, in view of the definition (3.3.7) of k small in terms of k large , and in view of thecommutator estimates of section 10.4.3, one easily checks that these terms are boundedin absolute value from above by (cid:16) | d ≤ s (Γ g ) | + r − | d ≤ s (Γ b ) | (cid:17) | d ≤ k small ψ | . We thus need, in view of Theorem 10.5.3, to estimate (cid:90) M ( τ ,τ ) r δ (cid:16) | d ≤ s (Γ g ) | + r − | d ≤ s (Γ b ) | (cid:17) | d ≤ k small ψ | + (cid:90) ( trap ) M ( τ ,τ ) | T d s ψ || d ≤ s (ˇΓ) || d ≤ k small ψ | (cid:46) sup M ( τ ,τ ) (cid:16) r | d ≤ k small ψ | (cid:17) (cid:90) M ( τ ,τ ) r − δ (cid:16) | d ≤ s (Γ g ) | + r − | d ≤ s (Γ b ) | (cid:17) + (cid:32) sup ( trap ) M ( τ ,τ ) u + δ dec | d ≤ k small ψ | (cid:33) (cid:32) sup τ ∈ [ τ ,τ ] E sδ [ ψ ]( τ ) (cid:33) (cid:18)(cid:90) ( trap ) M ( τ ,τ ) | d ≤ s ˇΓ | (cid:19) (cid:46) (cid:32) sup M ( τ ,τ ) ru + δ dec trap | d ≤ k small ψ | (cid:33) D s [Γ]+ (cid:32) sup M ( τ ,τ ) ru + δ dec trap | d ≤ k small ψ | (cid:33) (cid:32) sup τ ∈ [ τ ,τ ] E sδ [ ψ ]( τ ) (cid:33) (cid:112) D s [Γ]where we have used the definition of D s [Γ]. We infer (cid:90) M ( τ ,τ ) r δ (cid:16) | d ≤ s (Γ g ) | + r − | d ≤ s (Γ b ) | (cid:17) | d ≤ k small ψ | + (cid:90) ( trap ) M ( τ ,τ ) | T d s ψ || d ≤ s (ˇΓ) || d ≤ k small ψ | (cid:46) λ − (cid:32) sup M ( τ ,τ ) ru + δ dec trap | d ≤ k small ψ | (cid:33) D s [Γ] + λ sup τ ∈ [ τ ,τ ] E sδ [ ψ ]( τ )for any λ > λ > Proposition 10.5.4. Let ψ a reduced 2-scalar satisfying (cid:3) ψ = f ( r, m ) Y (0) ψ + (cid:101) N , where the function f is smooth and positive, and where the vectorfield Y (0) has beenintroduced in Proposition 10.1.29 in connection with the redshift vectorfield and is givenby Y (0) := (cid:18) m ( r − m ) + Υ (cid:19) e + (cid:18) m ( r − m ) (cid:19) e . Also, assume that the Ricci coefficients and curvature components associated to the globalnull frame we are using satisfy the estimates of section 10.4.1 for k ≤ k small derivatives.Then, for any ≤ s ≤ k large − , we have (cid:90) ( int ) M ( τ ,τ ) ( d s +1 ψ ) (cid:46) E sδ [ ψ ]( τ ) + (cid:90) ( ext ) M r ≤ m ( τ ,τ ) ( d s +1 ψ ) + D s [Γ] sup ( int ) M ( τ ,τ ) ∪ ( ext ) M r ≤ m r | d ≤ k small ψ | + (cid:90) ( int ) M ( τ ,τ ) ∪ ( ext ) M r ≤ m (cid:16) ( d ≤ s ψ ) + ( d ≤ s +1 (cid:101) N ) (cid:17) . Proof. Recall from Proposition 10.1.29 that the redshift vectorfield is given by Y H := κ H Y (0) , κ H := κ (cid:32) Υ δ H (cid:33) , where κ is a positive bump function κ = κ ( r ), supported in the region in [ − , 2] and equalto 1 for [ − , ψ in ( int ) M , we consider (cid:101) ψ := (cid:101) κ (cid:18) r − m (1 + 2 δ H )2 m δ H (cid:19) where (cid:101) κ is a positive bump function κ = κ ( r ), supported in the region in ( −∞ , 1] andequal to 1 for ( −∞ , ( int ) M is included in r ≤ m (1+2 δ H ), we infer in particular (cid:101) ψ = ψ on ( int ) M , supp( (cid:101) ψ ) ⊂ ( int ) M ( τ , τ ) ∪ ( ext ) M r ≤ m (1+3 δ H ) . CHAPTER 10. REGGE-WHEELER TYPE EQUATIONS Also, we have, in view of the wave equation for ψ , (cid:3) (cid:101) ψ = f ( r, m ) Y (0) (cid:101) ψ + (cid:101) N (cid:48) where (cid:101) N (cid:48) satisfies (cid:90) ( int ) M ( τ ,τ ) ∪ ( ext ) M r ≤ m ( d ≤ s +1 (cid:101) N (cid:48) ) (cid:17) (cid:46) (cid:90) ( ext ) M r ≤ m ( τ ,τ ) ( d ≤ s +1 ψ ) + (cid:90) ( int ) M ( τ ,τ ) ∪ ( ext ) M r ≤ m ( d ≤ s +1 (cid:101) N ) . Since (cid:101) ψ = ψ on ( int ) M , it thus suffices to prove for (cid:101) ψ the following estimate (cid:90) M ( τ ,τ ) ( d s +1 (cid:101) ψ ) (cid:46) E sδ [ (cid:101) ψ ]( τ ) + D s [Γ] sup ( int ) M ( τ ,τ ) ∪ ( ext ) M r ≤ m r | d ≤ k small (cid:101) ψ | + (cid:90) ( int ) M ( τ ,τ ) ∪ ( ext ) M r ≤ m (cid:16) ( d ≤ s (cid:101) ψ ) + ( d ≤ s +1 (cid:101) N (cid:48) ) (cid:17) . This estimate follows from first deriving the corresponding estimate for s = 0 by using theredshift as a multiplier, and then by recover derivatives one by one using commutationwith T , d / and the redshift vectorfield. Note that • (cid:101) ψ is supported on r ≤ m (1 + 2 δ H ) and hence is estimated on2 m (1 − δ H ) ≤ r ≤ m (1 + 2 δ H )so that the redshift vectorfield Y H has good properties, both as a multiplier and asa commutator, on the support of (cid:101) ψ . • The term f ( r, m ) Y (0) yields a good sign when using Y H as a multiplier since thefunction f ( r, m ) is positive, and since Y H = κ H Y (0) .This concludes the proof of the proposition. ppendix AAPPENDIX TO CHAPTER 2 A.1 Proof of Proposition 2.2.9 In a neighborhood of a given sphere S , we consider a ( u, s, θ, ϕ ) coordinates system, where θ is such that e ( θ ) = 0. Then, in this coordinates system, we have ∂ s = e . Since we have ∂ s (cid:18)(cid:90) S f (cid:19) = (cid:90) S (cid:16) ∂ s f + g ( D e θ ∂ s , e θ ) f + g ( D e ϕ ∂ s , e ϕ ) f (cid:17) , we infer e (cid:18)(cid:90) S f (cid:19) = (cid:90) S ( e ( f ) + κf ) . In particular, choosing f = 1, we deduce1 | S | e ( | S | ) = κ and since | S | = 4 πr , e ( r ) = rκ . APPENDIX A. APPENDIX TO CHAPTER 2 Next, let ∂ u the coordinates vectorfield in the ( u, s, θ, ϕ ) coordinates system. We have ∂ u (cid:18)(cid:90) S f (cid:19) = (cid:90) S (cid:16) ∂ u f + g ( D e θ ∂ u , e θ ) f + g ( D e ϕ ∂ u , e ϕ ) f (cid:17) = (cid:90) S (cid:16) ∂ u f + g ( D e θ ∂ u , e θ ) f − g ( ∂ u , D e ϕ e ϕ ) f (cid:17) = (cid:90) S (cid:16) ∂ u f + g ( D e θ ∂ u , e θ ) f + g ( ∂ u , D a (Φ) e a ) f (cid:17) On the other hand, we have we have, see (2.2.42), ∂ u = ς (cid:18) e − 12 Ω e − √ γbe θ (cid:19) . We infer g ( D e θ ∂ u , e θ ) + g ( ∂ u , D a (Φ) e a ) = 12 ςκ − ς Ω κ − d/ ( ς √ γb )and thus ς (cid:18) e − 12 Ω e (cid:19) (cid:18)(cid:90) S f (cid:19) = (cid:90) S (cid:32) ς (cid:18) e − 12 Ω e − √ γbe θ (cid:19) f + 12 ςκf − ς Ω κf − d/ ( √ γb ) f (cid:33) . We deduce e (cid:18)(cid:90) S f (cid:19) = Ω e (cid:18)(cid:90) S f (cid:19) + ς − (cid:90) S (cid:32) ςe f − ς Ω e f + ςκf − ς Ω κf − d/ ( ς √ γbf ) (cid:33) . Next, we use e (cid:18)(cid:90) S f (cid:19) = (cid:90) S ( e ( f ) + κf )and (cid:90) S d/ ( ς √ γbf ) = 0 . .1. PROOF OF PROPOSITION 2.2.9 e (cid:18)(cid:90) S f (cid:19) = Ω (cid:90) S ( e ( f ) + κf ) + ς − (cid:90) S (cid:32) ςe f − ς Ω e f + ςκf − ς Ω κf (cid:33) = ς − (cid:90) S ς ( e f + κf ) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:90) S ( e f + κf ) − ς − Ω (cid:90) S ˇ ς ( e f + κf ) − ς − (cid:90) S ˇΩ ς ( e f + κf ) . We further write, ς − (cid:90) S ς ( e f + κf ) = ς − ς (cid:90) S ( e f + κf ) + ς − (cid:90) S ˇ ς ( e f + κf )= (cid:90) S ( e f + κf ) + ( ς − ς − (cid:90) S ( e f + κf ) + ς − (cid:90) S ˇ ς ( e f + κf )= (cid:90) S ( e f + κf ) − ς − ˇ ς (cid:90) S ( e f + κf ) + ς − (cid:90) S ˇ ς ( e f + κf ) . Hence, e (cid:18)(cid:90) S f (cid:19) = (cid:90) S ( e f + κf ) + Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) , Err (cid:20) e (cid:18)(cid:90) S f (cid:19)(cid:21) = − ς − ˇ ς (cid:90) S ( e f + κf ) + ς − (cid:90) S ˇ ς ( e f + κf )+ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:90) S ( e f + κf ) − ς − Ω (cid:90) S ˇ ς ( e f + κf ) − ς − (cid:90) S ˇΩ ς ( e f + κf )as desired.In particular, choosing f = 1, we infer1 | S | e ( | S | ) = κ − ς − ˇ ς κ + ς − ˇ ςκ + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) κ − ς − Ω ˇ ς κ − ς − ˇΩ ς κ = κ − ς − ˇ ς κ + ς − ˇ ς ˇ κ + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) κ − ς − Ω ˇ ς ˇ κ − ς − ˇΩ ς κ. Hence, since | S | = 4 πr , recalling the definition of A ,2 e ( r ) r = κ − ς − ˇ ς κ + ς − ˇ ς ˇ κ + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) κ − ς − Ω ˇ ς ˇ κ − ς − ˇΩ ς κ = κ + A. This concludes the proof of Proposition 2.2.9.80 APPENDIX A. APPENDIX TO CHAPTER 2 A.2 Proof of Proposition 2.2.16 We start with the proof for e ( m ). Recall that the Hawking mass m is given by theformula mr = 1 + π (cid:82) S κκ . Differentiating in the e direction, we deduce,2 e ( m ) r − me ( r ) r = 116 π e (cid:18)(cid:90) S κκ (cid:19) = 116 π (cid:90) S (cid:16) e ( κκ ) + κκ (cid:17) . Now, making use of the e transport equations of Proposition 2.2.8, e ( κκ ) = κ (cid:18) − κ − ϑ (cid:19) + κ (cid:18) − κκ + 2 ρ − d/ ζ − ϑϑ + 2 ζ (cid:19) = − κκ + 2 κρ − κ d/ ζ − κϑ − κϑϑ + 2 κζ . we infer2 e ( m ) r − mr κ = 116 π (cid:90) S (cid:18) κρ − κ d/ ζ − κϑ − κϑϑ + 2 κζ (cid:19) = 18 π | S | κ ρ + 116 π (cid:90) S (cid:18) κ ˇ ρ + 2 e θ ( κ ) ζ − κϑ − κϑϑ + 2 κζ (cid:19) = r κ ρ + 116 π (cid:90) S (cid:18) κ ˇ ρ + 2 e θ ( κ ) ζ − κϑ − κϑϑ + 2 κζ (cid:19) and hence e ( m ) = r κ (cid:18) ρ + 2 mr (cid:19) + r π (cid:90) S (cid:18) − κϑ − κϑϑ + 2ˇ κ ˇ ρ + 2 e θ ( κ ) ζ + 2 κζ (cid:19) . Using the identity ρ = − mr + πr (cid:82) S ϑϑ (see (2.2.12) of Proposition 2.2.4), we deduce e ( m ) = r π (cid:90) S (cid:18) − κϑ − 12 ( κ − κ ) ϑϑ + 2ˇ κ ˇ ρ + 2 e θ ( κ ) ζ + 2 κζ (cid:19) = r π (cid:90) S (cid:18) − κϑ − 12 ˇ κϑϑ + 2ˇ κ ˇ ρ + 2 e θ ( κ ) ζ + 2 κζ (cid:19) = r π (cid:90) S Err as desired.In the same vein,2 e ( m ) r − me ( r ) r = 116 π e (cid:18)(cid:90) S κκ (cid:19) = 116 π (cid:90) S (cid:16) e ( κκ ) + κ κ (cid:17) + E , .2. PROOF OF PROPOSITION 2.2.16 E the error term defined in Proposition 2.2.9 E = 116 π Err (cid:20) e (cid:18)(cid:90) S κκ (cid:19)(cid:21) . We make use of the e transport equations of Proposition 2.2.8, e ( κκ ) = κ (cid:18) − κ κ + 2 ωκ + 2 d/ η + 2 ρ − ϑ ϑ + 2 η (cid:19) + κ (cid:18) − κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) = − κκ + 2 κ d/ η + 2 κ d/ ξ + 2 ρκ + κ (cid:18) η − ϑϑ (cid:19) + 2 κ (cid:0) η − ζ (cid:1) ξ − κϑ . Therefore, setting E = π (cid:82) S (cid:16) κ (cid:0) η − ϑϑ (cid:1) + 2 κ (cid:0) η − ζ (cid:1) ξ − κϑ (cid:17) ,2 e ( m ) r − mr ( κ + A ) = 116 π (cid:90) S (cid:16) κ d/ η + 2 κ d/ ξ + 2 ρκ (cid:17) + E + E = 116 π (cid:90) S (cid:16) − e θ ( κ ) η − e θ ( κ ) ξ + 2( ρ + ˇ ρ )( κ + ˇ κ ) (cid:17) + E + E = 12 r ρ κ + 116 π (cid:90) S (cid:16) − e θ ( κ ) η − e θ ( κ ) ξ + 2 ˇ ρ ˇ κ (cid:17) + E + E = 12 r κ (cid:16) − mr + 116 πr (cid:90) S ϑϑ (cid:17) + 116 π (cid:90) S (cid:16) − e θ ( κ ) η − e θ ( κ ) ξ + 2 ˇ ρ ˇ κ (cid:17) + E + E . We deduce2 e ( m ) r = 116 π (cid:90) S (cid:16) − e θ ( κ ) η − e θ ( κ ) ξ + 2 ˇ ρ ˇ κ + 12 κϑϑ (cid:17) + E + 116 π (cid:90) S (cid:18) κ (cid:18) η − ϑϑ (cid:19) + 2 κ (cid:0) η − ζ (cid:1) ξ − κϑ (cid:19) + mr A = 116 π (cid:90) S (cid:16) − e θ ( κ ) η + 2 κη − e θ ( κ ) ξ + 2 κηξ − κϑ (cid:17) + 116 π (cid:90) S (cid:16) ρ ˇ κ − κ ζ ξ − 12 ˇ κϑ ϑ (cid:17) + E + mr A, i.e., e ( m ) = r π (cid:90) S (cid:16) − e θ ( κ ) η + 2 κη − e θ ( κ ) ξ + 2 κηξ − κϑ (cid:17) + r π (cid:90) S (cid:16) ρ ˇ κ − κ ζ ξ − 12 ˇ κϑ ϑ (cid:17) + r (cid:16) E + mr A (cid:17) . APPENDIX A. APPENDIX TO CHAPTER 2 It remains to calculate E + mr A . Using the definitions of E and A and grouping similarterms appropriately we find E + mr A = − ς − ˇ ς (cid:20) π (cid:90) S ( e ( κκ ) + κκ ) + mr κ (cid:21) + ς − (cid:20) π (cid:90) S ˇ ς ( e ( κκ ) + κκ ) + mr ˇ ς ˇ κ (cid:21) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:20) π (cid:90) S ( e ( κκ ) + κ κ ) + mr κ (cid:21) − ς − Ω (cid:20) π (cid:90) S ˇ ς ( e ( κκ ) + κ κ ) + mr ˇ ς ˇ κ (cid:21) − ς − (cid:20) π (cid:90) S ˇΩ ς ( e ( κκ ) + κ κ ) + mr ˇΩ ςκ (cid:21) . Now, we have from above calculations e ( κκ ) + κκ = 2 κρ − κ d/ ζ + Err[ e ( κκ )] , Err[ e ( κκ )] = − κϑ − κϑϑ + 2 κζ ,e ( κκ ) + κκ = 2 ρκ + 2 κ d/ η + 2 κ d/ ξ + Err[ e ( κκ )] , Err[ e ( κκ )] = κ (cid:18) η − ϑϑ (cid:19) + 2 κ (cid:0) η − ζ (cid:1) ξ − κϑ . We infer E + mr A = − ς − ˇ ς (cid:20) π (cid:90) S (2 ρκ + 2 κ d/ η + 2 κ d/ ξ + Err[ e ( κκ )]) + mr κ (cid:21) + ς − (cid:20) π (cid:90) S ˇ ς (2 ρκ + 2 κ d/ η + 2 κ d/ ξ + Err[ e ( κκ )]) + mr ˇ ς ˇ κ (cid:21) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:20) π (cid:90) S (2 κρ − κ d/ ζ + Err[ e ( κκ )]) + mr κ (cid:21) − ς − Ω (cid:20) π (cid:90) S ˇ ς (2 κρ − κ d/ ζ + Err[ e ( κκ )]) + mr ˇ ς ˇ κ (cid:21) − ς − (cid:20) π (cid:90) S ˇΩ ς (2 κρ − κ d/ ζ + Err[ e ( κκ )]) + mr ˇΩ ςκ (cid:21) .3. PROOF OF LEMMA 2.2.17 E + mr A = − ς − ˇ ς (cid:20) π (cid:90) S (cid:18) ρ ˇ κ − e θ ( κ ) η − e θ ( κ ) ξ + 12 κϑϑ + Err[ e ( κκ )] (cid:19)(cid:21) + ς − (cid:20) π (cid:90) S ˇ ς (2 ρ ˇ κ + 2 ˇ ρκ + 2 ˇ ρ ˇ κ + 2 κ d/ η + 2 κ d/ ξ + Err[ e ( κκ )]) + mr ˇ ς ˇ κ (cid:21) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:20) π (cid:90) S (cid:18) κ ˇ ρ + 2 e θ ( κ ) ζ + 12 κϑϑ + Err[ e ( κκ )] (cid:19)(cid:21) − ς − Ω (cid:20) π (cid:90) S ˇ ς (2 ρ ˇ κ + 2 ˇ ρκ + 2 ˇ ρ ˇ κ − κ d/ ζ + Err[ e ( κκ )]) + mr ˇ ς ˇ κ (cid:21) − ς − (cid:20) π (cid:90) S ˇΩ ς (2 ρ ˇ κ + 2 ˇ ρκ + 2 ˇ ρ ˇ κ − κ d/ ζ + Err[ e ( κκ )]) + mr ˇΩ ςκ (cid:21) . We deduce e ( m ) = (cid:0) − ς − ˇ ς (cid:1) r π (cid:90) S Err + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) r π (cid:90) S Err + ς − r π (cid:90) S ˇ ς (cid:0) ρ ˇ κ + 2 ˇ ρκ + 2 κ d/ η + 2 κ d/ ξ + Err (cid:1) − ς − r π (cid:90) S (Ωˇ ς + ˇΩ ς ) (2 ρ ˇ κ + 2 ˇ ρκ − κ d/ ζ + Err ) − mr ς − (cid:104) − ˇ ς ˇ κ + Ω ˇ ς ˇ κ + ˇΩ ςκ (cid:105) , where we have introducedErr = 2ˇ κ ˇ ρ + 2 e θ ( κ ) ζ + 12 κϑϑ + Err[ e ( κκ )] , Err = 2 ˇ ρ ˇ κ − e θ ( κ ) η − e θ ( κ ) ξ + 12 κϑϑ + Err[ e ( κκ )] , Err = 2 ˇ ρ ˇ κ + Err[ e ( κκ )] , Err = 2 ˇ ρ ˇ κ + Err[ e ( κκ )] . In view of the definition of Err[ e ( κκ )] and Err[ e ( κκ )], this concludes the proof of Propo-sition 2.2.16. A.3 Proof of Lemma 2.2.17 Recall that we have e ( κ ) = − κ − ϑ . APPENDIX A. APPENDIX TO CHAPTER 2 We infer e ( κ ) = e ( κ ) + ˇ κ = − κ − ϑ + ˇ κ = − κ − ϑ + 12 ˇ κ and hence e (cid:18) κ − r (cid:19) = − κ − ϑ + 12 ˇ κ + 2 r e ( r ) r = − κ + 1 r κ − ϑ + 12 ˇ κ = − κ (cid:18) κ − r (cid:19) − ϑ + 12 ˇ κ . Next, using e ( ω ) = ρ + ζ (2 η + ζ )we infer that e ( ω ) = e ( ω ) + ˇ κ ˇ ω = ρ + ζ (2 η + ζ ) + ˇ κ ˇ ω, and hence e (cid:16) ω − mr (cid:17) = e ( ω ) + 2 me ( r ) r − e ( m ) r = ρ + 2 mr + mr (cid:18) κ − r (cid:19) − e ( m ) r + 3 ζ (2 η + ζ ) + ˇ κ ˇ ω as stated.Next, using e ( κ ) + 12 κκ − ωκ = 2 d/ η + 2 ρ − ϑϑ + 2 η we deduce e ( κ ) = − κκ + 2 ωκ + 2 ρ − ϑϑ + 2 η = − κ κ + 2 ω κ + 2 ρ + 2ˇ ω ˇ κ − 12 ˇ κ ˇ κ − ϑϑ + 2 η . Making use of Corollary 2.2.11 e ( κ ) = e ( κ ) + Err[ e κ ]= − κ κ + 2 ω κ + 2 ρ + 2ˇ ω ˇ κ − 12 ˇ κ ˇ κ − ϑϑ + 2 η + Err[ e κ ] .3. PROOF OF LEMMA 2.2.17 e (cid:18) κ − r (cid:19) = e ( κ ) + 2 r r κ + A )= − κ κ + 2 ω κ + 2 ρ + 2ˇ ω ˇ κ − 12 ˇ κ ˇ κ − ϑϑ + 2 η + 1 r κ + 1 r A + Err[ e κ ]= − κ (cid:18) κ − r (cid:19) + 2 ω κ + 2 ρ + 2ˇ ω ˇ κ − 12 ˇ κ ˇ κ − ϑϑ + 2 η + 1 r A + Err[ e κ ] . Now, 2 ω κ + 2 ρ = 2 ω (cid:18) κ − r (cid:19) + 4 r ω + 2 ρ = 2 ω (cid:18) κ − r (cid:19) + 4 r (cid:16) ω − mr (cid:17) + 2 (cid:18) ρ + 2 mr (cid:19) . Hence, e (cid:18) κ − r (cid:19) + 12 κ (cid:18) κ − r (cid:19) = 2 ω (cid:18) κ − r (cid:19) + 4 r (cid:16) ω − mr (cid:17) + 2 (cid:18) ρ + 2 mr (cid:19) + 2 η + 2ˇ ω ˇ κ − 12 ˇ κ ˇ κ − ϑϑ + 1 r A + Err[ e κ ] . In view of Corollary 2.2.11 the error term Err[ e ( κ )] is given byErr[ e ( κ )] = − ς − ˇ ς ( e κ + κκ − κκ ) + ς − (cid:16) ˇ ς ( e κ + κκ ) − ˇ ς ˇ κ κ (cid:17) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:16) e κ + κ − κ (cid:17) − ς − Ω (cid:16) ˇ ς ( e κ + κ ) − ˇ ς ˇ κ κ (cid:17) − ς − (cid:16) ˇΩ ς ( e κ + κ ) − ˇΩ ς κ κ (cid:17) + ˇ κ ˇ κ. Together with the null structure equations for e ( κ ) and e ( κ ), we inferErr[ e ( κ )] = − ς − ˇ ς (cid:18) κκ + 2 ωκ + 2 ρ + 2 d/ η − ϑϑ + 2 η − κ κ (cid:19) + ς − (cid:32) ˇ ς (cid:18) κκ + 2 ωκ + 2 ρ + 2 d/ η − ϑϑ + 2 η (cid:19) − ˇ ς ˇ κ κ (cid:33) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) κ − ϑ − κ (cid:19) − ς − Ω (cid:32) ˇ ς (cid:18) κ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) − ς − (cid:32) ˇΩ ς (cid:18) κ − ϑ (cid:19) − ˇΩ ς κ κ (cid:33) + ˇ κ ˇ κ. APPENDIX A. APPENDIX TO CHAPTER 2 and henceErr[ e ( κ )] = − ς − (cid:18) − κ κ + 2 ω κ + 2 ρ (cid:19) ˇ ς − κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) − ς − ˇ ς (cid:18) 12 ˇ κ ˇ κ + 2ˇ ω ˇ κ − ϑϑ + 2 η (cid:19) + ς − (cid:32) ˇ ς (cid:18) κκ + 2 ωκ + 2 ˇ ρ + 2 d/ η − ϑϑ + 2 η (cid:19) − ˇ ς ˇ κ κ (cid:33) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) 12 ˇ κ − ϑ (cid:19) − ς − Ω (cid:32) ˇ ς (cid:18) κ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) − ς − (cid:32) ˇΩ ς (cid:18) κ − ϑ (cid:19) − ˇΩ ς κ κ (cid:33) + ˇ κ ˇ κ (A.3.1)so that, in view of the definition of A , we obtain e (cid:18) κ − r (cid:19) + 12 κ (cid:18) κ − r (cid:19) = 2 ω (cid:18) κ − r (cid:19) + 4 r (cid:16) ω − mr (cid:17) + 2 (cid:18) ρ + 2 mr (cid:19) − ς − (cid:18) − κ κ + 2 ω κ + 2 ρ (cid:19) ˇ ς − κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) − r ς − κ ˇ ς + 1 r κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) + Err (cid:20) e (cid:18) κ − r (cid:19)(cid:21) , withErr (cid:20) e (cid:18) κ − r (cid:19)(cid:21) = 2 η + 2ˇ ω ˇ κ − 12 ˇ κ ˇ κ − ϑϑ + 1 r ς − ˇ ς ˇ κ − r ς − Ω ˇ ς ˇ κ − r ς − ˇΩ ςκ − ς − ˇ ς (cid:18) 12 ˇ κ ˇ κ + 2ˇ ω ˇ κ − ϑϑ + 2 η (cid:19) + ς − (cid:32) ˇ ς (cid:18) κκ + 2 ωκ + 2 ˇ ρ + 2 d/ η − ϑϑ + 2 η (cid:19) − ˇ ς ˇ κ κ (cid:33) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) 12 ˇ κ − ϑ (cid:19) − ς − Ω (cid:32) ˇ ς (cid:18) κ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) − ς − (cid:32) ˇΩ ς (cid:18) κ − ϑ (cid:19) − ˇΩ ς κ κ (cid:33) + ˇ κ ˇ κ. This concludes the proof of Lemma 2.2.17. .4. PROOF OF PROPOSITION 2.2.18 A.4 Proof of Proposition 2.2.18 In view of Corollary 2.2.11 applied to e ( κ ) + 12 κ = − ϑ , we deduce, e ˇ κ + κ ˇ κ = − 12 ˇ κ − 12 ˇ κ − 12 ( ϑ − ϑ ) . In view of Corollary 2.2.11 applied to e ( κ ) + 12 κκ = − d/ ζ + 2 ρ − ϑϑ + 2 ζ we deduce, e ˇ κ + 12 κ ˇ κ + 12 ˇ κκ = − 12 ˇ κ ˇ κ − 12 ˇ κ ˇ κ + F − F where, F − F = (cid:18) − d/ ζ + 2 ρ − ϑϑ + 2 ζ (cid:19) − (cid:18) − d/ ζ + 2 ρ − ϑϑ + 2 ζ (cid:19) = − d/ ζ + 2 ˇ ρ + (cid:18) − ϑϑ + 2 ζ (cid:19) − (cid:18) − ϑϑ + 2 ζ (cid:19) . Hence, e ˇ κ + 12 κ ˇ κ + 12 ˇ κκ = − d/ ζ + 2 ˇ ρ + Err[ e ˇ κ ]Err[ e ˇ κ ] : = − 12 ˇ κ ˇ κ − 12 ˇ κ ˇ κ + (cid:18) − ϑϑ + 2 ζ (cid:19) − (cid:18) − ϑϑ + 2 ζ (cid:19) . In view of Corollary 2.2.11 applied to e ( ω ) = ρ + 3 ζ we deduce, e ˇ ω = − ˇ κ ˇ ω + ( ρ + 3 ζ ) − ( ρ + 3 ζ ) = ˇ ρ − ˇ κ ˇ ω + 3( ζ − ζ ) . In view of Corollary 2.2.11 applied to e ( ρ ) + 32 κρ = d/ β − ϑα − ζβ APPENDIX A. APPENDIX TO CHAPTER 2 we deduce, e ˇ ρ + 32 κ ˇ ρ + 32 ρ ˇ κ = − 32 ˇ κ ˇ ρ + 12 ˇ κ ˇ ρ + d/ β − (cid:18) ϑα + ζβ (cid:19) + (cid:18) ϑα + ζβ (cid:19) .e µ + 32 κµ = Err[ e µ ] , we deduce e ˇ µ + 32 κ ˇ µ + 32 µ ˇ κ = − 32 ˇ κ ˇ µ + 12 ˇ κ ˇ µ + Err[ e µ ] − Err[ e µ ] . In view of Corollary 2.2.11 applied to e (Ω) = − ω we deduce, − e ( ˇΩ) = 2ˇ ω − ˇ κ ˇΩas stated.In view of Corollary 2.2.11 applied to the equation e ( κ ) + 12 κκ = 2 d/ η + 2 ρ + 2 η + 2 ωκ − ϑϑ to deduce, e (ˇ κ ) = e ( κ ) − e ( κ ) − Err[ e ( κ )]= − κκ + 2 d/ η + 2 ρ + 2 η + 2 ωκ − ϑϑ + 12 κκ − ρ − η − ωκ + 12 ϑϑ − Err[ e κ ]= 2 d/ η + 2 ˇ ρ − 12 ( κ ˇ κ + κ ˇ κ ) + 2 ( ω ˇ κ + κ ˇ ω )+2 (cid:16) η − η (cid:17) − 12 ˇ κ ˇ κ + 2ˇ ω ˇ κ − (cid:0) ϑϑ − ϑϑ (cid:1) − Err[ e κ ] . .4. PROOF OF PROPOSITION 2.2.18 e ( κ )] = − ς − (cid:18) − κ κ + 2 ω κ + 2 ρ (cid:19) ˇ ς − κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) − ς − ˇ ς (cid:18) 12 ˇ κ ˇ κ + 2ˇ ω ˇ κ − ϑϑ + 2 η (cid:19) + ς − (cid:32) ˇ ς (cid:18) κκ + 2 ωκ + 2 ˇ ρ + 2 d/ η − ϑϑ + 2 η (cid:19) − ˇ ς ˇ κ κ (cid:33) + (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) 12 ˇ κ − ϑ (cid:19) − ς − Ω (cid:32) ˇ ς (cid:18) κ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) − ς − (cid:32) ˇΩ ς (cid:18) κ − ϑ (cid:19) − ˇΩ ς κ κ (cid:33) + ˇ κ ˇ κ. We deduce e (ˇ κ ) = 2 d/ η + 2 ˇ ρ − 12 ( κ ˇ κ + κ ˇ κ ) + 2 ( ω ˇ κ + κ ˇ ω )+ ς − (cid:18) − κ κ + 2 ω κ + 2 ρ (cid:19) ˇ ς + 12 κ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) +2 (cid:16) η − η (cid:17) − 12 ˇ κ ˇ κ + 2ˇ ω ˇ κ − (cid:0) ϑϑ − ϑϑ (cid:1) + ς − ˇ ς (cid:18) 12 ˇ κ ˇ κ + 2ˇ ω ˇ κ − ϑϑ + 2 η (cid:19) − ς − (cid:32) ˇ ς (cid:18) κκ + 2 ωκ + 2 ˇ ρ + 2 d/ η − ϑϑ + 2 η (cid:19) − ˇ ς ˇ κ κ (cid:33) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) 12 ˇ κ − ϑ (cid:19) + ς − Ω (cid:32) ˇ ς (cid:18) κ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) + ς − (cid:32) ˇΩ ς (cid:18) κ − ϑ (cid:19) − ˇΩ ς κ κ (cid:33) − ˇ κ ˇ κ as desired.In view of Corollary 2.2.11 applied to the equation e ( κ ) + 12 κ = 2 d/ ξ − ω κ + 2( η − ζ ) ξ − ϑ we deduce, e (ˇ κ ) + κ ˇ κ = 2 d/ ξ − ω κ + ω ˇ κ ) − 12 ˇ κ − ω ˇ κ + 2( η − ζ ) ξ − η − ζ ) ξ − (cid:16) ϑ − ϑ (cid:17) − Err[ e κ ]90 APPENDIX A. APPENDIX TO CHAPTER 2 where, − Err[ e κ ] = ς − ˇ ς (cid:16) e κ + κ − κκ (cid:17) − ς − (cid:16) ˇ ς ( e κ + κ ) − ˇ ς ˇ κ κ (cid:17) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:16) e κ + κκ ) − κ κ (cid:17) + ς − Ω (cid:16) ˇ ς ( e κ + κκ ) − ˇ ς ˇ κ κ (cid:17) + ς − (cid:16) ˇΩ ς ( e κ + κκ ) − ˇΩ ς κ κ (cid:17) − ˇ κ . In view of the null structure equations for e ( κ ) and e ( κ ), we infer − Err[ e κ ] = ς − ˇ ς (cid:18) − κ − ω κ (cid:19) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) − κ κ + 2 ρ (cid:19) + ς − ˇ ς (cid:18) 12 ˇ κ − ω ˇ κ + 2( η − ζ ) ξ − ϑ (cid:19) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) 12 ˇ κ ˇ κ − ϑ ϑ + 2 ζ (cid:19) − ς − (cid:32) ˇ ς (cid:18) κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) + ς − Ω (cid:32) ˇ ς (cid:18) κ κ − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ (cid:19) − ˇ ς ˇ κ κ (cid:33) + ς − (cid:32) ˇΩ ς (cid:18) κ κ − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ (cid:19) − ˇΩ ς κ κ (cid:33) − ˇ κ and hence e (ˇ κ ) + κ ˇ κ = 2 d/ ξ − ω κ + ω ˇ κ ) + ς − ˇ ς (cid:18) − κ − ω κ (cid:19) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) − κ κ + 2 ρ (cid:19) + Err[ e (ˇ κ )] , Err[ e (ˇ κ )] = − 12 ˇ κ − ω ˇ κ + 2( η − ζ ) ξ − η − ζ ) ξ − (cid:16) ϑ − ϑ (cid:17) − ς − (cid:32) ˇ ς (cid:18) κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) − ˇ ς ˇ κ κ (cid:33) + ς − Ω (cid:32) ˇ ς (cid:18) κ κ − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ (cid:19) − ˇ ς ˇ κ κ (cid:33) + ς − (cid:32) ˇΩ ς (cid:18) κ κ − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ (cid:19) − ˇΩ ς κ κ (cid:33) − ˇ κ as desired. .4. PROOF OF PROPOSITION 2.2.18 e ( ρ ) + 32 κρ = d/ β − ϑα − ζβ + 2 ηβ + 2 ξβ we deduce, e ˇ ρ + 32 κ ˇ ρ + 32 ˇ κρ = d/ β − (cid:18) ϑα + ζβ − ηβ − ξβ (cid:19) + (cid:18) ϑα + ζβ − ηβ − ξβ (cid:19) − 32 ˇ κ ˇ ρ − Err[ e ρ ]where − Err[ e ( ρ )] = ς − ˇ ς ( e ρ + κρ − κρ ) − ς − (cid:16) ˇ ς ( e ρ + κρ ) − ˇ ς ˇ κ ρ (cid:17) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:16) e ρ + κρ ) − κ ρ (cid:17) + ς − Ω (cid:16) ˇ ς ( e ρ + κρ ) − ˇ ς ˇ κ ρ (cid:17) + ς − (cid:16) ˇΩ ς ( e ρ + κρ ) − ˇΩ ς κ (cid:17) − ˇ κ ˇ ρ. In view of the null structure equations for e ( ρ ) and e ( ρ ), we infer − Err[ e ( ρ )] = − κ ρς − ˇ ς + 32 κ ρ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) + ς − ˇ ς (cid:18) − 12 ˇ κ ˇ ρ − ϑ α − ζ β + 2( η β + ξ β ) (cid:19) − ς − (cid:32) ˇ ς (cid:18) − κρ + d/ β − ϑ α − ζ β + 2( η β + ξ β ) (cid:19) − ˇ ς ˇ κ ρ (cid:33) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) − 12 ˇ κ ˇ ρ − ϑ α − ζβ (cid:19) + ς − Ω (cid:32) ˇ ς (cid:18) − κρ + d/ β − ϑ α − ζβ (cid:19) − ˇ ς ˇ κ ρ (cid:33) + ς − (cid:32) ˇΩ ς (cid:18) − κρ + d/ β − ϑ α − ζβ (cid:19) − ˇΩ ς κ (cid:33) − ˇ κ ˇ ρ. APPENDIX A. APPENDIX TO CHAPTER 2 and hence e ˇ ρ + 32 κ ˇ ρ = − ρ ˇ κ + d/ β − κ ρς − ˇ ς + 32 κ ρ (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) + Err[ e ˇ ρ ] , Err[ e ˇ ρ ] = − (cid:18) ϑα + ζβ − ηβ − ξβ (cid:19) + (cid:18) ϑα + ζβ − ηβ − ξβ (cid:19) − 32 ˇ κ ˇ ρ + ς − ˇ ς (cid:18) − 12 ˇ κ ˇ ρ − ϑ α − ζ β + 2( η β + ξ β ) (cid:19) − ς − (cid:32) ˇ ς (cid:18) − κρ + d/ β − ϑ α − ζ β + 2( η β + ξ β ) (cid:19) − ˇ ς ˇ κ ρ (cid:33) − (cid:0) ˇΩ + ς − Ωˇ ς (cid:1) (cid:18) − 12 ˇ κ ˇ ρ − ϑ α − ζβ (cid:19) + ς − Ω (cid:32) ˇ ς (cid:18) − κρ + d/ β − ϑ α − ζβ (cid:19) − ˇ ς ˇ κ ρ (cid:33) + ς − (cid:32) ˇΩ ς (cid:18) − κρ + d/ β − ϑ α − ζβ (cid:19) − ˇΩ ς κ (cid:33) − ˇ κ ˇ ρ, which ends the proof of Proposition 2.2.18. A.5 Proof of Proposition 2.2.19 In view of the null structure equation for e ( ζ ), we have12 κξ + 2 d (cid:63) / ω = e ( ζ ) + 12 κ ( ζ + η ) − ω ( ζ − η ) − β + 12 ϑ ( ζ + η ) − ϑξ and hence 12 κξ + 2 d (cid:63) / ω = (cid:18) κ + 2 ω + 12 ϑ (cid:19) η + e ( ζ ) − β + 12 κζ − ωζ + 12 ϑζ − ϑξ which is the first desired identity.To prove the second identity we start with e ( κ ) + 12 κ κ − ωκ = 2 d/ η + 2 ρ − ϑ ϑ + 2 η . .5. PROOF OF PROPOSITION 2.2.19 e θ , e ( e θ ( κ )) + [ e θ , e ] κ + 12 κe θ ( κ ) + 12 κe θ ( κ ) − ωe θ ( κ ) − κe θ ( ω )= 2 e θ ( d/ η ) + 2 e θ ( ρ ) − e θ ( ϑ ϑ ) + 2 e θ ( η ) . Since [ e θ , e ] κ = ( κ + ϑ ) e θ κ + ( ζ − η ) e κ − ξe κ we deduce,2 e θ ( d/ η ) + ηe ( κ ) + 2 e θ ( η ) = − ξe ( κ ) − κe θ ( ω ) + e ( e θ ( κ )+ 12 ( κ + ϑ ) e θ ( κ ) + ζe ( κ ) + 12 κe θ ( κ ) + 12 κe θ ( κ ) − ωe θ ( κ ) − e θ ( ρ ) + 12 e θ ( ϑ ϑ ) , or, making use of the equations for e κ and e κ in Proposition 2.2.8,2 e θ ( d/ η ) + (cid:18) − κκ + 2 ωκ + 2 d/ η + 2 ρ − ϑϑ + 2 η (cid:19) η + 2 e θ ( η )= − (cid:18) − κ − ϑ (cid:19) ξ − κe θ ( ω ) + e ( e θ ( κ ))+ 12 ( κ + ϑ ) e θ ( κ ) + (cid:18) − κκ + 2 ωκ + 2 d/ η + 2 ρ − ϑϑ + 2 η (cid:19) ζ + 12 κe θ ( κ ) + 12 κe θ ( κ ) − ωe θ ( κ ) − e θ ( ρ ) + 12 e θ ( ϑ ϑ ) . Since e θ = − d (cid:63) / , d (cid:63) / d/ = d/ d (cid:63) / + 2 K and K = − ρ − κκ + ϑϑ , we infer that (cid:16) − d/ d (cid:63) / + 12 κκ + 2 ωκ + 2 d/ η + 6 ρ − ϑϑ + 2 η (cid:17) η + 2 e θ ( η )= κ (cid:18) κξ + 2 d (cid:63) / ω (cid:19) + e ( e θ ( κ ))+ 12 ( κ + ϑ ) e θ ( κ ) + (cid:18) − κκ + 2 ωκ + 2 d/ η + 2 ρ − ϑϑ + 2 η (cid:19) ζ + 12 κe θ ( κ ) + 12 κe θ ( κ ) − ωe θ ( κ ) − e θ ( ρ ) + 12 e θ ( ϑ ϑ ) + 12 ϑ ξ. Making use of the previously derived identity,2 d (cid:63) / ω + 12 κξ = (cid:18) κ + 2 ω + 12 ϑ (cid:19) η + e ( ζ ) − β + 12 κζ − ωζ + 12 ϑζ − ϑξ, APPENDIX A. APPENDIX TO CHAPTER 2 we infer that, (cid:16) − d/ d (cid:63) / + 12 κκ + 2 ωκ + 2 d/ η + 6 ρ − ϑϑ + 2 η (cid:17) η + 2 e θ ( η )= κ (cid:18)(cid:18) κ + 2 ω + 12 ϑ (cid:19) η + e ( ζ ) − β (cid:19) + κ (cid:18) κζ − ωζ + 12 ϑζ − ϑξ (cid:19) + e ( e θ ( κ ))+ 12 ( κ + ϑ ) e θ ( κ ) + (cid:18) − κκ + 2 ωκ + 2 d/ η + 2 ρ − ϑϑ + 2 η (cid:19) ζ + 12 κe θ ( κ ) + 12 κe θ ( κ ) − ωe θ ( κ ) − e θ ( ρ ) + 12 e θ ( ϑ ϑ ) + 12 ϑ ξ, or, (cid:18) − d/ d (cid:63) / + 6 ρ + 2 d/ η − κϑ − ϑϑ + 2 η (cid:19) η + 2 e θ ( η )= κ (cid:0) e ( ζ ) − β (cid:1) + e ( e θ ( κ ))+ κ (cid:18) κζ − ωζ + 12 ϑζ − ϑξ (cid:19) + 12 ( κ + ϑ ) e θ ( κ ) + (cid:18) − κκ + 2 ωκ + 2 d/ η + 2 ρ − ϑϑ + 2 η (cid:19) ζ + 12 κe θ ( κ ) + 12 κe θ ( κ ) − ωe θ ( κ ) − e θ ( ρ ) + 12 e θ ( ϑ ϑ ) + 12 ϑ ξ and hence (cid:18) d/ d (cid:63) / − d/ η + 12 κϑ − η (cid:19) η − e θ ( η )= κ (cid:0) − e ( ζ ) + β (cid:1) − e ( e θ ( κ )) − κ (cid:18) κζ − ωζ + 12 ϑζ − ϑξ (cid:19) + 6 ρη − ϑϑη − 12 ( κ + ϑ ) e θ ( κ ) − (cid:18) − κκ + 2 ωκ + 2 d/ η + 2 ρ − ϑϑ + 2 η (cid:19) ζ − κe θ ( κ ) − κe θ ( κ ) + 2 ωe θ ( κ ) + 2 e θ ( ρ ) − e θ ( ϑ ϑ ) − ϑ ξ which is the second desired identity.To prove the third identity we start with, e ( κ ) + 12 κ + 2 ω κ = 2 d/ ξ + 2( η − ζ ) ξ − ϑ . .5. PROOF OF PROPOSITION 2.2.19 e θ = − d (cid:63) / and using d (cid:63) / d/ = d/ d (cid:63) / + 2 K as before, e ( e θ ( κ )) + [ e θ , e ] κ + κe θ ( κ ) + 2 ωe θ ( κ ) + 2 κe θ ( ω )= − d (cid:63) / d/ ξ + 2 e θ (cid:16) ( η − ζ ) ξ (cid:17) − e θ ( ϑ )= − d/ d (cid:63) / + 2 K ) ξ + 2 e θ (cid:16) ( η − ζ ) ξ (cid:17) − e θ ( ϑ ) . Thus, since [ e θ , e ] κ = ( κ + ϑ ) e θ κ + ζ − η ) e κ − ξe κ , − d/ d (cid:63) / + 2 K ) ξ = e ( e θ ( κ )) + 2 κe θ ( ω ) + 12 ( κ + ϑ ) e θ κ + ( ζ − η ) e κ − ξe κ + κe θ ( κ ) + 2 ωe θ ( κ ) − e θ (cid:16) ( η − ζ ) ξ (cid:17) + 12 e θ ( ϑ ) . Making use of the equations for e κ, e κ in Proposition 2.2.82( d/ d (cid:63) / + 2 K ) ξ = 2 κ d (cid:63) / ω − e ( e θ ( κ ) − 12 ( κ + ϑ ) e θ κ − ( ζ − η ) (cid:18) − κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) + ξ (cid:18) − κ κ − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ (cid:19) − κe θ ( κ ) − ωe θ ( κ ) + 2 e θ (cid:16) ( η − ζ ) ξ (cid:17) − e θ ( ϑ ) . We deduce,2( d/ d (cid:63) / + K ) ξ = 2 κ d (cid:63) / ω + η (cid:18) − κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) + 2 e θ (cid:16) ( η − ζ ) ξ (cid:17) − e ( e θ ( κ )) − e θ ( ϑ ) − 12 ( κ + ϑ ) e θ ( κ ) − ζ (cid:18) − κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) + ξ (cid:18) − κ κ − K − d/ ζ + 2 ρ − ϑ ϑ + 2 ζ (cid:19) − κe θ ( κ ) − ωe θ ( κ ) − ηζξ − e θ ( ζξ ) . Making use of K = − ρ − κκ + ϑϑ and reorganizing we deduce,2( d/ d (cid:63) / + K ) ξ = 2 κ d (cid:63) / ω + η (cid:18) − κ − ω κ + 2 d/ ξ + 2 ηξ − ϑ (cid:19) + 2 e θ ( ηξ ) − e ( e θ ( κ )) − e θ ( ϑ ) − 12 ( κ + ϑ ) e θ ( κ ) − ζ (cid:18) − κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) + ξ (cid:16) ρ − ϑϑ − d/ ζ + 2 ζ (cid:17) − κe θ ( κ ) − ωe θ ( κ ) − ηζξ − e θ ( ζξ ) . APPENDIX A. APPENDIX TO CHAPTER 2 We make use again of the identity,2 d (cid:63) / ω + 12 κξ = (cid:18) κ + 2 ω + 12 ϑ (cid:19) η + e ( ζ ) − β + 12 κζ − ωζ + 12 ϑζ − ϑξ, to derive,2( d/ d (cid:63) / + K ) ξ = κ (cid:18) − κξ + (cid:18) κ + 2 ω + 12 ϑ (cid:19) η + e ( ζ ) − β (cid:19) + κ (cid:18) κζ − ωζ + 12 ϑζ − ϑξ (cid:19) + η (cid:18) − κ − ω κ + 2 d/ ξ + 2 ηξ − ϑ (cid:19) + 2 e θ ( ηξ ) − e ( e θ ( κ )) − e θ ( ϑ ) − 12 ( κ + ϑ ) e θ ( κ ) − ζ (cid:18) − κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) + ξ (cid:16) ρ − ϑϑ − d/ ζ + 2 ζ (cid:17) − κe θ ( κ ) − ωe θ ( κ ) − ηζξ − e θ ( ζξ ) . Grouping terms and using once more the identity K = − ρ − κκ + ϑϑ we deduce,2 d/ d (cid:63) / ξ = − e ( e θ ( κ )) + κ (cid:0) e ( ζ ) − β (cid:1) + (cid:18) d/ ξ + 12 κ ϑ + 2 ηξ − ϑ (cid:19) η + 2 e θ ( ηξ ) − e θ ( ϑ )+ κ (cid:18) κζ − ωζ + 12 ϑζ − ϑξ (cid:19) − 12 ( κ + ϑ ) e θ ( κ ) − ϑϑξ − ζ (cid:18) − κ − ω κ + 2 d/ ξ + 2( η − ζ ) ξ − ϑ (cid:19) + ξ (cid:16) ρ − ϑϑ − d/ ζ + 2 ζ (cid:17) − κe θ ( κ ) − ωe θ ( κ ) − ηζξ − e θ ( ζξ )which is the third desired identity. This concludes the proof of Proposition 2.2.19. A.6 Proof of Proposition 2.3.4 The proof follows by straightforward calculations using the definition of Ricci coefficientsand curvature components with respect to the two frames. Recall the transformation .6. PROOF OF PROPOSITION 2.3.4 e (cid:48) = λ (cid:18) e + f e θ + 14 f e (cid:19) ,e (cid:48) θ = (cid:18) f f (cid:19) e θ + 12 f e + 12 f (cid:18) f f (cid:19) e ,e (cid:48) = λ − (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19) . We first derive the transformation formulae for κ . We have, under a transformation oftype (2.3.3), χ (cid:48) = g ( D e (cid:48) θ e (cid:48) , e θ (cid:48) )= g (cid:18) D e (cid:48) θ (cid:18) λ (cid:18) e + f e θ + 14 f e (cid:19)(cid:19) , e (cid:48) θ (cid:19) = λg (cid:18) D e (cid:48) θ (cid:18) e + f e θ + 14 f e (cid:19) , e (cid:48) θ (cid:19) = λg (cid:0) D e (cid:48) θ e , e (cid:48) θ (cid:1) + λe (cid:48) θ ( f ) g ( e θ , e (cid:48) θ ) + λ e (cid:48) θ ( f ) g ( e , e (cid:48) θ ) + λf g (cid:0) D e (cid:48) θ e θ , e (cid:48) θ (cid:1) + λ f g ( D e (cid:48) θ e , e (cid:48) θ )= λg (cid:0) D e (cid:48) θ e , e (cid:48) θ (cid:1) + λ (cid:18) f f (cid:19) e (cid:48) θ ( f ) − λ f e (cid:48) θ ( f ) + λf g (cid:0) D e (cid:48) θ e θ , e (cid:48) θ (cid:1) + λ f χ + l.o.t.We recall that the lower order terms we denote by l.o.t., here and throughout the proof, arelinear with respect Γ = { ξ, ξ, ϑ, κ, η, η, ζ, κ, ϑ } and quadratic or higher order in f, f , anddo not contain derivatives of these latter. We also recall that χ = ( κ + ϑ ), χ = ( κ + ϑ ).Next, we compute g (cid:0) D e (cid:48) θ e , e (cid:48) θ (cid:1) = g (cid:18) D e (cid:48) θ e , (cid:18) f f (cid:19) e θ + 12 f e (cid:19) + l.o.t.= (cid:18) f f (cid:19) g (cid:16) D ( ff ) e θ + fe + fe e , e θ (cid:17) + 12 f g (cid:16) D e θ + fe + fe e , e (cid:17) + l.o.t.= (cid:0) f f (cid:1) χ + f ξ + f η + f ζ + f f ω − f ω + l.o.t. , and f g (cid:0) D e (cid:48) θ e θ , e (cid:48) θ (cid:1) = 12 f f g (cid:0) D e (cid:48) θ e θ , e (cid:1) + 12 f g (cid:0) D e (cid:48) θ e θ , e (cid:1) = − f f χ − f χ + l.o.t.98 APPENDIX A. APPENDIX TO CHAPTER 2 This yields χ (cid:48) = λg (cid:0) D e (cid:48) θ e , e (cid:48) θ (cid:1) + λ (cid:18) f f (cid:19) e (cid:48) θ ( f ) − λ f e (cid:48) θ ( f ) + λf g (cid:0) D e (cid:48) θ e θ , e (cid:48) θ (cid:1) + λ f χ + l.o.t.= λ (cid:32) χ + (cid:18) f f (cid:19) e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + f ( ζ + η ) + f ξ − f χ + 12 f f χ + f f ω − f ω +l.o.t. (cid:33) . Hence, κ (cid:48) = χ (cid:48) + e (cid:48) Φ = χ (cid:48) + λ (cid:18) e + f e θ + 14 f e (cid:19) Φ= λ (cid:32) κ + e (cid:48) θ ( f ) + e θ (Φ) f + 18 ( κ − ϑ ) f + 12 f f e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + f ( ζ + η ) + f ξ − f χ + 12 f f χ + f f ω − f ω + l.o.t. (cid:33) = λ (cid:18) κ + d/ (cid:48) ( f ) + 12 f f e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) − f κ + f ( ζ + η ) + f ξ + f f ω − f ω + l.o.t. (cid:19) and ϑ (cid:48) = χ (cid:48) − e (cid:48) Φ = χ (cid:48) − λ (cid:18) e + f e θ + 14 f e (cid:19) Φ= λ (cid:32) ϑ + e (cid:48) θ ( f ) − e θ (Φ) f − 18 ( κ − ϑ ) f + 12 f f e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + f ( ζ + η ) + f ξ − f χ + 12 f f χ + f f ω − f ω + l.o.t. (cid:33) = λ (cid:18) ϑ − d (cid:63) / (cid:48) ( f ) + 12 f f e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + 14 f f κ + f ( ζ + η ) + f ξ + f f ω − f ω + l.o.t. (cid:19) . This yields κ (cid:48) = λ ( κ + d/ (cid:48) ( f )) + λ Err( κ, κ (cid:48) ) , Err( κ, κ (cid:48) ) = 12 f f e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + f ( ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t.= f ( ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t. .6. PROOF OF PROPOSITION 2.3.4 ϑ (cid:48) = λ ( ϑ − d (cid:63) / (cid:48) ( f )) + λ Err( ϑ, ϑ (cid:48) ) , Err( ϑ, ϑ (cid:48) ) = 12 f f e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + f ( ζ + η ) + f ξ + 14 f f κ + f f ω − f ω + l.o.t.= f ( ζ + η ) + f ξ + 14 f f κ + f f ω − f ω + l.o.t.Next, we derive the transformation formula for κ and ϑ . We have, under a transformationof type (2.3.3), χ (cid:48) = g ( D e (cid:48) θ e (cid:48) , e θ (cid:48) )= g (cid:18) D e (cid:48) θ (cid:18) λ − (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19)(cid:19) , e (cid:48) θ (cid:19) = λ − g (cid:18) D e (cid:48) θ (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19) , e (cid:48) θ (cid:19) = λ − e (cid:48) θ (cid:18) f f + 18 f f (cid:19) g ( e , e (cid:48) θ ) + λ − e (cid:48) θ (cid:18) f (cid:18) f f (cid:19)(cid:19) g ( e θ , e (cid:48) θ ) + λ − e (cid:48) θ (cid:0) f (cid:1) g ( e , e (cid:48) θ )+ λ − (cid:18) f f + 116 f f (cid:19) g (cid:0) D e (cid:48) θ e , e (cid:48) θ (cid:1) + λ − f (cid:18) f f (cid:19) g (cid:0) D e (cid:48) θ e θ , e (cid:48) θ (cid:1) + 14 λ − f g (cid:0) D e (cid:48) θ e , e (cid:48) θ (cid:1) = − λ − f e (cid:48) θ (cid:18) f f + 18 f f (cid:19) + λ − (cid:18) f f (cid:19) e (cid:48) θ (cid:18) f (cid:18) f f (cid:19)(cid:19) − λ − f (cid:18) f f (cid:19) e (cid:48) θ (cid:0) f (cid:1) + λ − (cid:18) f f (cid:19) g (cid:0) D e (cid:48) θ e , e (cid:48) θ (cid:1) + λ − f g (cid:0) D e (cid:48) θ e θ , e (cid:48) θ (cid:1) + 14 λ − f χ + l.o.t.Then, we easily derive by symmetry from the formula for κ and ϑκ (cid:48) = λ − (cid:0) κ + d/ (cid:48) ( f ) (cid:1) + λ − Err( κ, κ (cid:48) ) , Err( κ, κ (cid:48) ) = − f e (cid:48) θ (cid:18) f f + 18 f f (cid:19) + (cid:18) f f + 18 ( f f ) (cid:19) e (cid:48) θ ( f ) + 14 (cid:18) f f (cid:19) f e (cid:48) θ (cid:0) f f (cid:1) − f (cid:18) f f (cid:19) e (cid:48) θ (cid:0) f (cid:1) + f ( − ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t.= − f e (cid:48) θ ( f ) + f ( − ζ + η ) + f ξ − f κ + f f ω − f ω + l.o.t.00 APPENDIX A. APPENDIX TO CHAPTER 2 and ϑ (cid:48) = λ (cid:0) ϑ − d (cid:63) / (cid:48) ( f ) (cid:1) + λ − Err( ϑ, ϑ (cid:48) ) , Err( ϑ, ϑ (cid:48) ) = − f e (cid:48) θ (cid:18) f f + 18 f f (cid:19) + (cid:18) f f + 18 ( f f ) (cid:19) e (cid:48) θ ( f ) + 14 (cid:18) f f (cid:19) f e (cid:48) θ (cid:0) f f (cid:1) − f (cid:18) f f (cid:19) e (cid:48) θ (cid:0) f (cid:1) + f ( − ζ + η ) + f ξ + 14 f f κ + f f ω − f ω + l.o.t.= − f e (cid:48) θ ( f ) + f ( − ζ + η ) + f ξ + 14 f f κ + f f ω − f ω + l.o.t.Next, we derive the transformation formula for ζ . We have, under a transformation oftype (2.3.3),2 ζ (cid:48) = g ( D e (cid:48) θ e (cid:48) , e (cid:48) )= g (cid:18) D e (cid:48) θ (cid:18) λ (cid:18) e + f e θ + 14 f e (cid:19)(cid:19) , e (cid:48) (cid:19) = − e (cid:48) θ (log( λ )) + λg (cid:18) D e (cid:48) θ (cid:18) e + f e θ + 14 f e (cid:19) , e (cid:48) (cid:19) = − e (cid:48) θ (log( λ )) + λe (cid:48) θ ( f ) g ( e θ , e (cid:48) ) + 14 λe (cid:48) θ ( f ) g ( e , e (cid:48) ) + λg (cid:0) D e (cid:48) θ e , e (cid:48) (cid:1) + λf g (cid:0) D e (cid:48) θ e θ , e (cid:48) (cid:1) + 14 λf g (cid:0) D e (cid:48) θ e , e (cid:48) (cid:1) = − e (cid:48) θ (log( λ )) + f (cid:18) f f (cid:19) e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + λg (cid:0) D e (cid:48) θ e , e (cid:48) (cid:1) + λf g (cid:0) D e (cid:48) θ e θ , e (cid:48) (cid:1) + l.o.t.We compute λg (cid:0) D e (cid:48) θ e , e (cid:48) (cid:1) = g (cid:0) D e (cid:48) θ e , e + f e θ (cid:1) + l.o.t.= g (cid:16) D e θ + fe + fe e , e (cid:17) + f g (cid:16) D e θ + fe + fe e , e θ (cid:17) + l.o.t.= 2 ζ + 2 ωf − ωf + f χ + l.o.t.and λf g (cid:0) D e (cid:48) θ e θ , e (cid:48) (cid:1) = f g (cid:0) D e (cid:48) θ e θ , e (cid:1) + l.o.t.= f g (cid:16) D e θ + fe + fe e θ , e (cid:17) + l.o.t.= − f χ + l.o.t. .6. PROOF OF PROPOSITION 2.3.4 ζ (cid:48) = − e (cid:48) θ (log( λ )) + f (cid:18) f f (cid:19) e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + λg (cid:0) D e (cid:48) θ e , e (cid:48) (cid:1) + λf g (cid:0) D e (cid:48) θ e θ , e (cid:48) (cid:1) + l.o.t.= 2 ζ − e (cid:48) θ (log( λ )) + f (cid:18) f f (cid:19) e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + 2 ωf − ωf + f χ − f χ + l.o.t.and hence ζ (cid:48) = ζ − e (cid:48) θ (log( λ )) + 14 ( − f κ + f κ ) + f ω − f ω + Err( ζ, ζ (cid:48) ) , Err( ζ, ζ (cid:48) ) = 12 f (cid:18) f f (cid:19) e (cid:48) θ ( f ) − f e (cid:48) θ ( f ) + 14 ( − f ϑ + f ϑ ) + l.o.t.= 12 f e (cid:48) θ ( f ) + 14 ( − f ϑ + f ϑ ) + l.o.t.Next, we derive the transformation formulae for η . We have, under a transformation oftype (2.3.3),2 η (cid:48) = g (cid:0) D e (cid:48) e (cid:48) , e (cid:48) θ (cid:1) = g (cid:18) D e (cid:48) (cid:18) λ (cid:18) e + f e θ + 14 f e (cid:19)(cid:19) , e (cid:48) θ (cid:19) = λg (cid:18) D e (cid:48) (cid:18) e + f e θ + 14 f e (cid:19) , e (cid:48) θ (cid:19) = λg (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λe (cid:48) ( f ) g ( e θ , e (cid:48) θ ) + λf g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + 14 λe (cid:48) ( f ) g ( e , e (cid:48) θ ) + 14 λf g ( D e (cid:48) e , e (cid:48) θ )= λ (cid:18) f f (cid:19) e (cid:48) ( f ) − λf e (cid:48) ( f ) + λg (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λf g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + l.o.t.We compute λg (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) = λg (cid:18) D e (cid:48) e , e θ + 12 f e (cid:19) + l.o.t.= g (cid:16) D e + fe θ e , e θ (cid:17) + 12 f g ( D e e , e ) + l.o.t.= 2 η + f χ − ωf + l.o.t.and λf g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) = l.o.t.02 APPENDIX A. APPENDIX TO CHAPTER 2 This yields2 η (cid:48) = λ (cid:18) f f (cid:19) e (cid:48) ( f ) − λf e (cid:48) ( f ) + λg (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λf g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + l.o.t.= λ (cid:18) f f (cid:19) e (cid:48) ( f ) − λf e (cid:48) ( f ) + 2 η + f χ − ωf + l.o.t.and hence η (cid:48) = η + 12 λe (cid:48) ( f ) + 14 κf − f ω + Err( η, η (cid:48) ) , Err( η, η (cid:48) ) = 14 λf f e (cid:48) ( f ) − λf e (cid:48) ( f ) + 14 f ϑ + l.o.t.= 14 f ϑ + l.o.t.Next, we derive the transformation formulae for η . We have, under a transformation oftype (2.3.3),2 η (cid:48) = g (cid:0) D e (cid:48) e (cid:48) , e (cid:48) θ (cid:1) = g (cid:18) D e (cid:48) (cid:18) λ − (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19)(cid:19) , e (cid:48) θ (cid:19) = λ − g (cid:18) D e (cid:48) (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19) , e (cid:48) θ (cid:19) = 12 λ − e (cid:48) (cid:18) f f + 18 f f (cid:19) g ( e , e (cid:48) θ ) + λ − (cid:18) f f + 116 f f (cid:19) g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λ − e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) g ( e θ , e (cid:48) θ ) + λ − f (cid:18) f f (cid:19) g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + 14 λ − e (cid:48) ( f ) g ( e , e (cid:48) θ ) + 14 λ − f g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) = − λ − f e (cid:48) (cid:18) f f + 18 f f (cid:19) + λ − (cid:18) f f (cid:19) e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) − λ − f (cid:18) f f (cid:19) e (cid:48) ( f ) + λ − g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + l.o.t.We compute λ − g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) = λ − g (cid:18) D e (cid:48) e , e θ + 12 f e (cid:19) + l.o.t.= 2 η + f χ − f ω + l.o.t. .6. PROOF OF PROPOSITION 2.3.4 λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) = l.o.t.This yields2 η (cid:48) = − λ − f e (cid:48) (cid:18) f f + 18 f f (cid:19) + λ − (cid:18) f f (cid:19) e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) − λ − f (cid:18) f f (cid:19) e (cid:48) ( f ) + λ − g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + l.o.t.= − λ − f e (cid:48) (cid:18) f f + 18 f f (cid:19) + λ − (cid:18) f f (cid:19) e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) − λ − f (cid:18) f f (cid:19) e (cid:48) ( f ) + 2 η + f χ − f ω + l.o.t.and hence η (cid:48) = η + 12 λ − e (cid:48) ( f ) + 14 κf − f ω + Err( η, η (cid:48) ) , Err( η, η (cid:48) ) = 14 λ − f f e (cid:48) ( f ) − λ − f e (cid:48) (cid:18) f f + 18 f f (cid:19) + λ − (cid:18) f f (cid:19) e (cid:48) (cid:0) f f (cid:1) − λ − f (cid:18) f f (cid:19) e (cid:48) ( f ) + 14 f ϑ + l.o.t.= − f λ − e (cid:48) ( f ) + 14 f ϑ + l.o.t.Next, we derive the transformation formulae for ξ . We have, under a transformation oftype (2.3.3),2 ξ (cid:48) = g (cid:0) D e (cid:48) e (cid:48) , e (cid:48) θ (cid:1) = g (cid:18) D e (cid:48) (cid:18) λ (cid:18) e + f e θ + 14 f e (cid:19)(cid:19) , e (cid:48) θ (cid:19) = λg (cid:18) D e (cid:48) (cid:18) e + f e θ + 14 f e (cid:19) , e (cid:48) θ (cid:19) = λg (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λe (cid:48) ( f ) g ( e θ , e (cid:48) θ ) + λf g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + 14 λe (cid:48) ( f ) g ( e , e (cid:48) θ ) + 14 λf g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) = λ (cid:18) f f (cid:19) e (cid:48) ( f ) − λf e (cid:48) ( f ) + λg (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λf g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + l.o.t.04 APPENDIX A. APPENDIX TO CHAPTER 2 We compute λg (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) = λg (cid:18) D e (cid:48) e , e θ + 12 f e (cid:19) + l.o.t.= λ g ( D e + fe θ e , e θ ) + 12 λ f g ( D e e , e ) + l.o.t.= 2 λ ξ + λ f χ + 2 λ f ω + l.o.t.and λf g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) = l.o.t.This yields2 ξ (cid:48) = λ (cid:18) f f (cid:19) e (cid:48) ( f ) − λf e (cid:48) ( f ) + λg (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λf g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + l.o.t.= λ (cid:18) f f (cid:19) e (cid:48) ( f ) − λf e (cid:48) ( f ) + 2 λ ξ + λ f χ + 2 λ f ω + l.o.t.and hence ξ (cid:48) = λ (cid:18) ξ + 12 λ − e (cid:48) ( f ) + ωf + 14 f κ (cid:19) + λ Err( ξ, ξ (cid:48) ) , Err( ξ, ξ (cid:48) ) = 14 λ − f f e (cid:48) ( f ) − λ − f e (cid:48) ( f ) + 14 f ϑ + l.o.t.= 14 f ϑ + l.o.t.In the particular case when λ = 1 , f = 0, see Remark 2.3.5, the error term takes the form,Err( ξ, ξ (cid:48) ) = 14 f ϑ + 14 f (cid:0) η + 2 ζ − η (cid:1) − f (cid:18) ω + 12 χ (cid:19) − f ξ. Next, we derive the transformation formulae for ξ . We have, under a transformation of .6. PROOF OF PROPOSITION 2.3.4 ξ (cid:48) = g (cid:0) D e (cid:48) e (cid:48) , e (cid:48) θ (cid:1) = g (cid:18) D e (cid:48) (cid:18) λ − (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19)(cid:19) , e (cid:48) θ (cid:19) = λ − g (cid:18) D e (cid:48) (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19) , e (cid:48) θ (cid:19) = 12 λ − e (cid:48) (cid:18) f f + 18 f f (cid:19) g ( e , e (cid:48) θ ) + λ − g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λ − e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) g ( e θ , e (cid:48) θ ) + λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + 14 λ − e (cid:48) ( f ) g ( e , e (cid:48) θ ) + l.o.t.= − λ − f e (cid:48) (cid:18) f f + 18 f f (cid:19) + λ − (cid:18) f f (cid:19) e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) − λ − f (cid:18) f f (cid:19) e (cid:48) ( f ) + λ − g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + l.o.t.We compute λ − g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) = λ − g (cid:18) D e (cid:48) e , e θ + 12 f e (cid:19) + l.o.t.= λ − g (cid:16) D e + fe θ e , e θ (cid:17) + 12 λ − f g ( D e e , e ) + l.o.t.= 2 λ − ξ + λ − f χ + 2 λ − f ω + l.o.t.and λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) = l.o.t.This yields2 ξ (cid:48) = − λ − f e (cid:48) (cid:18) f f + 18 f f (cid:19) + λ − (cid:18) f f (cid:19) e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) − λ − f (cid:18) f f (cid:19) e (cid:48) ( f ) + λ − g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + l.o.t.= − λ − f e (cid:48) (cid:18) f f + 18 f f (cid:19) + λ − (cid:18) f f (cid:19) e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) − λ − f (cid:18) f f (cid:19) e (cid:48) ( f ) + 2 λ − ξ + λ − f χ + 2 λ − f ω + l.o.t.06 APPENDIX A. APPENDIX TO CHAPTER 2 and hence ξ (cid:48) = λ − (cid:18) ξ + 12 λe (cid:48) ( f ) + ω f + 14 f κ (cid:19) + λ − Err( ξ, ξ (cid:48) ) , Err( ξ, ξ (cid:48) ) = − λf e (cid:48) (cid:18) f f + 18 f f (cid:19) + 14 λf f e (cid:48) ( f ) + 18 λ (cid:18) f f (cid:19) e (cid:48) (cid:0) f f (cid:1) − λf (cid:18) f f (cid:19) e (cid:48) ( f ) + 14 f ϑ + l.o.t.= − λf e (cid:48) ( f ) + 14 f ϑ + l.o.t.Next, we derive the transformation formulae for ω . We have, under a transformation oftype (2.3.3),4 ω (cid:48) = g (cid:0) D e (cid:48) e (cid:48) , e (cid:48) (cid:1) = g (cid:18) D e (cid:48) (cid:18) λ (cid:18) e + f e θ + 14 f e (cid:19)(cid:19) , e (cid:48) (cid:19) = − e (cid:48) (log λ ) + λg (cid:18) D e (cid:48) (cid:18) e + f e θ + 14 f e (cid:19) , e (cid:48) (cid:19) = − e (cid:48) (log λ ) + λg (cid:0) D e (cid:48) e , e (cid:48) (cid:1) + λe (cid:48) ( f ) g ( e θ , e (cid:48) ) + λf g (cid:0) D e (cid:48) e θ , e (cid:48) (cid:1) + 14 λe (cid:48) ( f ) g ( e , e (cid:48) ) + 14 λf g (cid:0) D e (cid:48) e , e (cid:48) (cid:1) = − e (cid:48) (log λ ) + f (cid:18) f f (cid:19) e (cid:48) ( f ) − f e (cid:48) ( f ) + λg (cid:0) D e (cid:48) e , e (cid:48) (cid:1) + λf g (cid:0) D e (cid:48) e θ , e (cid:48) (cid:1) + l.o.t.We compute λg (cid:0) D e (cid:48) e , e (cid:48) (cid:1) = g (cid:18) D e (cid:48) e , (cid:18) f f (cid:19) e + f e θ (cid:19) + l.o.t.= λ (cid:18) f f (cid:19) g (cid:16) D e + fe θ + f e e , e (cid:17) + λf g ( D e + fe θ e , e θ ) + l.o.t.= 4 λ (cid:18) f f (cid:19) ω + 2 λf ζ − λf ω + 2 λf ξ + λf f χ + l.o.t.and λf g (cid:0) D e (cid:48) e θ , e (cid:48) (cid:1) = f g (cid:0) D e (cid:48) e θ , e (cid:1) + l.o.t.= λf g ( D e + fe θ e θ , e ) + l.o.t.= − λf η − λf χ + l.o.t. .6. PROOF OF PROPOSITION 2.3.4 ω (cid:48) = − e (cid:48) (log λ ) + f (cid:18) f f (cid:19) e (cid:48) ( f ) − f e (cid:48) ( f ) + λg (cid:0) D e (cid:48) e , e (cid:48) (cid:1) + λf g (cid:0) D e (cid:48) e θ , e (cid:48) (cid:1) + l.o.t.= − e (cid:48) (log λ ) + f (cid:18) f f (cid:19) e (cid:48) ( f ) − f e (cid:48) ( f ) + 4 λ (cid:18) f f (cid:19) ω + 2 λf ζ − λf ω +2 λf ξ + λf f χ − λf η − λf χ + l.o.t.and hence ω (cid:48) = λ (cid:18) ω − λ − e (cid:48) (log( λ )) (cid:19) + λ Err( ω, ω (cid:48) ) , Err( ω, ω (cid:48) ) = 14 f (cid:18) f f (cid:19) e (cid:48) ( f ) − f e (cid:48) ( f ) + 12 ωf f − f η + 12 f ξ + 12 f ζ − κf + 18 f f κ − ωf + l.o.t.= 14 f e (cid:48) ( f ) + 12 ωf f − f η + 12 f ξ + 12 f ζ − κf + 18 f f κ − ωf + l.o.t.In the particular case, see Remark 2.3.5, when λ = 1 , f = 0 we have the more preciseformula, ω (cid:48) = ω + 12 f ( ζ − η ) − f (cid:0) ω + κ + ϑ + f ξ (cid:1) Next, we derive the transformation formulae for ω . We have, under a transformation oftype (2.3.3),4 ω (cid:48) = g (cid:0) D e (cid:48) e (cid:48) , e (cid:48) (cid:1) = g (cid:18) D e (cid:48) (cid:18) λ − (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19)(cid:19) , e (cid:48) (cid:19) = 2 e (cid:48) (log( λ ))+ λ − g (cid:18) D e (cid:48) (cid:18)(cid:18) f f + 116 f f (cid:19) e + f (cid:18) f f (cid:19) e θ + 14 f e (cid:19) , e (cid:48) (cid:19) = 2 e (cid:48) (log( λ )) + 12 λ − e (cid:48) (cid:18) f f + 18 f f (cid:19) g ( e , e (cid:48) ) + λ − (cid:18) f f (cid:19) g (cid:0) D e (cid:48) e , e (cid:48) (cid:1) + λ − e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) g ( e θ , e (cid:48) ) + λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) (cid:1) + 14 λ − e (cid:48) ( f ) g ( e , e (cid:48) ) + l.o.t.= 2 e (cid:48) (log( λ )) − e (cid:48) (cid:18) f f + 18 f f (cid:19) + f e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) − f e (cid:48) ( f )+ λ − (cid:18) f f (cid:19) g (cid:0) D e (cid:48) e , e (cid:48) (cid:1) + λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) (cid:1) + l.o.t.08 APPENDIX A. APPENDIX TO CHAPTER 2 We compute λ − (cid:18) f f (cid:19) g (cid:0) D e (cid:48) e , e (cid:48) (cid:1) = (cid:18) f f (cid:19) g (cid:0) D e (cid:48) e , e + f e θ (cid:1) = λ − (cid:18) f f (cid:19) g (cid:16) D ( ff ) e + fe θ + f e e , e + f e θ (cid:17) + l.o.t.= 4 λ − (cid:18) f f (cid:19) ω − λ − f ζ − λ − f ω + 2 λ − f f ω +2 λ − f ξ + λ − f f χ + l.o.t.and λ − f g (cid:0) D e (cid:48) e θ , e (cid:48) (cid:1) = f g (cid:0) D e (cid:48) e θ , e (cid:1) + l.o.t.= λ − f g (cid:16) D e + fe θ e θ , e (cid:17) + l.o.t.= − λ − f η − λ − f χ + l.o.t.This yields4 ω (cid:48) = 2 e (cid:48) (log( λ )) − e (cid:48) (cid:18) f f + 18 f f (cid:19) + f e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) − f e (cid:48) ( f )+4 λ − (cid:18) f f (cid:19) ω − λ − f ζ − λ − f ω + 2 λ − f f ω +2 λ − f ξ + λ − f f χ − λ − f η − λ − f χ + l.o.t.and hence ω (cid:48) = λ − (cid:18) ω + 12 λe (cid:48) (log( λ )) (cid:19) + λ − Err( ω, ω (cid:48) ) , Err( ω, ω (cid:48) ) = − e (cid:48) (cid:18) f f + 18 f f (cid:19) + 14 f e (cid:48) (cid:18) f (cid:18) f f (cid:19)(cid:19) − f e (cid:48) ( f )+ ωf f − f η + 12 f ξ − f ζ − κf + 18 f f κ − ωf + l.o.t.= − f e (cid:48) ( f ) + ωf f − f η + 12 f ξ − f ζ − κf + 18 f f κ − ωf + l.o.t.Next we derive the formula for α . We have α (cid:48) = R ( e (cid:48) , e (cid:48) ) = λ R (cid:18) e + f e θ + 14 f e , e + f e θ + 14 f e (cid:19) = λ (cid:18) R + 2 f R θ + f R θθ + 12 f R (cid:19) + l.o.t.= λ (cid:18) α + 2 f β + 32 f ρ (cid:19) + l.o.t. .6. PROOF OF PROPOSITION 2.3.4 α (cid:48) = λ α + λ Err( α, α (cid:48) ) , Err( α, α (cid:48) ) = 2 f β + 32 f ρ + l.o.t.The formula for α is easily derived by symmetry from the one on α .Next we derive the formula for β . We have β (cid:48) = R ( e (cid:48) , e (cid:48) θ ) = λR (cid:18) e + f e θ + 14 f e , (cid:18) f f (cid:19) e θ + 12 ( f e + f e ) (cid:19) + l.o.t.= λ (cid:18) R θ + f R θθ + 12 f R + 12 f R (cid:19) + l.o.t.= λ (cid:18) β + 32 f ρ + 12 f α (cid:19) + l.o.t.and hence β (cid:48) = λ (cid:18) β + 32 f ρ (cid:19) + λ Err( β, β (cid:48) ) , Err( β, β (cid:48) ) = 12 f α + l.o.t.The formula for β is easily derived by symmetry from the one on β .Finally, we derive the formula for ρ . We have ρ (cid:48) = R ( e (cid:48) , e (cid:48) ) = R (cid:18) e + f e θ + 14 f e , (cid:18) f f (cid:19) e + f e θ + 14 f e (cid:19) + l.o.t.= R + 12 f f R + f R θ + f R θ + f f R θθ + l.o.t.= ρ + 32 ρf f + f β + f β + l.o.t.and hence ρ (cid:48) = ρ + Err( ρ, ρ (cid:48) ) , Err( ρ, ρ (cid:48) ) = 32 ρf f + f β + f β + l.o.t.This concludes the proof of Proposition 2.3.4.10 APPENDIX A. APPENDIX TO CHAPTER 2 A.7 Proof of Lemma 2.3.6 For ξ (cid:48) and ω (cid:48) , we need more precise transformation formula than the ones of Proposition2.3.4. We have2 ξ (cid:48) = g ( D e (cid:48) e (cid:48) , e (cid:48) θ )= λ g ( D λ − e (cid:48) ( λ − e (cid:48) ) , e (cid:48) θ )= λ g (cid:18) D λ − e (cid:48) ( λ − e (cid:48) ) , e θ + 12 f e (cid:19) + λ f g (cid:18) D λ − e (cid:48) ( λ − e (cid:48) ) , e + f e θ + 14 f e (cid:19) = λ g (cid:18) D λ − e (cid:48) ( λ − e (cid:48) ) , e θ + 12 f e (cid:19) + λ f g (cid:0) D λ − e (cid:48) ( λ − e (cid:48) ) , λ − e (cid:48) (cid:1) = λ g (cid:18) D λ − e (cid:48) ( λ − e (cid:48) ) , e θ + 12 f e (cid:19) . Also, we have4 ω (cid:48) = g ( D e (cid:48) e (cid:48) , e (cid:48) )= − e (cid:48) (log( λ )) + λg ( D λ − e (cid:48) ( λ − e (cid:48) ) , λe (cid:48) )= − e (cid:48) (log( λ )) + λg ( D λ − e (cid:48) ( λ − e (cid:48) ) , e ) + λf g (cid:18) D λ − e (cid:48) ( λ − e (cid:48) ) , e θ + 12 f e (cid:19) + 14 λf g (cid:18) D λ − e (cid:48) ( λ − e (cid:48) ) , e + f e θ + 14 f e (cid:19) = − e (cid:48) (log( λ )) + λg ( D λ − e (cid:48) ( λ − e (cid:48) ) , e ) + λf g (cid:18) D λ − e (cid:48) ( λ − e (cid:48) ) , e θ + 12 f e (cid:19) + 14 λf g (cid:0) D λ − e (cid:48) ( λ − e (cid:48) ) , λ − e (cid:48) (cid:1) = − e (cid:48) (log( λ )) + λg ( D λ − e (cid:48) ( λ − e (cid:48) ) , e ) + λf g (cid:18) D λ − e (cid:48) ( λ − e (cid:48) ) , e θ + 12 f e (cid:19) . In view of the change of frame formula for ξ (cid:48) , we infer4 ω (cid:48) = − e (cid:48) (log( λ )) + λg ( D λ − e (cid:48) ( λ − e (cid:48) ) , e ) + λ − f ξ (cid:48) . .7. PROOF OF LEMMA 2.3.6 g (cid:0) D λ − e (cid:48) ( λ − e (cid:48) ) , e θ (cid:1) = g (cid:18) D λ − e (cid:48) (cid:18) e + f e θ + 14 f e (cid:19) , e θ (cid:19) = g (cid:0) D λ − e (cid:48) e , e θ (cid:1) + λ − e (cid:48) ( f ) + 14 f g (cid:0) D λ − e (cid:48) e , e θ (cid:1) = g (cid:16) D e + fe θ + f e e , e θ (cid:17) + (cid:18) e + f e θ + 14 f e (cid:19) f + 14 f g (cid:16) D e + fe θ + f e e , e θ (cid:17) = 2 ξ + f χ + 12 f η + (cid:18) e + f e θ + 14 f e (cid:19) f + 12 f η + 14 f χ + 18 f ξ. Also, we have g (cid:0) D λ − e (cid:48) ( λ − e (cid:48) ) , e (cid:1) = g (cid:18) D λ − e (cid:48) (cid:18) e + f e θ + 14 f e (cid:19) , e (cid:19) = g (cid:0) D λ − e (cid:48) e , e (cid:1) + f g (cid:0) D λ − e (cid:48) e θ , e (cid:1) = g (cid:16) D e + fe θ + f e e , e (cid:17) + f g (cid:16) D e + fe θ + f e e θ , e (cid:17) = 4 ω + 2 f ζ − f ω − ηf − f χ − f ξ. We deduce2 ξ (cid:48) = λ g (cid:18) D λ − e (cid:48) ( λ − e (cid:48) ) , e θ + 12 f e (cid:19) = λ (cid:40) ξ + f χ + 12 f η + (cid:18) e + f e θ + 14 f e (cid:19) f + 12 f η + 14 f χ + 18 f ξ + 12 f (cid:18) ω + 2 f ζ − f ω − ηf − f χ − f ξ (cid:19) (cid:41) = λ (cid:40) ξ + (cid:18) e + f e θ + 14 f e (cid:19) f + f χ + 2 f ω + 12 f η − f η + f ζ − f χ − f ω − f ξ (cid:41) APPENDIX A. APPENDIX TO CHAPTER 2 and4 ω (cid:48) = − e (cid:48) (log( λ )) + λg ( D λ − e (cid:48) ( λ − e (cid:48) ) , e ) + λ − f ξ (cid:48) = λ (cid:40) ω − (cid:18) e + f e θ + 14 f e (cid:19) log( λ ) + 2 f ζ − f ω − ηf − f χ − f ξ (cid:41) + λ − f ξ (cid:48) . If ξ (cid:48) = 0, we infer2 ξ + (cid:18) e + f e θ + 14 f e (cid:19) f + f χ + 2 f ω + 12 f η − f η + f ζ − f χ − f ω − f ξ = 0and hence λ − e (cid:48) ( f ) + (cid:16) κ ω (cid:17) f = − ξ − ϑf − f η + 12 f η − f ζ + 18 f κ + 12 f ω + 18 f ϑ + 18 f ξ which yields the desired transport equation for fλ − e (cid:48) ( f ) + (cid:16) κ ω (cid:17) f = − ξ + E ( f, Γ) ,E ( f, Γ) = − ϑf − f η + 12 f η − f ζ + 18 f κ + 12 f ω + 18 f ϑ + 18 f ξ. Also, if ξ (cid:48) = 0 and ω (cid:48) = 0, we infer0 = 4 ω − (cid:18) e + f e θ + 14 f e (cid:19) log( λ ) + 2 f ζ − f ω − ηf − f χ − f ξ and hence λ − e (cid:48) (log( λ )) = 2 ω + f ζ − f ω − ηf − f κ − f ϑ − f ξ which yields the desired transport equation for log( λ ) λ − e (cid:48) (log( λ )) = 2 ω + E ( f, Γ) ,E ( f, Γ) = f ζ − f ω − ηf − f κ − f ϑ − f ξ. .8. PROOF OF COROLLARY 2.3.7 f . In view of the transformation formulasof Proposition 2.3.4 for ζ (cid:48) and η (cid:48) , and the fact that we assume ζ (cid:48) + η (cid:48) = 0, we have12 λ − e (cid:48) ( f ) = − ( ζ + η ) + e (cid:48) θ (log( λ )) − f κ + f ω − f e (cid:48) θ ( f ) + 18 f λ − e (cid:48) ( f ) − f ϑ + l.o.t.Together with the above identity for λ − e (cid:48) ( f ), we infer λ − e (cid:48) ( f ) + κ f = − ζ + η ) + 2 e (cid:48) θ (log( λ )) + 2 f ω + E ( f, f , Γ) ,E ( f, f , Γ) = − f e (cid:48) θ ( f ) − f ϑ + l.o.t. , which yields the third identity of the statement. This concludes the proof of Lemma 2.3.6. A.8 Proof of Corollary 2.3.7 In view of Lemma 2.3.6 and the fact that ( e , e , e θ ) emanates from an outgoing geodesicfoliation and hence ξ = 0 , ω = 0 , ζ + η = 0 , we have λ − e (cid:48) ( f ) + κ f = E ( f, Γ) ,λ − e (cid:48) (log( λ )) = E ( f, Γ) ,λ − e (cid:48) ( f ) + κ f = 2 e (cid:48) θ (log( λ )) + 2 f ω + E ( f, f , Γ) . The second equation is the desired identity for log( λ ).We still need to derive the first and the third identities. We start with the first one. Wehave λ − e (cid:48) ( rf ) = r (cid:16) − κ f + E ( f, Γ) (cid:17) + λ − e (cid:48) ( r ) f = − r (cid:18) κ − λ − e (cid:48) ( r ) r (cid:19) f + rE ( f, Γ) .λ − e (cid:48) = e + f e θ + f e , APPENDIX A. APPENDIX TO CHAPTER 2 we infer λ − e (cid:48) ( r ) = r κ + f e ( r )and hence λ − e (cid:48) ( rf ) = − r (cid:18) ˇ κ − e ( r )2 r f (cid:19) f + rE ( f, Γ)as desired.Next, we have λ − e (cid:48) (cid:16) rf − r e (cid:48) θ (log( λ )) + rf Ω (cid:17) = r (cid:16) − κ f + 2 e (cid:48) θ (log( λ )) + 2 f ω + E ( f, f , Γ) (cid:17) − r e (cid:48) θ ( E ( f, Γ)) + r Ω (cid:16) − κ f + E ( f, Γ) (cid:17) + λ − e (cid:48) ( r ) f − r [ λ − e (cid:48) , e (cid:48) θ ] log( λ ) − rλ − e (cid:48) ( r ) e (cid:48) θ (log( λ )) + rλ − e (cid:48) (Ω) f + λ − e (cid:48) ( r ) f Ω= − r (cid:18) κ − λ − e (cid:48) ( r ) r (cid:19) f + 2 r (cid:16) − λ − e (cid:48) ( r ) (cid:17) e (cid:48) θ (log( λ )) − r λ − [ e (cid:48) , e (cid:48) θ ] log( λ )+ r (cid:16) λ − e (cid:48) (Ω) + 2 ω (cid:17) f − r (cid:18) κ − λ − e (cid:48) ( r ) r (cid:19) Ω f − r e (cid:48) θ (log( λ )) λ − e (cid:48) (log( λ ))+ rE ( f, f , Γ) − r e (cid:48) θ ( E ( f, Γ)) + r Ω E ( f, Γ) . Since we have λ − e (cid:48) = e + f e θ + f e , we infer λ − e (cid:48) ( r ) = r κ + f e ( r ) ,λ − e (cid:48) (Ω) = e (Ω) + f e θ (Ω) + f e (Ω)= − ω + f e θ (Ω) + f e (Ω) . Together with the transport equation for log( λ ) and the commutator identity for [ e (cid:48) , e (cid:48) θ ],we infer λ − e (cid:48) (cid:16) rf − r e (cid:48) θ (log( λ )) + rf Ω (cid:17) = − r (cid:18) ˇ κ − e ( r )2 r f (cid:19) f + 2 r (cid:18) − rκ − e ( r )2 f (cid:19) e (cid:48) θ (log( λ )) + r λ − ( κ (cid:48) + ϑ (cid:48) ) e (cid:48) θ (log( λ ))+ r (cid:18) e θ (Ω) + f e (Ω) (cid:19) f − r (cid:18) ˇ κ − e ( r )2 r f (cid:19) Ω f − r e (cid:48) θ (log( λ )) E ( f, Γ) + rE ( f, f , Γ) − r e (cid:48) θ ( E ( f, Γ)) + r Ω E ( f, Γ) . .9. PROOF OF LEMMA 2.3.5 λ − κ (cid:48) = κ + d/ (cid:48) ( f ) + Err( κ, κ (cid:48) ) , Err( κ, κ (cid:48) ) = f ( ζ + η ) − f κ − f ω + l.o.t.We infer λ − e (cid:48) (cid:16) rf − r e (cid:48) θ (log( λ )) + rf Ω (cid:17) = − r (cid:18) ˇ κ − e ( r )2 r f (cid:19) f + r (cid:18) ˇ κ − (cid:18) κ − r (cid:19) − e ( r ) r f (cid:19) e (cid:48) θ (log( λ ))+ r (cid:16) d/ (cid:48) ( f ) + Err( κ, κ (cid:48) ) + λ − ϑ (cid:48) (cid:17) e (cid:48) θ (log( λ ))+ r (cid:18) e θ (Ω) + f e (Ω) (cid:19) f − r (cid:18) ˇ κ − e ( r )2 r f (cid:19) Ω f − r e (cid:48) θ (log( λ )) E ( f, Γ) + rE ( f, f , Γ) − r e (cid:48) θ ( E ( f, Γ)) + r Ω E ( f, Γ) . This concludes the proof of Corollary 2.3.7. A.9 Proof of Lemma 2.3.5 Recall that we have obtained in section A.72 ξ (cid:48) = λ (cid:40) ξ + (cid:18) e + f e θ + 14 f e (cid:19) f + (cid:18) κ + 2 ω (cid:19) f + 12 ϑf + 12 f η − f η + f ζ − f χ − f ω − f ξ (cid:41) , ω (cid:48) = λ (cid:40) ω − (cid:18) e + f e θ + 14 f e (cid:19) log( λ ) + 2 f ζ − f ω − ηf − f χ − f ξ (cid:41) + λ − f ξ (cid:48) . In the case where λ = 1 and f = 0, we immediately infer2 ξ (cid:48) = 2 ξ + (cid:18) e + f e θ + 14 f e (cid:19) f + (cid:18) κ + 2 ω (cid:19) f + 12 ϑf + 12 f η − f η + f ζ − f χ − f ω − f ξ, ω (cid:48) = 4 ω + 2 f ζ − f ω − ηf − f χ − f ξ, APPENDIX A. APPENDIX TO CHAPTER 2 and hence ξ (cid:48) = ξ + 12 e (cid:48) ( f ) + (cid:18) κ + ω (cid:19) f + 14 ϑf + 14 f η − f η + 12 f ζ − f χ − f ω − f ξ,ω (cid:48) = ω + 12 f ζ − f ω − ηf − f κ − f ϑ − f ξ. Finally, we compute the change of frame formula for ζ (cid:48) and η (cid:48) when λ = 1 , f = 0. Wehave in this case e (cid:48) = e + f e θ + 14 f e ,e (cid:48) θ = e θ + 12 f e ,e (cid:48) = e , and hence 2 ζ (cid:48) = g (cid:0) D e (cid:48) θ e (cid:48) , e (cid:48) (cid:1) = g (cid:18) D e (cid:48) θ (cid:18) e + f e θ + 14 f e (cid:19) , e (cid:19) = g (cid:0) D e (cid:48) θ e , e (cid:1) + f g (cid:0) D e (cid:48) θ e θ , e (cid:1) = g (cid:16) D e θ + fe e , e (cid:17) + f g (cid:16) D e θ + fe e θ , e (cid:17) = 2 ζ − ωf − χf − ξf = 2 ζ − (cid:18) κ + 2 ω (cid:19) f − f (cid:18) ϑ + f ξ (cid:19) and2 η (cid:48) = g (cid:0) D e (cid:48) e (cid:48) , e (cid:48) θ (cid:1) = g (cid:18) D e (cid:48) (cid:18) e + f e θ + 14 f e (cid:19) , e (cid:48) θ (cid:19) = g (cid:0) D e (cid:48) e , e (cid:48) θ (cid:1) + e (cid:48) ( f ) g ( e θ , e (cid:48) θ ) + f g (cid:0) D e (cid:48) e θ , e (cid:48) θ (cid:1) + 14 e (cid:48) ( f ) g ( e , e (cid:48) θ ) + 14 f g ( D e (cid:48) e , e (cid:48) θ )= e (cid:48) ( f ) + g (cid:18) D e e , e θ + 12 f e (cid:19) + f g (cid:18) D e e θ , f e (cid:19) + 14 f g ( D e e , e θ )= 2 η + e (cid:48) ( f ) − f ω − f ξ which yields the desired change of frame formula for ζ (cid:48) and η (cid:48) . This concludes the proofof Lemma 2.3.5. .10. PROOF OF PROPOSITION 2.3.13 A.10 Proof of Proposition 2.3.13 Recall that we have q = r (cid:18) e ( e ( α )) + (2 κ − ω ) e ( α ) + (cid:18) − e ( ω ) + 8 ω − ω κ + 12 κ (cid:19) α (cid:19) , which we write in the form q = r J where, J = e ( e ( α )) + (2 κ − ω ) e ( α ) + V α,V = − e ( ω ) + 8 ω − ω κ + 12 κ . We make use of the general Bianchi equations, see Proposition 2.2.2 e α + 12 κα = − d (cid:63) / β + 4 ωα − ϑρ + Err[ e ( α )] ,e β + κβ = − d (cid:63) / ρ + 2 ωβ + 3 ηρ + Err[ e ( β )] ,e ρ + 32 κρ = d/ β + Err[ e ( ρ )]as well as the null structure equations (see Proposition 2.2.1) e ϑ + 12 κ ϑ − ωϑ = − d (cid:63) / η − κ ϑ + Err[ e ( ϑ )] e ( κ ) + 12 κ + 2 ω κ = 2 d/ ξ + Err[ e ( κ )]where Err[ e ( α )] , Err[ e ( β )] , +Err[ e ( ρ )] , Err[ e ( ϑ )] , Err[ e ( κ )] denote the correspondingquadratic terms in each equation. We also make use of the commutation formula (seeLemma 2.1.51)[ e , d (cid:63) / β ] = − κ d (cid:63) / β − Com ∗ ( β ) Com ∗ ( β ) = − ϑ d/ β − ( ζ − η ) e β − ηe Φ β + ξ ( e β − e (Φ) β ) − β · β In an arbitrary Z -invariant frame. APPENDIX A. APPENDIX TO CHAPTER 2 Thus, J = e (cid:18) − κα − d (cid:63) / β + 4 ωα − ϑρ + Err[ e ( α )] (cid:19) + (2 κ − ω ) e ( α ) + V α = ( 32 κ − ω ) e α + (cid:0) − e κ + 4 e ( ω ) + V (cid:1) α − d (cid:63) / e β − [ e , d (cid:63) / ] β − ϑe ρ − ρe ϑ + e Err[ e ( α )]= ( 32 κ − ω ) (cid:18) − κα − d (cid:63) / β + 4 ωα − ϑρ + Err[ e ( α )] (cid:19) + (cid:0) − e κ + 4 e ( ω ) + V (cid:1) α − d (cid:63) / e β + 12 κ d (cid:63) / β − ϑe ρ − ρe ϑ + e Err[ e ( α )] + Com ∗ ( β )= − d (cid:63) / e β + ( − κ + 2 ω ) d (cid:63) / β + (cid:16) − e κ + 4 e ( ω ) + V + ( 32 κ − ω )( − κ + 4 ω ) (cid:17) α − (cid:18) ϑe ρ + ρe ϑ + ϑρ ( 32 κ − ω ) (cid:19) + e Err[ e ( α )] + ( 32 κ − ω )Err[ e ( α )] + Com ∗ ( β )Hence, J = − d (cid:63) / e β + ( − κ + 2 ω ) d (cid:63) / β − (cid:18) ϑe ρ + ρe ϑ + ϑρ ( 32 κ − ω ) (cid:19) + W α + EW : = − e κ + 4 e ( ω ) + V + ( 32 κ − ω )( − κ + 4 ω ) E : = e Err[ e ( α )] + ( 32 κ − ω )Err[ e ( α )] + Com ∗ ( β ) (A.10.1)Now, ignoring cubic and higher order terms, − d (cid:63) / e β + ( − κ + 2 ω ) d (cid:63) / β = − d (cid:63) / ( − κβ − d (cid:63) / ρ + 2 ωβ + 3 ηρ + Err[ e ( β )]) + ( − κ + 2 ω ) d (cid:63) / β = d (cid:63) / d (cid:63) / ρ − ρ d (cid:63) / η + β d (cid:63) / ( κ − ω ) − η d (cid:63) / ρ − d (cid:63) / Err[ e ( β )]Also, ϑe ρ + ρe ϑ + ϑρ ( 32 κ − ω ) = ϑ (cid:18) − κρ + d/ β (cid:19) + ϑρ ( 32 κ − ω )+ ρ (cid:18) − κ ϑ + 2 ωϑ − d (cid:63) / η − κ ϑ + Err[ e ( ϑ )] (cid:19) = − κρϑ − κρϑ − ρ d (cid:63) / η + ϑ d/ β + ρ Err[ e ( ρ )] .10. PROOF OF PROPOSITION 2.3.13 W = − e κ + 4 e ( ω ) + (cid:18) − e ( ω ) + 8 ω − ω κ + 12 κ (cid:19) + ( 32 κ − ω )( − κ + 4 ω )= − e κ + (cid:18) ω − ω κ + 12 κ (cid:19) + (cid:18) − κ − ω + 7 ωκ (cid:19) = − e κ − κ − ωκ = − (cid:18) − κ − ω κ + 2 d/ ξ + Err[ e ( κ )] (cid:19) − κ − ωκ = − d/ ξ − 12 Err[ e ( κ )]Thus, back to (A.10.1), J = d (cid:63) / d (cid:63) / ρ − ρ d (cid:63) / η + β d (cid:63) / ( κ − ω ) − η d (cid:63) / ρ − d (cid:63) / Err[ e ( β )] − (cid:0) − κρϑ − κρϑ − ρ d (cid:63) / η + ϑ d/ β + ρ Err[ e ( ρ )] (cid:17) − d/ ξα + E = d (cid:63) / d (cid:63) / ρ + 34 ρ ( κϑ + κϑ ) + β d (cid:63) / ( κ − ω ) − η d (cid:63) / ρ − ϑ d/ β − d/ ξα − ρ Err[ e ( ρ )] + E In other words, J = d (cid:63) / d (cid:63) / ρ + 34 ρ ( κϑ + κϑ ) + ErrErr : = β d (cid:63) / ( κ − ω ) − η d (cid:63) / ρ − ϑ d/ β − d/ ξα − ρ Err[ e ( ρ )] + e Err[ e ( α )] + ( 32 κ − ω )Err[ e ( α )] + Com ∗ ( β ) + l.o.t.It remains to analyze the lower order terms according to our convention in Definition 2.3.8Note that we can write the first line in the expression of ErrErr = r − Γ b · β + r − Γ g · d / Γ g + r − Γ g d / Γ b + r − d / Γ b · α = r − d / ≤ Γ b · β + r − Γ g d / Γ b + l.o.t.On the other hand,Err[ e ( ρ )] = − ϑ α − ζ β + 2( η β + ξ β ) = Γ g · Γ b + Γ b · β Err[ e ( α )] = ( ζ + 4 η ) β = Γ g · βe Err[ e ( α )] = e ( ζ + 4 η ) β + ( ζ + 4 η ) e ( β )= e ( ζ + 4 η ) · β + ( ζ + 4 η )( − κβ − d (cid:63) / ρ + 2 ωβ + 3 ηρ )= e ( ζ + 4 η ) · β + r − Γ g β + r − Γ g d / Γ g + r − Γ g · Γ g APPENDIX A. APPENDIX TO CHAPTER 2 Com ∗ ( β ) = − ϑ d/ β − ( ζ − η ) e β − ηe Φ β + ξ ( e β − e (Φ) β ) − β · β = − ϑ d/ β − ( ζ − η ) ( − κβ − d (cid:63) / ρ + 2 ω β + 3 ηρ ) − ηe (Φ) β + ξ ( − κβ + d (cid:63) / α − ωβ ) − ξe Φ β − β · β + l.o.t.= r − Γ b · d / ≤ β + r − Γ g · d / Γ b + l.o.t.Therefore, schematically,Err = e ( ζ + 4 η ) · β + r − Γ b d / ≤ β + r − Γ g d / Γ b + l.o.t.and therefore,Err[ q ] = r Err = r (cid:0) e ( ζ + 4 η ) · β + r − Γ b · d / ≤ β + r − Γ g · d / Γ b (cid:1) + l.o.t.Since e ζ ∈ r − d Γ b and β ∈ r − Γ g we rewrite in the form,Err[ q ] = r e η · β + r d ≤ (cid:0) Γ b · Γ g ) . This concludes the proof of Proposition 2.3.13. A.11 Proof of Proposition 2.3.14 We start with the formula (2.3.11) r q = r (cid:18) d (cid:63) / d (cid:63) / ρ + 34 κρϑ + 34 κρϑ (cid:19) + r Err[ q ] . with Err[ q ] given by (2.3.12). Taking the e derivative we deduce, e ( r q ) = r L + 5 e ( r ) q + e ( r Err[ q ]) − e ( r )Err[ q ] ,L : = e (cid:26) d (cid:63) / d (cid:63) / ρ + 34 e ( κρϑ ) + 34 e ( κρϑ ) (cid:27) (A.11.1)We calculate L as follows, L = e d (cid:63) / d (cid:63) / ρ + 34 e ( κρϑ ) + 34 e ( κρϑ )= d (cid:63) / d (cid:63) / e ( ρ ) + [ e , d (cid:63) / d (cid:63) / ] ρ + 34 e ( κρϑ ) + 34 e ( κρϑ ) . .11. PROOF OF PROPOSITION 2.3.14 e ( κρϑ ) = κρe ( ϑ ) + e ( κ ) ρϑ + κe ( ρ ) ϑ = κρ (cid:18) − κ ϑ + 2 ωϑ − d (cid:63) / η − κ ϑ + Err[ e ϑ ] (cid:19) + (cid:18) − κ − ω κ + 2 d/ ξ + Err[ e κ ] (cid:19) ρϑ + κ (cid:18) − κρ + d/ β − ϑ α + Err[ e ( ρ )] (cid:19) ϑ = κρ (cid:18) − κ ϑ − d (cid:63) / η − κ ϑ (cid:19) + κρ Err[ e ϑ ] + 2 d/ ξ ρϑ + κ ( d/ β ) ϑ We deduce, e ( κρϑ ) = κρ (cid:18) − κ ϑ − d (cid:63) / η − κ ϑ (cid:19) + E E = 2 d/ ξ ρϑ + κ ( d/ β ) ϑ + κρ Err[ e ( ϑ )] (A.11.2)where Err[ e ( ϑ )], Err[ e ( κ )] , Err[ e ( ρ )] denote the quadratic error terms in the correspond-ing equations. Also, e ( κρϑ ) = κρe ( ϑ ) + e ( κ ) ρϑ + κe ( ρ ) ϑ = κρ (cid:16) − κ ϑ − ω ϑ − α − d (cid:63) / ξ + Err[ e ( ϑ )] (cid:17) + (cid:18) − κ κ + 2 ωκ + 2 d/ η + 2 ρ + Err[ e ( κ )] (cid:19) ρϑ + (cid:18) − κρ + d/ β + Err[ e ( ρ )] (cid:19) κϑ Hence, ignoring the higher order terms, e ( κρϑ ) = κρ (cid:16) − κ ϑ − α − d (cid:63) / ξ (cid:17) + 2 ρ ϑ + E E : = 2 ρ d/ ηϑ + κ d/ βϑ + κρ Err[ e ( ϑ )] (A.11.3)Also, we have d (cid:63) / d (cid:63) / e ( ρ ) = d (cid:63) / d (cid:63) / (cid:18) − κρ + d/ β + Err[ e ( ρ )] (cid:19) = d (cid:63) / d (cid:63) / d/ β − ρ d (cid:63) / d (cid:63) / κ − κ d (cid:63) / d (cid:63) / ρ + E E = d (cid:63) / d (cid:63) / Err[ e ( ρ )] − d (cid:63) / κ · d (cid:63) / ρ APPENDIX A. APPENDIX TO CHAPTER 2 i.e., e ( κρϑ ) = κρ (cid:18) − κ ϑ − d (cid:63) / η − κ ϑ (cid:19) + E ,e ( κρϑ ) = κρ (cid:16) − κ ϑ − α − d (cid:63) / ξ (cid:17) + 2 ρ ϑ + E d (cid:63) / d (cid:63) / e ( ρ ) = d (cid:63) / d (cid:63) / d/ β − ρ d (cid:63) / d (cid:63) / κ − κ d (cid:63) / d (cid:63) / ρ + E (A.11.4)Now, in view of Lemma 2.1.51 we have (for f = d (cid:63) / ρ ∈ s − ),[ e , d (cid:63) / ] d (cid:63) / ρ = − κ d (cid:63) / d (cid:63) / ρ − Com ∗ ( d (cid:63) / ρ )and [ e , d (cid:63) / ] ρ = − [ e , e θ ] ρ = − κ d (cid:63) / ρ − ϑe θ ρ + ( ζ − η ) e ρ − ξe ρ = − κ d (cid:63) / ρ − ϑe θ ρ + ( ζ − η ) (cid:0) − κρ + d/ β + Err[ e ( ρ )] (cid:1) − ξ (cid:0) − κρ + d/ β + Err[ e ( ρ )] (cid:1) = − κ d (cid:63) / ρ − ρ (cid:2) ( ζ − η ) κ − ξκ (cid:3) + E E = ( ζ − η ) d/ β − ξ d/ β + ( ζ − η ) e [( ρ )] − ϑe θ ρ − ξ Err[ e ( ρ )]We deduce d (cid:63) / [ e , d (cid:63) / ] ρ = d (cid:63) / (cid:16) − κ d (cid:63) / ρ − ρ (cid:2) κ ( ζ − η ) − κξ (cid:3) + E (cid:17) = − κ d (cid:63) / d (cid:63) / ρ − ρ (cid:16) κ d (cid:63) / ( ζ − η ) − κ d (cid:63) / ξ (cid:17) + E E = d (cid:63) / E − d (cid:63) / κ · d (cid:63) / ρ − 32 ( ζ − η ) d (cid:63) / ( κρ ) + 32 ξ d (cid:63) / ( κρ )Hence, since [ e , d (cid:63) / d (cid:63) / ] ρ = [ e , d (cid:63) / ] d (cid:63) / ρ + d (cid:63) / [ e , d (cid:63) / ] ρ ,[ e , d (cid:63) / d (cid:63) / ] ρ = − κ d (cid:63) / d (cid:63) / ρ − (cid:16) d (cid:63) / ( ζ − η ) κρ − d (cid:63) / ξκρ (cid:17) − Com ∗ ( d (cid:63) / ρ ) + E . (A.11.5) .11. PROOF OF PROPOSITION 2.3.14 L = d (cid:63) / d (cid:63) / e ( ρ ) + [ e , d (cid:63) / d (cid:63) / ] ρ + 34 e ( κρϑ ) + 34 e ( κρϑ )= d (cid:63) / d (cid:63) / d/ β − ρ d (cid:63) / d (cid:63) / κ − κ d (cid:63) / d (cid:63) / ρ + E − κ d (cid:63) / d (cid:63) / ρ − (cid:16) d (cid:63) / ( ζ − η ) κρ − d (cid:63) / ξκρ (cid:17) − Com ∗ ( d (cid:63) / ρ ) + E + 34 (cid:18) κρ (cid:18) − κ ϑ − d (cid:63) / η − κ ϑ (cid:19)(cid:19) + E + 34 (cid:16) κρ (cid:16) − κ ϑ − α − d (cid:63) / ξ (cid:17) + 2 ρ ϑ + Err (cid:17) = d (cid:63) / d (cid:63) / d/ β − κ d (cid:63) / d (cid:63) / ρ − ρ d (cid:63) / d (cid:63) / κ − ρκ d (cid:63) / ζ + 34 κρ (cid:18) − κ ϑ − κ ϑ (cid:19) + 34 κρ (cid:16) − κ ϑ − α (cid:17) + 32 ρ ϑ + E + E + E + E − Com ∗ ( d (cid:63) / ρ )i.e., L = d (cid:63) / d (cid:63) / d/ β − κ d (cid:63) / d (cid:63) / ρ − ρ d (cid:63) / d (cid:63) / κ − ρκ d (cid:63) / ζ − κρ (cid:18) κ ϑ + 12 κ ϑ (cid:19) − κρ (3 κ ϑ + 2 α ) + 32 ρ ϑ + EE = E + E + E + E − Com ∗ ( d (cid:63) / ρ ) (A.11.6)On the other hand, in view of (2.3.11), writing e r = r ( κ + A ),5 e ( r ) q = r κ q + 5 rA q = 52 r κ (cid:26) d (cid:63) / d (cid:63) / ρ + 34 κρϑ + 34 κρϑ + Err (cid:27) + 5 rA q APPENDIX A. APPENDIX TO CHAPTER 2 Hence, in view of (A.11.1) and (A.11.6), e ( r q ) = r L + 5 e ( r ) q + e ( r Err) − e ( r )Err= r (cid:40) (cid:18) d (cid:63) / d (cid:63) / d/ β − κ d (cid:63) / d (cid:63) / ρ − ρ d (cid:63) / d (cid:63) / κ − ρκ d (cid:63) / ζ (cid:19) − κρ (cid:18) κ ϑ + 12 κ ϑ (cid:19) − κρ (3 κ ϑ + 2 α ) + 32 ρ ϑ (cid:41) + 52 r κ (cid:26) d (cid:63) / d (cid:63) / ρ + 34 κρϑ + 34 κρϑ (cid:27) + r E + e ( r Err) − e rr r Err + 5 rA q = r (cid:18) d (cid:63) / d (cid:63) / d/ β − ρ d (cid:63) / d (cid:63) / κ − κρ d (cid:63) / ζ − κρα + 34 (2 ρ − κκρ ) ϑ (cid:19) + Err[ e ( r q )] . where, Err[ e ( r q )] = e ( r Err[ q ]) − e r Err[ q ] + 5 rA q + r E (A.11.7)= re (Err[ q ]) + Err[ q ] + 5 rA q + r E (A.11.8)and E = E + E + E + E − Com ∗ ( d (cid:63) / ρ )with, E = 2 d/ ξ ρϑ + κ ( d/ β ) ϑ + κρ Err[ e ( κ )] E = 2 ρ d/ ηϑ + κ d/ βϑ + κρ Err[ e ( ϑ )] E = d (cid:63) / d (cid:63) / Err[ e ( ρ )] − d (cid:63) / κ · d (cid:63) / ρE = d (cid:63) / E − d (cid:63) / κ · d (cid:63) / ρ − 32 ( ζ − η ) d (cid:63) / ( κρ ) + 32 ξ d (cid:63) / ( κρ ) E = ( ζ − η ) d/ β − ξ d/ β − ϑe θ ρCom ∗ ( d (cid:63) / ρ ) = − ϑ d/ d (cid:63) / ρ − ( ζ − η ) e d (cid:63) / ρ − ηe Φ d (cid:63) / ρ + ξ ( e d (cid:63) / ρ − e (Φ) d (cid:63) / ρ ) − β d (cid:63) / ρ. Note also that,( ζ − η ) e d (cid:63) / ρ = ( ζ − η ) · d (cid:63) / e ρ + ( ζ − η ) (cid:0) − κ d (cid:63) / ρ − ϑe θ ρ + ( ζ − η ) e ρ − ξe ˇ ρ (cid:1) Using our schematic notationErr[ e ( κ )] = Γ b · Γ b + l.o.t.Err[ e ( ϑ )] = Γ b · Γ b + l.o.t.Err[ e ( ρ )] = Γ g · α + l.o.t. = Γ g · Γ b + l.o.t. .12. PROOF OF THE TEUKOLSKY-STAROBINSKI IDENTITY E = r − Γ b · d / ≤ Γ b + l.o.t. E = r − Γ b · d / ≤ Γ b + r − Γ b · β + l.o.t. E = r − d / (Γ g · Γ b ) + r − ( d / Γ g ) · ( d / Γ g ) E = r − Γ g · ( d / Γ b ) + r − Γ b · d /β + r − Γ b d / · Γ g = r − d / (Γ g · Γ b ) + l.o.t. E = r − d / (Γ g · Γ b ) + l.o.t. Com ∗ ( d (cid:63) / ρ ) = r − Γ b · d ≤ Γ g + r − d / Γ b · Γ g + l.o.t.and, since r − Γ b can be replaced by Γ g and d /β can be replaced by r − Γ g , E = r − d / (Γ g · Γ b ) + l.o.t.Taking into account the expression of Err[ q ] in Proposition 2.3.13 we write re (Err[ q ]) + Err[ q ] = re (cid:104) r e η · β + r d ≤ (cid:0) Γ b · Γ g ) (cid:105) + r e η · β + r d ≤ (cid:0) Γ b · Γ g )= r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) and therefore, back to (A.11.7),Err[ e ( r q )] = e ( r Err[ q ]) + Err[ q ] + rA q + r E = r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) + r Γ b q + r d / (Γ g · Γ b ) + l.o.t.= r Γ b q + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) . This concludes the proof of Proposition 2.3.14. A.12 Proof of the Teukolsky-Starobinski identity According to Proposition 2.3.14 we have e ( r q ) = r (cid:40) d (cid:63) / d (cid:63) / d/ β − ρ d (cid:63) / d (cid:63) / κ − κρ d (cid:63) / ζ − κρα + 34 (2 ρ − κκρ ) ϑ (cid:41) + Err[ e q ] . We infer that e ( r e ( r q )) = r (cid:40) e d (cid:63) / d (cid:63) / d/ β − e ( ρ d (cid:63) / d (cid:63) / κ ) − e ( κρ d (cid:63) / ζ ) − e ( κρα )+ 34 e (cid:16) (2 ρ − κκρ ) ϑ (cid:17)(cid:41) + 7 re ( r ) e ( r q )+ r e (cid:0) Err[ e q ] (cid:1) + r Err[ e q ] + l.o.t. (A.12.1)26 APPENDIX A. APPENDIX TO CHAPTER 2 We first compute e d (cid:63) / d (cid:63) / d/ β = d (cid:63) / d (cid:63) / d/ ( e β ) + [ e , d (cid:63) / ] d (cid:63) / d/ β + d (cid:63) / [ e , d (cid:63) / ] d/ β + d (cid:63) / d (cid:63) / [ e , d/ ] β = d (cid:63) / d (cid:63) / d/ ( e β ) + (cid:18) − κ d (cid:63) / + 12 ϑ d/ + ( ζ − η ) e + ηe Φ − ξ ( e − e (Φ)) + β (cid:19) d (cid:63) / d/ β + d (cid:63) / (cid:18)(cid:18) − κ d (cid:63) / − ϑ d (cid:63) / + ( ζ − η ) e − ξe (cid:19) d/ β (cid:19) + d (cid:63) / d (cid:63) / (cid:18)(cid:18) − κ d/ + 12 ϑ d (cid:63) / − ( ζ − η ) e + ηe Φ + ξ ( e + e (Φ)) + β (cid:19) β (cid:19) In view of our general commutation formulas in Lemma 2.1.51 and our notation conventionfor error terms we have ,[ e , d (cid:63) / ] d (cid:63) / d/ β = (cid:18) − κ d (cid:63) / + 12 ϑ d/ + ( ζ − η ) e + ηe Φ − ξ ( e − e (Φ)) + β (cid:19) d (cid:63) / d/ β = − κ d (cid:63) / d (cid:63) / d/ β + r − Γ b · d / ≤ Γ b + l.o.t. d (cid:63) / [ e , d (cid:63) / ] d/ β = d (cid:63) / (cid:18)(cid:18) − κ d (cid:63) / − ϑ d (cid:63) / + ( ζ − η ) e − ξe (cid:19) d/ β (cid:19) = − κ d (cid:63) / d (cid:63) / d/ β + r − (cid:0) Γ b · d / ≤ Γ b + Γ ≤ b · d / ≤ Γ b (cid:1) + l.o.t. d (cid:63) / d (cid:63) / [ e , d/ ] β = d (cid:63) / d (cid:63) / (cid:18)(cid:18) − κ d/ + 12 ϑ d (cid:63) / − ( ζ − η ) e + ηe Φ + ξ ( e + e (Φ)) + β (cid:19) β (cid:19) = − κ d (cid:63) / d (cid:63) / d/ β + r − (cid:0) Γ b · d / ≤ Γ b + Γ ≤ b · d / ≤ Γ b (cid:1) + l.o.t.Hence, schematically, e d (cid:63) / d (cid:63) / d/ β = d (cid:63) / d (cid:63) / d/ ( e β ) − κ d (cid:63) / d (cid:63) / d/ β + r − d / ≤ (Γ b · Γ b )Using the Bianchi identity e β = d/ α − κ + ω ) β + ( − ζ + η ) α + 3 ξρ we further deduce, d (cid:63) / d (cid:63) / d/ ( e β ) = d (cid:63) / d (cid:63) / d/ d/ α − κ + ω ) d (cid:63) / d (cid:63) / d/ β + 3 ρ d (cid:63) / d (cid:63) / d/ ξ + r − d / ≤ (Γ b · Γ b )i.e., e d (cid:63) / d (cid:63) / d/ β = d (cid:63) / d (cid:63) / d/ d/ α − κ + ω ) d (cid:63) / d (cid:63) / d/ β + 3 ρ d (cid:63) / d (cid:63) / d/ ξ − κ d (cid:63) / d (cid:63) / d/ β + r − d / ≤ (Γ b · Γ b ) (A.12.2) In particular we write β ∈ r − Γ b . We also commute once more e and e with d (cid:63) / , d/ , d (cid:63) / , d/ and use Bianchi. .12. PROOF OF THE TEUKOLSKY-STAROBINSKI IDENTITY e ( ρ d (cid:63) / d (cid:63) / κ ) on the right hand side of (A.12.1) e ( ρ d (cid:63) / d (cid:63) / κ ) = ρ d (cid:63) / d (cid:63) / e κ + ρ [ e , d (cid:63) / ] d (cid:63) / κ + ρ d (cid:63) / [ e , d (cid:63) / ] κ + e ( ρ ) d (cid:63) / d (cid:63) / κ. Using the equation for e κ in Proposition 2.2.1 we derive, ρ d (cid:63) / d (cid:63) / e κ = ρ d (cid:63) / d (cid:63) / (cid:18) − κ − ω κ + 2 d/ ξ + Γ b · Γ b (cid:19) , = − ρ ( κ + 2 ω ) d (cid:63) / d (cid:63) / κ − ρκ d (cid:63) / d (cid:63) / ω + 2 ρ d (cid:63) / d (cid:63) / d/ ξ + r − d / ≤ Γ b · Γ b Also,[ e , d (cid:63) / ] d (cid:63) / κ = (cid:18) − κ d (cid:63) / + 12 ϑ d/ + ( ζ − η ) e + ηe Φ − ξ ( e − e (Φ)) + β (cid:19) d (cid:63) / κ = − κ d (cid:63) / d (cid:63) / κ + r − Γ b · d / ≤ Γ g ,d (cid:63) / [ e , d (cid:63) / ] κ = d (cid:63) / (cid:18)(cid:18) − κ d (cid:63) / − ϑ d (cid:63) / + ( ζ − η ) e − ξe (cid:19) κ (cid:19) , = − κ d (cid:63) / d (cid:63) / κ + d (cid:63) / ( ζ − η ) e κ − d (cid:63) / ξe κ + r − d / ≤ (Γ b · Γ g )Using also. the Bianchi equation e ρ = − κρ + d/ β − ϑ α − ζ β + 2( η β + ξ β )We deduce, e ( ρ d (cid:63) / d (cid:63) / κ ) = − ρ ( κ + 2 ω ) d (cid:63) / d (cid:63) / κ − ρκ d (cid:63) / d (cid:63) / ω + 2 ρ d (cid:63) / d (cid:63) / d/ ξ − ρκ d (cid:63) / d (cid:63) / κ + ρ (cid:0) d (cid:63) / ( ζ − η ) e κ − d (cid:63) / ( ξ ) e κ (cid:1) − ρκ d (cid:63) / d (cid:63) / κ + r − d / ≤ (Γ b · Γ g )i.e., e ( ρ d (cid:63) / d (cid:63) / κ ) = − ρκ d (cid:63) / d (cid:63) / ( κ ) − ρω d (cid:63) / d (cid:63) / κ − ρκ d (cid:63) / d (cid:63) / ω + 2 ρ d (cid:63) / d (cid:63) / d/ ξ + ρ (cid:0) d (cid:63) / ( ζ − η ) e κ − d (cid:63) / ( ξ ) e κ (cid:1) + r − d / ≤ (Γ b · Γ g ) . (A.12.3)28 APPENDIX A. APPENDIX TO CHAPTER 2 Now, d (cid:63) / ( ζ − η ) e κ − d (cid:63) / ( ξ ) e κ = d (cid:63) / ( ζ − η ) (cid:18) − κ − ω κ + 2 d/ ξ + Γ b · Γ b (cid:19) − d (cid:63) / ξ (cid:18) − κκ + 2 ωκ + 2 d/ η + 2 ρ + Γ g · Γ b (cid:19) = d (cid:63) / ( ζ − η ) (cid:18) − κ − ω κ (cid:19) − d (cid:63) / ξ (cid:18) − κκ + 2 ωκ + 2 ρ (cid:19) + r − d / ≤ Γ b · d / ≤ Γ b . Therefore, back to (A.12.3), e ( ρ d (cid:63) / d (cid:63) / κ ) = − ρκ d (cid:63) / d (cid:63) / ( κ ) − ρω d (cid:63) / d (cid:63) / κ − ρκ d (cid:63) / d (cid:63) / ω + 2 ρ d (cid:63) / d (cid:63) / d/ ξ + ρ d (cid:63) / ( ζ − η ) (cid:18) − κ − ω κ (cid:19) − ρ d (cid:63) / ξ (cid:18) − κκ + 2 ωκ + 2 ρ (cid:19) + r − d / ≤ (Γ b · Γ g ) .. (A.12.4)We next estimate the third term e ( κρ d (cid:63) / ζ ) on the right hand side of (A.12.1), e ( κρ d (cid:63) / ζ ) = κρ d (cid:63) / ( e ζ ) + κρ [ e , d (cid:63) / ] ζ + e ( κ ) ρ d (cid:63) / ζ + κe ( ρ ) d (cid:63) / ζ Using again the equations e ( κ ) = − κ − ω κ + 2 d/ ξ + Γ b · Γ b e ρ = − κρ + d/ β + Γ g · Γ b i.e., e ( κ ) ρ d (cid:63) / ζ = ( − κ − ω κ ) ρ d (cid:63) / ζ + r − d / Γ b · d / Γ g κe ρ d (cid:63) / ζ = − κ ρ + r − d / Γ b · d / Γ g the equation, e ζ = − κ ( ζ + η ) + 2 ω ( ζ − η ) + β + 2 d (cid:63) / ω + 2 ωξ + 12 κξ + Γ b · Γ b .12. PROOF OF THE TEUKOLSKY-STAROBINSKI IDENTITY e , d (cid:63) / ] ζ = (cid:18) − κ d (cid:63) / + 12 ϑ d/ + ( ζ − η ) e + ηe Φ − ξ ( e − e (Φ)) + β (cid:19) ζ = − κ d (cid:63) / ζ + r − Γ b · d / ≤ Γ g we deduce e ( κρ d (cid:63) / ζ ) = κρ d (cid:63) / (cid:18) − κ ( ζ + η ) + 2 ω ( ζ − η ) + β + 2 d (cid:63) / ω + 2 ωξ + 12 κξ + Γ b · Γ b (cid:19) − κ ρ d (cid:63) / ζ + r − Γ b · d / ≤ Γ g − κ ρ d (cid:63) / ζ − ω κρ d (cid:63) / ζ + r − d / ≤ Γ b · d / ≤ Γ g − κ ρ d (cid:63) / ζ + r − d / Γ b · d / Γ g = κρ (cid:18) − κ d (cid:63) / ζ − κ d (cid:63) / η − ω d (cid:63) / η + d (cid:63) / β + 2 d (cid:63) / d (cid:63) / ω + 2 ω d (cid:63) / ξ + 12 κ d (cid:63) / ξ (cid:19) + r − d / ≤ Γ b · d / ≤ Γ g + r − d / ≤ Γ b · d / ≤ Γ b i.e., e ( κρ d (cid:63) / ζ ) = κρ (cid:18) − κ d (cid:63) / ζ − κ d (cid:63) / η − ω d (cid:63) / η + d (cid:63) / β + 2 d (cid:63) / d (cid:63) / ω + 2 ω d (cid:63) / ξ + 12 κ d (cid:63) / ξ (cid:19) + r − d / ≤ Γ b · d / ≤ Γ g + r − d / ≤ Γ b · d / ≤ Γ b (A.12.5)For the fourth term on the right hand side of (A.12.1) we have e ( κρα ) = κρe ( α ) + e ( κ ) ρα + κe ( ρ ) α = κρe ( α ) + (cid:18) − κ κ + 2 ωκ + 2 d/ η + 2 ρ − ϑ ϑ + 2( ξ ξ + η η ) (cid:19) ρα + κ (cid:18) − κρ + d/ β − ϑ α − ζ β + 2( η β + ξ β ) (cid:19) α = κρe ( α ) + ( − κκ + 2 ωκ + 2 ρ ) ρα + (cid:0) r − d / Γ b + r − Γ b · Γ b ) · α i.e., e ( κρα ) = κρe ( α ) + ( − κκ + 2 ωκ + 2 ρ ) ρα + (cid:0) r − d / Γ b + r − Γ b · Γ b ) · α. (A.12.6)Finally, for the fifth term on the right hand side of (A.12.1), using the e equations for30 APPENDIX A. APPENDIX TO CHAPTER 2 ϑ, ρ, κ, κ , e (cid:16) (2 ρ − κκρ ) ϑ (cid:17) = (2 ρ − κκρ ) e ϑ + 4 ρe ( ρ ) ϑ − e ( κ ) κρϑ − κe ( κ ) ρϑ − κκe ( ρ ) ϑ = (2 ρ − κκρ ) (cid:16) − κ ϑ − ω ϑ − α − d (cid:63) / ξ + Γ b · Γ b (cid:17) +4 ρ (cid:18) − κρ + d/ β + Γ g · Γ b (cid:19) ϑ − (cid:18) − κ κ + 2 ωκ + 2 ρ + 2 d/ η + Γ b · Γ b (cid:19) κρϑ − κ (cid:18) − κ − ω κ + 2 d/ ξ + Γ b · Γ b (cid:19) ρϑ − κκ (cid:18) − κρ + d/ β + Γ g · Γ b (cid:19) ϑ i.e., e (cid:16) (2 ρ − κκρ ) ϑ (cid:17) = (2 ρ − κκρ ) (cid:16) − κ ϑ − ω ϑ − α − d (cid:63) / ξ (cid:17) − κρ ϑ − (cid:18) − κ κ + 2 ωκ + 2 ρ (cid:19) κρϑ + κ (cid:18) κ + 2 ω κ (cid:19) ρϑ + 32 κ κρ + r − d / ≤ Γ b · Γ b from which, e (cid:16) (2 ρ − κκρ ) ϑ (cid:17) = (2 ρ − κκρ ) (cid:16) − α − d (cid:63) / ξ (cid:17) + (cid:18) κκ ρ − κρ + 2 κκρω − ωρ (cid:19) ϑ + r − d / ≤ Γ b · Γ b . (A.12.7)Recalling (A.12.1) r − e ( r e ( r q )) = e d (cid:63) / d (cid:63) / d/ β − e ( ρ d (cid:63) / d (cid:63) / κ ) − e ( κρ d (cid:63) / ζ ) − e ( κρα )+ 34 e (cid:16) (2 ρ − κκρ ) ϑ (cid:17) + 7 r − e ( r ) e ( r q )+ r e (cid:0) Err[ e q ] (cid:1) + r Err[ e q ] + l.o.t. .12. PROOF OF THE TEUKOLSKY-STAROBINSKI IDENTITY r − e ( r e ( r q )) = d (cid:63) / d (cid:63) / d/ d/ α − κ + ω ) d (cid:63) / d (cid:63) / d/ β + 3 ρ d (cid:63) / d (cid:63) / d/ ξ − κ d (cid:63) / d (cid:63) / d/ β − (cid:40) − ρκ d (cid:63) / d (cid:63) / ( κ ) − ρω d (cid:63) / d (cid:63) / κ − ρκ d (cid:63) / d (cid:63) / ω + 2 ρ d (cid:63) / d (cid:63) / d/ ξ + ρ d (cid:63) / ( ζ − η ) (cid:18) − κ − ω κ (cid:19) − ρ d (cid:63) / ξ (cid:18) − κκ + 2 ωκ + 2 ρ (cid:19) (cid:41) − (cid:40) κρ (cid:18) − κ d (cid:63) / ζ − κ d (cid:63) / η − ω d (cid:63) / η + d (cid:63) / β + 2 d (cid:63) / d (cid:63) / ω + 2 ω d (cid:63) / ξ + 12 κ d (cid:63) / ξ (cid:19) (cid:41) − (cid:40) κρe ( α ) + ( − κκ + 2 ωκ + 2 ρ ) ρα + (cid:0) r − d / Γ b + r − Γ b · Γ b ) · α (cid:41) + 34 (cid:40) (2 ρ − κκρ ) (cid:16) − α − d (cid:63) / ξ (cid:17) + (cid:18) κκ ρ − κρ + 2 κκρω − ωρ (cid:19) ϑ (cid:41) + 7 r − e ( r ) e ( r q ) + Err + r − (cid:16) r e (cid:0) Err[ e q ] (cid:1) + r Err[ e q ] (cid:17) + l.o.t. . where, the error term Err is given byErr = (cid:0) r − d / Γ b + r − Γ b · Γ b ) · α + r − d / ≤ Γ b · d / ≤ Γ g + r − d / ≤ Γ b · d / ≤ Γ b (A.12.8)Denoting the expression of left hand side of the identity (2.3.15) by I , i.e. I := e ( r e ( r q )) + 2 ωr e ( r q )we deduce, r − I = d (cid:63) / d (cid:63) / d/ d/ α − (cid:18) κ + 2 ω (cid:19) d (cid:63) / d (cid:63) / d/ β + 32 (cid:18) κ + 2 ω (cid:19) ρ d (cid:63) / d (cid:63) / κ + 32 (cid:18) κ + 2 ω (cid:19) κρ d (cid:63) / ζ − κρ d (cid:63) / β − κρe ( α ) − (cid:16) − κκ + 2 ωκ + 4 ρ (cid:17) ρα + 34 (cid:18) κκ ρ − κρ + 2 κκρω − ωρ (cid:19) ϑ + r − (cid:2) re ( r ) e ( r q ) + 2 ωr e ( r q ) (cid:3) + (cid:102) Err . where the new error term (cid:102) Err is given by (cid:102) Err = Err + r − (cid:16) r e (cid:0) Err[ e q ] (cid:1) + r Err[ e q ] (cid:17) + 2 ωr − Err[ e q ]32 APPENDIX A. APPENDIX TO CHAPTER 2 To calculate the term J := 7 re ( r ) e ( r q ) + 2 ωr e ( r q ) in the last row we make use oncemore of the identity of Lemma 2.3.14 to derive J = r (cid:18) κ + 2 ω (cid:19) e ( r q ) + 7 r (cid:16) e ( r ) − r κ (cid:17) e ( r q )= r (cid:18) κ + 2 ω (cid:19) e ( r q ) + r Γ b e ( r q )= r (cid:18) κ + 2 ω (cid:19) (cid:26) d (cid:63) / d (cid:63) / d/ β − ρ d (cid:63) / d (cid:63) / κ − κρ d (cid:63) / ζ − κρα + 34 (2 ρ − κκρ ) ϑ (cid:27) + r (cid:18) κ + 2 ω (cid:19) Err[ e ( r q )] + r Γ b e ( r q )i.e., r − J = (cid:18) κ + 2 ω (cid:19) (cid:26) d (cid:63) / d (cid:63) / d/ β − ρ d (cid:63) / d (cid:63) / κ − κρ d (cid:63) / ζ − κρα + 34 (2 ρ − κκρ ) ϑ (cid:27) + r − (cid:18) κ + 2 ω (cid:19) Err[ e ( r q )] + r − Γ b e ( r q )Combining and simplifying, r − I = d (cid:63) / d (cid:63) / d/ d/ α − κρ d (cid:63) / β − κρe ( α ) − (cid:18) κκ + 4 ωκ + 4 ρ (cid:19) ρα − κρ ϑ + (cid:102)(cid:102) Err . where, (cid:102)(cid:102) Err = (cid:102) Err + r − (cid:18) κ + 2 ω (cid:19) Err[ e ( r q )] + r − Γ b e ( r q )Using Bianchi to replace d (cid:63) / β , we deduce r − I = d (cid:63) / d (cid:63) / d/ d/ α − κρ (cid:18) − e α − κα + 4 ωα − ρϑ + r − Γ g · Γ b (cid:19) − κρe ( α ) − (cid:18) κκ + 4 ωκ + 4 ρ (cid:19) ρα − κρ ϑ + r − Γ g · Γ b + l.o.t.= d (cid:63) / d (cid:63) / d/ d/ α + 32 κρe α − κρe ( α ) − (cid:16) κω + ωκ + ρ (cid:17) ρα + r − Γ g · Γ b .I = e ( r e ( r q )) + 2 ωr e ( r q )= r (cid:40) d (cid:63) / d (cid:63) / d/ d/ α + 32 κρe α − κρe ( α ) (cid:111) + Err[ ST ] .13. PROOF OF PROPOSITION 2.4.6 ST ] = r (cid:102) Err + r (cid:18) κ + 2 ω (cid:19) Err[ e ( r q )] + r Γ b e ( r q ) + r Γ g · Γ b + r Err + (cid:16) r e (cid:0) Err[ e q ] (cid:1) + r Err[ e q ] (cid:17) + 2 ωr Err[ e q ] + r Γ b e ( r q ) + r Γ g · Γ b Recall that, see (A.12.8),Err = (cid:0) r − d / Γ b + r − Γ b · Γ b ) · α + r − d / ≤ Γ b · d / ≤ Γ g + r − d / ≤ Γ b · d / ≤ Γ b Hence,Err[ ST ] = r (cid:0) d / Γ b + r Γ b · Γ b ) · α + r d / ≤ Γ b · d / ≤ Γ g + r d / ≤ Γ b · d / ≤ Γ b + r Γ b e ( r q )+ r e (cid:0) Err[ e q ] (cid:1) + r Err[ e q ] + 2 ωr Err[ e q ]Recall that, see Proposition 2.3.14,Err[ e ( r q )] = r Γ b q + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) . Therefore, E = r e (cid:0) Err[ e q ] (cid:1) + r Err[ e q ] + 2 ωr Err[ e q ]= r (cid:0) Γ b e ( r q ) + e (Γ b ) r q (cid:17) + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) + r Γ b q + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) = r (cid:0) Γ b e ( r q ) + ( d ≤ Γ b ) r q (cid:17) + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) Thus, Err[ T S ] = r (cid:0) d / Γ b + r Γ b · Γ b ) · α + r (cid:0) Γ b e ( r q ) + ( d ≤ Γ b ) r q (cid:17) + r d ≤ (cid:0) e η · β (cid:1) + r d ≤ (cid:0) Γ b · Γ g (cid:1) which end the proof of Proposition 2.3.15. A.13 Proof of Proposition 2.4.6 In this section we give a proof of Proposition 2.4.6, i.e. we derive the wave equation forthe extreme curvature component α , (cid:3) g α = − ωe ( α ) + (2 κ + 4 ω ) e ( α ) + V α + Err( (cid:3) g α ) ,V : = (cid:18) − e ( ω ) + 12 κκ − κω + 2 κω − ωω − ρ + 4 e θ (Φ) (cid:19) α, (A.13.1)34 APPENDIX A. APPENDIX TO CHAPTER 2 whereErr( (cid:3) g α ) = 12 ϑe ( α ) + 34 ϑ ρ + e θ (Φ) ϑβ − κ ( ζ + 4 η ) β − ( ζ + η ) e ( β ) − ξe ( β )+ e θ (Φ)(2 ζ + η ) α + β + e (Φ) ηβ + e (Φ) ξβ − ( ζ + 4 η ) e ( β ) − ( e ( ζ ) + 4 e ( η )) β − κ + ω )( ζ + 4 η ) β + 2 e θ ( κ + ω ) β − e θ ((2 ζ + η ) α ) − ξe θ ( ρ ) + 2 ηe θ ( α )+ 32 ϑ ( e θ ( β ) + e θ (Φ) β ) + 3 ρ ( η + η + 2 ζ ) ξ + ( e θ ( η ) + e θ (Φ) η ) α + 14 κϑα − ωϑα − ϑϑα + ξξα + η α + 32 ϑζβ + 3 ϑ ( ηβ + ξβ ) − ϑ ( ζ + 4 η ) β. The equation for α can then be easily inferred by symmetry. Proof. We make use of the Bianchi identities e θ ( β ) − e θ (Φ) β = e ( α ) + (cid:16) κ − ω (cid:17) α + 32 ϑ ρ − ( ζ + 4 η ) β,e ( β ) + 2( κ + ω ) β = e θ ( α ) + 2 e θ (Φ) α + (2 ζ + η ) α + 3 ξρ. to infer that e ( e ( α )) = e ( e θ ( β )) − e θ (Φ) e ( β ) − e ( e θ (Φ)) β − (cid:16) κ − ω (cid:17) e ( α ) − (cid:18) e ( κ )2 − e ( ω ) (cid:19) α − ϑe ( ρ ) − e ( ϑ ) ρ + ( ζ + 4 η ) e ( β ) + ( e ( ζ ) + 4 e ( η )) β = e ( e θ ( β )) − e θ (Φ) (cid:16) e θ ( α ) + 2 e θ (Φ) α − κ + ω ) β + (2 ζ + η ) α + 3 ξρ (cid:17) − ( D D θ Φ + D D e θ Φ) β − (cid:16) κ − ω (cid:17) e ( α ) − (cid:18) e ( κ )2 − e ( ω ) (cid:19) α − ϑe ( ρ ) − e ( ϑ ) ρ + ( ζ + 4 η ) e ( β ) + ( e ( ζ ) + 4 e ( η )) β. Hence, e ( e ( α )) = e ( e θ ( β )) − e θ (Φ)( e θ ( α ) + 2 e θ (Φ) α ) + 2 e θ (Φ)( κ + ω ) β − e θ (Φ) ξρ + e (Φ) e θ (Φ) β − (cid:16) κ − ω (cid:17) e ( α ) − (cid:18) e ( κ )2 − e ( ω ) (cid:19) α − ϑe ( ρ ) − e ( ϑ ) ρ − e θ (Φ)(2 ζ + η ) α − β − e (Φ) ηβ − e (Φ) ξβ +( ζ + 4 η ) e ( β ) + ( e ( ζ ) + 4 e ( η )) β .13. PROOF OF PROPOSITION 2.4.6 e θ ( e θ ( α )) = e θ ( e ( β )) + 2( κ + ω ) e θ ( β ) + 2 e θ ( κ + ω ) β − e θ (Φ) e θ ( α ) − e θ ( e θ (Φ)) α − e θ ((2 ζ + η ) α ) − e θ ( ξρ )= e θ ( e ( β )) + 2( κ + ω ) (cid:16) e θ (Φ) β + e ( α ) + (cid:16) κ − ω (cid:17) α + 32 ϑ ρ − ( ζ + 4 η ) β (cid:17) +2 e θ ( κ + ω ) β − e θ (Φ) e θ ( α ) − D θ D θ Φ + D D θ e θ Φ) α − e θ ((2 ζ + η ) α ) − e θ ( ξρ )= e θ ( e ( β )) + 2( κ + ω ) e θ (Φ) β + 2( κ + ω ) e ( α ) + 2( κ + ω ) (cid:16) κ − ω (cid:17) α + 3( κ + ω ) ϑ ρ − e θ (Φ) e θ ( α ) − (cid:18) ρ − e θ (Φ) + 12 χe (Φ) + 12 χe (Φ) (cid:19) α − e θ ( ξ ) ρ − κ + ω )( ζ + 4 η ) β + 2 e θ ( κ + ω ) β − e θ ((2 ζ + η ) α ) − ξe θ ( ρ ) . In view of Lemma 2.4.1, we have (cid:3) g f = − e ( e ( f )) + e θ ( e θ ( f )) − κe ( f ) + (cid:18) − κ + 2 ω (cid:19) e ( f ) + e θ (Φ) e θ ( f ) + 2 ηe θ ( f ) . We infer (cid:3) g α = − e ( e ( α )) + e θ ( e θ ( α )) − κe ( α ) + (cid:18) − κ + 2 ω (cid:19) e ( α ) + e θ (Φ) e θ ( α ) + 2 ηe θ ( α )= [ e θ , e ]( β ) − e (Φ) e θ (Φ) β + 32 ϑe ( ρ ) + 32 e ( ϑ ) ρ + 3( κ + ω ) ϑ ρ − e θ ( ξ ) − e θ (Φ) ξ ) ρ − ωe ( α ) + (cid:18) κ + 4 ω (cid:19) e ( α )+ (cid:18) e ( κ )2 − e ( ω ) + κκ − κω + κω − ωω − ρ + 4 e θ (Φ) − χe (Φ) − χe (Φ) (cid:19) α + e θ (Φ)(2 ζ + η ) α + β + e (Φ) ηβ + e (Φ) ξβ − ( ζ + 4 η ) e ( β ) − ( e ( ζ ) + 4 e ( η )) β − κ + ω )( ζ + 4 η ) β + 2 e θ ( κ + ω ) β − e θ ((2 ζ + η ) α ) − ξe θ ( ρ ) + 2 ηe θ ( α ) . Next, we have[ e θ , e ]( β ) = χe θ ( β ) − ( ζ + η ) e ( β ) − ξe ( β )= χ (cid:18) e θ (Φ) β + e ( α ) + (cid:16) κ − ω (cid:17) α + 32 ϑ ρ − ( ζ + 4 η ) β (cid:19) − ( ζ + η ) e ( β ) − ξe ( β )36 APPENDIX A. APPENDIX TO CHAPTER 2 and hence (cid:3) g α = − ωe ( α ) + (cid:18) κ + χ + 4 ω (cid:19) e ( α ) + V α + 32 ϑe ( ρ ) + 32 e ( ϑ ) ρ + 3( κ + ω ) ϑ ρ − e θ ( ξ ) − e θ (Φ) ξ ) ρ + 32 χϑ ρ + Err where, V := e ( κ )2 − e ( ω ) + κκ − κω + κω − ωω − ρ + 4 e θ (Φ) − χe (Φ) − χe (Φ)+ χ κ − χω, Err := e θ (Φ) ϑβ − χ ( ζ + 4 η ) β − ( ζ + η ) e ( β ) − ξe ( β )+ e θ (Φ)(2 ζ + η ) α + β + e (Φ) ηβ + e (Φ) ξβ − ( ζ + 4 η ) e ( β ) − ( e ( ζ ) + 4 e ( η )) β − κ + ω )( ζ + 4 η ) β + 2 e θ ( κ + ω ) β − e θ ((2 ζ + η ) α ) − ξe θ ( ρ ) + 2 ηe θ ( α ) . Next, we make use of e ( ϑ ) + κϑ + 2 ωϑ = − α + 2( e θ ( ξ ) − e θ (Φ) ξ ) + 2( η + η + 2 ζ ) ξ,e ( ρ ) + 32 κρ = e θ ( β ) + e θ (Φ) β − ϑα + ζβ + 2( ηβ + ξβ ) , to calculate the term I : = 32 ϑe ( ρ ) + 32 e ( ϑ ) ρ + 3( κ + ω ) ϑ ρ − e θ ( ξ ) − e θ (Φ) ξ ) ρ + 32 χϑ ρ = 32 ϑ (cid:18) − κρ + d/ β (cid:19) + 32 ρ ( − κϑ − ωϑ − α + 2( e θ ( ξ ) − e θ (Φ) ξ )) + 3( κ + ω ) ϑ ρ − e θ ( ξ ) − e θ (Φ) ξ ) ρ + 32 κ + ϑ ϑ ρ + l.o.t.= − ρα + 32 ϑ d/ β + 34 ϑ ρ. Hence, (cid:3) g α = − ωe ( α ) + (cid:18) κ + χ + 4 ω (cid:19) e ( α ) + ( V − ρ ) α + Err + 32 ϑ d/ β + 34 ϑ ρ. Using also, e ( κ ) + 12 κκ − ωκ = 2( e θ ( η ) + e θ (Φ) η ) + 2 ρ − ϑϑ + 2( ξξ + η ) .14. PROOF OF THEOREM 2.4.7 χ = κ + ϑ , as well as 2 χ = κ + ϑ , we finally obtain (cid:3) g α = − ωe ( α ) + (2 κ + 4 ω ) e ( α ) + V α + Err[ (cid:3) α ]as desired.We write schematically the error term,Err[ (cid:3) α ] = 12 ϑe ( α ) + 34 ϑ ρ + e θ (Φ) ϑβ − κ ( ζ + 4 η ) β − ( ζ + η ) e ( β ) − ξe ( β )+ e θ (Φ)(2 ζ + η ) α + β + e (Φ) ηβ + e (Φ) ξβ − ( ζ + 4 η ) e ( β ) − ( e ( ζ ) + 4 e ( η )) β − κ + ω )( ζ + 4 η ) β + 2 e θ ( κ + ω ) β − e θ ((2 ζ + η ) α ) − ξe θ ( ρ ) + 2 ηe θ ( α )+ 32 ϑ ( e θ ( β ) + e θ (Φ) β ) + 3 ρ ( η + η + 2 ζ ) ξ + ( e θ ( η ) + e θ (Φ) η ) α + 14 κϑα − ωϑα − ϑϑα + ξξα + η α + 32 ϑζβ + 3 ϑ ( ηβ + ξβ ) − ϑ ( ζ + 4 η ) β as follows, Err[ (cid:3) g α ] = (cid:18) r Γ g + 1 r d / Γ g (cid:19) α + Γ g e ( α ) + 1 r Γ g d /α + (cid:18) β + 1 r Γ g + 1 r d Γ g (cid:19) β + 1 r Γ g d ( β ) + Γ g e ( β )+ (Γ g ) ρ + 1 r Γ g d / ( ρ )= Γ g e ( α, β ) + r − Γ ≤ g · d ≤ ( α, β, ˇ ρ ) + β + Γ g ρ. This concludes the proof of Proposition 2.4.6. A.14 Proof of Theorem 2.4.7 Recall the symbolic notation used in the statement of the theorem.Γ g = (cid:110) ϑ, η, η, ζ, A (cid:111) , Γ b = (cid:110) ϑ, ξ, A (cid:111) , d Γ g = (cid:110) d ϑ, re θ ( κ ) , d η, d η, d ζ, d A (cid:111) , d Γ b = (cid:110) d ϑ, e θ ( κ ) , d ξ, d A (cid:111) , where A = r e ( r ) − κ, A = r e ( r ) − κ. We also denote, for s ≥ d s Γ g = d s − d Γ g , d s Γ b = d s − d Γ b , APPENDIX A. APPENDIX TO CHAPTER 2 for higher derivatives with respect to d = ( e , re , d / ) (see definition 2.1.37 for the notation d / and d / s ).We also recall Remark 2.3.9. Remark A.14.1. According to the main bootstrap assumptions BA-E , BA-D (see sec-tion 3.4.1.) the terms Γ b behave worse in powers of r than the terms in Γ g . Thus, inthe symbolic expressions below, we replace the terms of the form Γ g + Γ b by Γ b . We alsoreplace r − Γ b by Γ g . We will denote l.o.t. all cubic and higher error terms in ˇΓ , ˇ R . Wealso include in l.o.t. terms which decay faster in powers of r that those taking into accountby the main quadratic terms. Recall that q = r Q ( α ) , (A.14.1)where Q is the operator Q := e e + (2 κ − ω ) e + W, W := − e ( ω ) + 8 ω − ω κ + 12 κ . (A.14.2) Lemma A.14.2. The quantity q is fully invariant with respect to the conformal frametransformations e (cid:48) = λ − e , e = λe , e (cid:48) θ = e θ . Proof. The proof is an immediate consequence of Definition A.14.3 and Lemmas A.14.5,A.14.4 below.We recall that under the above mentioned frame transformation we have α (cid:48) = λ α, β (cid:48) = λβ, ρ (cid:48) = ρ, κ (cid:48) = λ − κ, κ (cid:48) = λκ, η (cid:48) = η, η (cid:48) = η,ω (cid:48) = λ − (cid:18) ω + 12 e (log λ ) (cid:19) , ω (cid:48) = λ (cid:18) ω − e (log λ ) (cid:19) , ζ (cid:48) = ζ − e θ (log λ ) . Definition A.14.3. We say that a reduced tensor is conformal invariant of type a , i.e. a -conformal invariant, if under the conformal change of frames e (cid:48) = λ − , e (cid:48) = λe ittransforms by f (cid:48) = λ a f. Note that for a given Ricci or curvature coefficient a coincides with the signature of the component. .14. PROOF OF THEOREM 2.4.7 Lemma A.14.4. Let f be an a -conformal invariant tensor.1. The tensor ∇ f : = e f − aωf (A.14.3) is a − conformal invariant.2. The tensor ∇ f : = e f + 2 aωf (A.14.4) is a + 1 conformal invariant.3. The tensor, ∇ / ( c ) A f = ∇ / A f + αζ A f (A.14.5) is a -conformal invariant.Proof. Immediate verification. Lemma A.14.5. We have Q ( α ) = ∇ ( ∇ α ) + 2 κ ∇ α + 12 κ α. Proof. We have, ∇ ( ∇ α ) = ∇ ( e α − ωα ) = e ( e α − ωα ) − ω ( e α − ωα )= e e α − e ωα − ωe α − ωe α + 8 ω α. Hence, Q ( α ) = ∇ ( ∇ α ) + 2 κ ∇ α + 12 κα = e e α − e ωα − ωe α − ωe α + 8 ω α + 2 κ ( e α − ωα ) + 12 κ α = e e α + (2 κ − ω ) e α + (cid:18) − e ω + 8 ω − κ ω + 12 κ (cid:19) as stated.40 APPENDIX A. APPENDIX TO CHAPTER 2 Remark A.14.6. Using the definitions of ∇ , ∇ the null structure equations for κ, κ take the form, ∇ κ + 12 κ = 2 d/ ξ + Γ b · Γ b = r − d / + l.o.t. , ∇ κ + 12 κ κ = 2 d/ η + 2 ρ + Γ g · Γ b = 2 ρ + r − d / Γ g , ∇ κ + 12 κ κ = 2 d/ η + 2 ρ + Γ g · Γ b = 2 ρ + r − d / Γ g , ∇ κ + 12 κ = 2 d/ ξ + Γ g · Γ g = r − d / Γ g . (A.14.6) Also, since ρ is -conformal ∇ ρ + 32 κρ = d/ β + Γ g · Γ b = r − d / Γ g . (A.14.7) Definition A.14.7. Given f an a -conformal S -tangent tensor we define its a-conformalLaplacian to be ( c ) (cid:52) / f = ( c ) ∇ / A ( c ) ∇ / A f. Lemma A.14.8. The following formula holds true for a -conformal tensor f ( c ) (cid:52) / f = (cid:52) / f + 4 ζ ∇ / f + 2 (cid:0) div ζ + 2 | ζ | (cid:1) f. In particular we have, ( c ) (cid:52) / f = (cid:52) / f + r − d / ≤ (Γ g · f ) . Proof. Immediate verification.The goal of this section is to prove Theorem 2.4.7 which we recall below for the convenienceof the reader. Theorem A.14.9. The invariant scalar quantity q defined in (2.3.10) verifies the equa-tion, (cid:3) q + κκ q = Err [ (cid:3) q ] (A.14.8) where, schematically,Err [ (cid:3) q ] := r d ≤ (Γ g · ( α, β )) + e (cid:16) r d ≤ (Γ g · ( α, β )) (cid:17) + d ≤ (Γ g · q ) + l.o.t. (A.14.9) .14. PROOF OF THEOREM 2.4.7 Definition A.14.10. Given a quadratic or higher order E we say the following1. E ∈ Good if r E can be expressed in the form (2.4.8) .2. E ∈ Good if after applying r e or r it can be expressed in the form (2.4.8) .3. E ∈ Good if after applying r e e , r e or r it can be expressed in the form (2.4.8) . In view of the definition we note that,( e + r − )Good = Good , Q Good = Good . To prove the theorem we have to check that Err[ (cid:3) q ] = r Good. A.14.1 The Teukolsky equation for α We recall below Proposition 2.4.6. Lemma A.14.11. We have (cid:3) α = − ωe ( α ) + (4 ω + 2 κ ) e ( α ) + V α + Err [ (cid:3) g α ] ,V = − ρ − e ( ω ) − ωω + 2 ω κ − κ ω + 12 κ κ, where Err [ (cid:3) g α ] is given schematically byErr ( (cid:3) g α ) := Γ g e ( α ) + r − d ≤ (cid:16) ( η, Γ g )( α, β ) (cid:17) + ξ ( e ( β ) , r − d ˇ ρ ) . Remark A.14.12. Since ξ vanishes for r ≥ m , η ∈ Γ g and e α = r − d α we deduce,Err ( (cid:3) g α ) ∈ Good . Lemma A.14.13. The Teukolsky equation for α can be written in the form, L ( α ) = Good (A.14.10) where L is the operator L α = −∇ ∇ α + ( c ) (cid:52) / α − κ ∇ α − κ ∇ α − (cid:18) − ρ + 12 κκ (cid:19) α. (A.14.11) We also note that, for a -conformal tensor f , (cid:3) f = −∇ ∇ f + ( c ) (cid:52) / f − κ ∇ f − κ ∇ f + r − Γ g · d /f. (A.14.12)42 APPENDIX A. APPENDIX TO CHAPTER 2 Proof. Recall that we have (see Definition 2.4.2) (cid:3) α = − e ( e ( α )) + (cid:52) / α − κe ( α ) + (cid:18) − κ + 2 ω (cid:19) e ( α ) + 2 ηe θ ( α ) . Therefore, L ( α ) = − e ( e ( α )) + (cid:52) / α − κe ( α ) + (cid:18) − κ + 2 ω (cid:19) e ( α ) + 2 ηe θ ( α )+ 4 ωe ( α ) − (4 ω + 2 κ ) e ( α ) − V α = − e ( e ( α )) + (cid:52) / α − (cid:18) κ − ω (cid:19) e α − (cid:18) κ + 2 ω (cid:19) e α + 2 ηe θ ( α ) − V α = − e ( e ( α )) + ( c ) (cid:52) / α − (cid:18) κ − ω (cid:19) e α − (cid:18) κ + 2 ω (cid:19) e α − V α + Good . On the other hand, ∇ ( ∇ ( α )) = ∇ (cid:0) e α − ωα (cid:1) = e (cid:0) e α − ωα (cid:1) + 2 ω (cid:0) e α − ωα (cid:1) = e e α − ωe α − e ωα + 2 ωe α − ωωα. Hence, −∇ ∇ α − κ ∇ α − κ ∇ α = − e e α + 4 ωe α + 4 e ωα − ωe α + 8 ωωα − κ (cid:0) e α − ωα ) − κ ( e α + 4 ωα )= − e e α − 12 ( κ − ω ) e α − (cid:18) κ + 2 ω (cid:19) e α + (cid:0) e ω + 8 ωω + 10 κω − ωκ (cid:1) α. We deduce, with V (cid:48) = − ρ + κκ , −∇ ∇ α − κ ∇ α − κ ∇ − V (cid:48) α = − e e α − 12 ( κ − ω ) e α − (cid:18) κ + 2 ω (cid:19) e α + (cid:18) e ω + 8 ωω + 10 κω − ωκ + 4 ρ − κκ (cid:19) α = − e e α − 12 ( κ − ω ) e α − (cid:18) κ + 2 ω (cid:19) e α − V α. Hence, L α = −∇ ∇ α + ( c ) (cid:52) / α − κ ∇ α − κ ∇ α − (cid:18) − ρ + 12 κκ (cid:19) α ∈ Good as desired. The proof of the second part of the lemma follows in the same manner. .14. PROOF OF THEOREM 2.4.7 A.14.2 Commutation lemmas The goal of the following lemmas is to calculate the commutator of Q with L . Lemma A.14.14. Give f an a -conformal tensor we have, [ ∇ , ∇ ] f = 2 aρf + r − Γ g d / ≤ f. (A.14.13) Proof. We have[ ∇ , ∇ ] f = ∇ ∇ f − ∇ ∇ f = (cid:0) e − a + 1) ω (cid:1)(cid:0) e f + 2 aωf (cid:1) − (cid:0) e + 2( a − ω (cid:1) ( e f − aωf (cid:1) = e e f − a + 1) ωe f + 2 ae ( ωf ) − a ( a + 1) ωωf − e e f − a − ωe f + 2 ae ( ωf ) + 4 a ( a − ωω = [ e , e ] f − ωe f + 2 ωe ( f ) + 2 a (cid:16) e ω + e ω − ωω (cid:17) f. Recall that, [ e , e ] = − ωe + 2 ωe + 2( η − η ) e θ ,e ω + e ω − ωω = ρ + Γ g · Γ b . We deduce , [ ∇ , ∇ ] f = 2 aρ + r − Γ g d / ≤ f as stated. Lemma A.14.15. Assume f a-conformal and g is b- conformal. Then f g is a + b -conformal and ∇ ( f g ) = f ∇ g + g ∇ f, ∇ ( f g ) = f ∇ g + g ∇ f. Proof. Indeed ∇ ( f g ) = e ( f g ) − a + b ) ωf g = f e g + ge f − a + b ) ωf g = f ∇ g + g ∇ f as stated. Recall that η ∈ Γ g in the frame we are using. APPENDIX A. APPENDIX TO CHAPTER 2 Lemma A.14.16. We have, [ Q, ∇ ] α = κ ∇ α + 12 κ α + Good , [ Q, ∇ ] α = (cid:0) ρ + κκ (cid:1) ∇ α + 12 κκ α + Good . (A.14.14) Also, [ Q, ∇ ∇ ] α = (cid:0) − ρ + κκ (cid:1) ∇ ∇ α + κ ∇ ∇ α + 12 κ ∇ α + (cid:18) ρκ − κκ (cid:19) ∇ α + 32 κ (cid:18) − κκ + 2 ρ (cid:19) α + Good . (A.14.15) Proof. We have ,[ Q, ∇ ] α = (cid:18) ∇ ∇ + 2 κ ∇ + 12 κ (cid:19) ∇ α − ∇ (cid:18)(cid:18) ∇ ∇ + 2 κ ∇ + 12 κ (cid:19) α (cid:19) = − ∇ ( κ ) ∇ α − κ ( ∇ κ ) α = − (cid:18) − κ + r − Γ b (cid:19) ∇ α − κ (cid:18) − κ + r − Γ b (cid:19) α = κ ∇ α + 12 κ α + r − Γ g d ≤ α and,[ Q, ∇ ] α = (cid:18) ∇ ∇ + 2 κ ∇ + 12 κ (cid:19) ∇ α − ∇ (cid:18)(cid:18) ∇ ∇ + 2 κ ∇ + 12 κ (cid:19) α (cid:19) = (cid:0) ∇ ∇ ∇ − ∇ ∇ ∇ (cid:1) α + 2 κ (cid:0) ∇ ∇ − ∇ ∇ (cid:1) α − ∇ κ ∇ α − κ ( ∇ κ ) α = ∇ (cid:16)(cid:2) ∇ , ∇ ] α (cid:17) + [ ∇ , ∇ ] ∇ α + 2 κ [ ∇ , ∇ ] α − ∇ κ ∇ α − κ ( ∇ κ ) α. In view of Lemma A.14.14 we have, (cid:2) ∇ , ∇ ] α = 4 ρα + r − Γ g · d / ≤ α, (cid:2) ∇ , ∇ ] ∇ α = 2 ρ ∇ α + r − Γ g · d / ≤ ∇ α. Hence,[ Q, ∇ ] α = ∇ (cid:16) ρα + r − Γ g d / ≤ α (cid:17) + (cid:16) ρ + r − Γ g d / ≤ (cid:17) ∇ α + 2 κ (cid:16) ρα + r − Γ g d / ≤ α (cid:17) − ∇ κ ∇ α − κ ( ∇ κ ) α = (cid:0) ρ − ∇ κ (cid:1) ∇ α + (cid:0) ∇ ρ + 8 κρ − κ ∇ κ (cid:1) α + Good . Recall that r − Γ b = Γ g . .14. PROOF OF THEOREM 2.4.7 ∇ ρ and ∇ κ ,4 ∇ ρ + 8 κρ − κ ∇ κ = 4 (cid:18) − κρ + r − d / Γ g (cid:19) + 8 κρ − κ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) = 12 κκ + r − d / Γ g ρ − ∇ κ = 6 ρ − (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) = 2 ρ + κκ + r − d / Γ g . Hence, [ Q, ∇ ] α = (cid:0) ρ + κκ (cid:1) ∇ α + 12 κκ α + Good as stated.Also, [ Q, ∇ ∇ ] α = [ Q, ∇ ] ∇ α + ∇ (cid:16) [ Q, ∇ ] α (cid:17) . (A.14.16)We first calculate, as above, for f = ∇ α [ Q, ∇ ] f = (cid:18) ∇ ∇ + 2 κ ∇ + 12 κ (cid:19) ∇ f − ∇ (cid:18)(cid:18) ∇ ∇ + 2 κ ∇ + 12 κ (cid:19) f (cid:19) = (cid:0) ∇ ∇ ∇ − ∇ ∇ ∇ (cid:1) f + 2 κ (cid:0) ∇ ∇ − ∇ ∇ (cid:1) f − ∇ κ ∇ α − κ ( ∇ κ ) f = ∇ (cid:16)(cid:2) ∇ , ∇ ] f (cid:17) + [ ∇ , ∇ ] ∇ f + 2 κ [ ∇ , ∇ ] f − ∇ κ ∇ f − κ ( ∇ κ ) f. In view of Lemma A.14.14, since f = ∇ α is 1-conformal and ∇ f is 0-conformal, wehave (cid:2) ∇ , ∇ ] f = 2 ρf + r − Γ g d / ≤ f, (cid:2) ∇ , ∇ ] ∇ f = r − Γ g d / ≤ ∇ f. Hence[ Q, ∇ ] f = ∇ (cid:16) ρf + r − Γ g d / ≤ f (cid:17) + (cid:16) r − Γ g d / ≤ (cid:17) ∇ f + 2 κ (cid:16) ρf + r − Γ g d / ≤ f (cid:17) − ∇ κ ∇ f − κ ( ∇ κ ) f = (cid:0) ρ − ∇ κ (cid:1) ∇ f + (cid:0) ∇ ρ + 4 κρ − κ ∇ κ (cid:1) f + r − Γ g d / ≤ ∇ f + r − Γ g d / ≤ f. APPENDIX A. APPENDIX TO CHAPTER 2 Therefore,[ Q, ∇ ] ∇ α = (cid:0) ρ − ∇ κ (cid:1) ∇ ∇ α + (cid:0) ∇ ρ + 4 κρ − κ ∇ κ (cid:1) ∇ α + r − Γ g d ≤ α. As above, 2 ρ − ∇ κ = 2 ρ − (cid:18) − κκ + 2 ρ + r − Γ g (cid:19) = − ρ + κκ + r − Γ g , ∇ ρ + 4 κρ − κ ∇ κ = 2 (cid:18) − κρ + r − Γ g (cid:19) + 4 κρ − κ (cid:18) − κκ + 2 ρ + r − Γ g (cid:19) = 12 κκ − ρκ + r − Γ g . Hence, since r − Γ g ( ∇ ∇ α, ∇ α ) = r − Γ g · d ≤ α = Good,[ Q, ∇ ] ∇ α = (cid:0) − ρ + κκ (cid:1) ∇ ∇ α + (cid:18) κκ − ρκ (cid:19) ∇ α + Good . (A.14.17)We deduce,[ Q, ∇ ∇ ] α = [ Q, ∇ ] ∇ α + ∇ (cid:16) [ Q, ∇ ] α (cid:17) = (cid:0) − ρ + κκ (cid:1) ∇ ∇ α + (cid:18) κκ − ρκ (cid:19) ∇ α + ∇ (cid:18) κ ∇ α + 12 κ α + Good (cid:19) + Good= (cid:0) − ρ + κκ (cid:1) ∇ ∇ α + κ ∇ ∇ α + 12 κ ∇ α + (cid:18) ∇ ( κ ) + 12 κκ − ρκ (cid:19) ∇ α + 32 κ ∇ κα + Good . Note that ∇ ( κ ) + 12 κκ − ρκ = 2 κ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) + 12 κκ − ρκ = 3 ρκ − κκ + r − d / Γ g , κ ∇ κ = 32 κ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) . Hence, [ Q, ∇ ∇ ] α = (cid:0) − ρ + κκ (cid:1) ∇ ∇ α + κ ∇ ∇ α + 12 κ ∇ α + (cid:18) ρκ − κκ (cid:19) ∇ α + 32 κ (cid:18) − κκ + 2 ρ (cid:19) α + Goodas stated. .14. PROOF OF THEOREM 2.4.7 Lemma A.14.17. Given f a -conformal tensor in s we have [ ∇ , ( c ) (cid:52) / ] f = − κ ( c ) (cid:52) / f + r − d ≤ (Γ b · f ) . Proof. Recall that for a 2-conformal spacetime tensor f we have ( c ) (cid:52) / f = (cid:52) / f + r − d / ≤ (Γ g · f ) . Hence, [ ∇ , ( c ) (cid:52) / ] f = [ ∇ , (cid:52) / ] f + ∇ (cid:0) r − d / ≤ (Γ g · f ) (cid:1) + r − d / ≤ (Γ g · ∇ f ) . On the other hand, since ∇ / ω = r − d / Γ b , ∇ / ω = r − d / Γ b ,[ ∇ , (cid:52) / ] f = [ e − ω, (cid:52) / ] f = [ e , (cid:52) / ] f + r − d / ≤ (Γ b · f ) . We deduce, [ ∇ , ( c ) (cid:52) / ] f = [ e , (cid:52) / ] f + r − d ≤ (Γ b · f ) + e (cid:0) r − d / ≤ (Γ g · f ) (cid:1) . In the reduced form, for an s tensor f ,[ ∇ , ( c ) (cid:52) / ] f = [ e , (cid:52) / ] f + r − d ≤ (Γ b · f ) + e (cid:0) r − d / ≤ (Γ g · f ) (cid:1) . We now recall that (cid:52) / = − d (cid:63) / d/ + 2 K . Hence, applying the commutation Lemma (cid:52) / , e ] f = [ − d (cid:63) / d/ + 2 K, e ] f = − d (cid:63) / [ d/ , e ] f − [ d (cid:63) / , e ] d/ f − e ( K ) f = − d (cid:63) / (cid:18) κ d/ + Com ( f ) (cid:19) − (cid:18) κ d (cid:63) / + Com ∗ ( d/ f ) (cid:19) − e ( K )= − κ d (cid:63) / d/ f − e ( K ) f + e θ ( κ ) d/ f − d (cid:63) / ( Com ( f )) − Com ∗ ( d/ f )= − κ d (cid:63) / d/ f − e ( K ) f + r − d ≤ (Γ b · f ) + r − d ≤ (Γ g · e f )= κ (cid:52) / f − e K + κK ) f + r − d ≤ (Γ b · f ) + r − d ≤ (Γ g · e f ) . Recall that we have Com ( f ) = − ϑ d (cid:63) / f + ( ζ − η ) e f − ηe Φ f − ξ ( e f + ke (Φ) f ) − βf,Com ∗ ( f ) = − ϑ d/ f − ( ζ − η ) e f − ηe Φ f + ξ ( e f − e (Φ) f ) − βf. APPENDIX A. APPENDIX TO CHAPTER 2 Note that, ignoring the quadratic terms, e K + κK = − e (cid:18) ρ + 14 κκ (cid:19) − κ (cid:18) ρ + 14 κκ (cid:19) = − e ρ − κρ − (cid:16) e ( κκ ) + κκ (cid:17) = 12 κρ − d/ β − (cid:26) κ (cid:18) − κ − ω κ (cid:19) + κ (cid:18) − κκ + 2 ωκ + 2 d/ η + 2 ρ (cid:19) + κκ (cid:27) = − d/ β − κ d/ η. We deduce, [ e , (cid:52) / ] = − [ (cid:52) / , e ] f = − κ (cid:52) / f + r − d ≤ (Γ b · f ) . Consequently, [ ∇ , ( c ) (cid:52) / ] f = − κ ( c ) (cid:52) / f + r − d ≤ (Γ b · f )as stated. Lemma A.14.18. We have, [ Q, ( c ) (cid:52) / ] α = − κ ∇ c ) (cid:52) / α − κ c ) (cid:52) / α + Good . (A.14.18) Proof. We have[ Q, ( c ) (cid:52) / ] α = (cid:20) ∇ ∇ + 2 κ ∇ + 12 κ (cid:21) ( c ) (cid:52) / α − ( c ) (cid:52) / (cid:20) ∇ ∇ α + 2 κ ∇ α + 12 κ α (cid:21) = ∇ [ ∇ , ( c ) (cid:52) / ] α + [ ∇ , ( c ) (cid:52) / ] e α + [2 κ ∇ , ( c ) (cid:52) / ] α + (cid:20) κ , ( c ) (cid:52) / (cid:21) α. Note that [2 κ ∇ , ( c ) (cid:52) / ] α = 2 κ [ ∇ , ( c ) (cid:52) / ] α + Good , (cid:20) κ , ( c ) (cid:52) / (cid:21) α = Good . Hence, using the previous commutation Lemma,[ Q, ( c ) (cid:52) / ] α = ∇ [ ∇ , ( c ) (cid:52) / ] α + [ ∇ , ( c ) (cid:52) / ] e α + 2 κ [ ∇ , ( c ) (cid:52) / ] α + Good= ∇ (cid:16) − κ ( c ) (cid:52) / α + r − d ≤ (Γ b · α ) (cid:17) + (cid:16) − κ ( c ) (cid:52) / ∇ α + r − d ≤ (Γ b · ∇ α ) (cid:17) + 2 κ (cid:16) − κ ( c ) (cid:52) / α + r − d ≤ (Γ b · α ) (cid:17) + Good= − κ (cid:0) ∇ c ) (cid:52) / α + ( c ) (cid:52) / ∇ α (cid:1) − (cid:0) ∇ κ + 2 κ (cid:1) ( c ) (cid:52) / α + Good= − κ (cid:0) ∇ c ) (cid:52) / α − [ ∇ , ( c ) (cid:52) / ] α (cid:1) − (cid:0) ∇ κ + 2 κ (cid:1) ( c ) (cid:52) / α + Good= − κ ∇ c ) (cid:52) / α − (cid:0) ∇ κ + 3 κ (cid:1) ( c ) (cid:52) / α + Good . .14. PROOF OF THEOREM 2.4.7 ∇ κ + 3 κ ) ( c ) (cid:52) / α = (cid:18) κ + r − d / Γ b (cid:19) ( c ) (cid:52) / α = 52 κ + r − d / Γ g · d / ≤ α. Hence, [ Q, ( c ) (cid:52) / ] α = − κ ∇ c ) (cid:52) / α − κ c ) (cid:52) / α + Goodas stated. Lemma A.14.19. We have Q ( f g ) = Q ( f ) g + f Q ( g ) + 2 ∇ f ∇ g − κ f g. Also, [ Q, f e ] g = Q ( f ) ∇ g + f [ Q, e ] g + 2 ∇ f ∇ ∇ g − κ f ∇ g, [ Q, f ∇ ] g = Q ( f ) ∇ g + f [ Q, ∇ ] g + 2 ∇ f ∇ ∇ g − κ f ∇ g. Proof. Recall that, Q = ∇ ∇ + 2 κ ∇ + 12 κ . Hence, Q ( f g ) = (cid:20) ∇ ∇ + 2 κ ∇ + 12 κ (cid:21) ( f g )= ( ∇ ∇ f ) g + f ( ∇ ∇ g ) + 2 ∇ f ∇ g + 2 κ ( ∇ f g + f ∇ g ) + 12 κ f g = (cid:16) ∇ ∇ f + 2 κ ∇ f (cid:17) g + 2 ∇ f ∇ g + f Q ( g )= Q ( f ) g + f Q ( g ) + 2 ∇ f ∇ g − κ f g. Also,[ Q, f ∇ ] g = Q ( f ∇ g ) − f ∇ Q ( g ) = Q ( f ) ∇ g + f Q ∇ ( g ) + 2 ∇ f ∇ ∇ g − κ f ∇ g − f ∇ Q ( g )= (cid:18) Q ( f ) − κf (cid:19) ∇ g + f [ Q, ∇ ] g + 2 ∇ f ∇ ∇ g. Similarly, [ Q, f ∇ ] g = (cid:18) Q ( f ) − κ f (cid:19) ∇ g + f [ Q, ∇ ] g + 2 ∇ f ∇ ∇ g as stated.50 APPENDIX A. APPENDIX TO CHAPTER 2 A.14.3 Main commutation Proposition A.14.20. The following identity holds true. [ Q, L ] α = − κ ∇ Q ( α ) + C Q Q ( α ) + Good , (A.14.19) where, C Q = − ρ − κκ. Proof. In view of Lemma A.14.13, we have L α = −∇ ∇ α + ( c ) (cid:52) / α − κ ∇ α − κ ∇ α − (cid:18) − ρ + 12 κκ (cid:19) α = Good . Hence, we infer[ Q, L ] α = − [ Q, ∇ ∇ ] α + [ Q, (cid:52) / ] α − 12 [ Q, κ ∇ ] α − 52 [ Q, κ ∇ ] α + (cid:20) Q, ρ − κκ (cid:21) α = I + J + K + L + M (A.14.20)with I, J, K, L, M denoting each of the commutators on the left of (A.14.20). Expression for I In view of Lemma A.14.16 we have, for I = − [ Q, ∇ ∇ ] α , I = (2 ρ − κκ (cid:1) ∇ ∇ α − κ ∇ ∇ α − κ ∇ α − (cid:18) − κκ + 3 ρκ (cid:19) ∇ α − κ (cid:18) − κκ + 2 ρ (cid:19) α + Good . (A.14.21) Expression for J Using Lemma A.14.18, J = [ Q, ( c ) (cid:52) / ] α = − κ ∇ c ) (cid:52) / α − κ c ) (cid:52) / α. .14. PROOF OF THEOREM 2.4.7 L and the fact that L α = Good we write, (cid:52) / α = ∇ ∇ α + 52 κ ∇ α + 12 κ ∇ α + (cid:18) − ρ + 12 κκ (cid:19) α + Good . Hence, J = − κ ∇ (cid:18) ∇ ∇ α + 52 κ ∇ α + 12 κ ∇ α + (cid:18) − ρ + 12 κκ (cid:19) α (cid:19) − κ (cid:18) ∇ ∇ α + 52 κ ∇ α + 12 κ ∇ α + (cid:18) − ρ + 12 κκ (cid:19) α (cid:19) = − κ ∇ ∇ ∇ α − κκ ∇ ∇ α − κ ∇ ∇ α − κ (cid:18) − ρ + 12 κκ (cid:19) ∇ α − κ (cid:18) ∇ κ ∇ α + 12 ∇ κ ∇ α + ∇ (cid:18) − ρ + 12 κκ (cid:19) α (cid:19) − κ (cid:18) ∇ ∇ α + 52 κ ∇ α + 12 κ ∇ α + (cid:18) − ρ + 12 κκ (cid:19) α (cid:19) . According to Lemma A.14.14 ∇ ∇ ∇ α = ∇ ∇ ∇ α + [ ∇ , ∇ ] ∇ α = ∇ ∇ ∇ α + 2 ρ ∇ α + r − Γ g d / ≤ ∇ α = ∇ ∇ ∇ α + 2 ρ ∇ α + Good , ∇ ∇ α = ∇ ∇ α + 4 ρα + Good . We deduce, modulo Good error terms, J = − κ (cid:0) ∇ ∇ ∇ α + 2 ρ ∇ α (cid:1) − κκ ∇ ∇ α − κ (cid:0) ∇ ∇ α + 4 ρα (cid:1) − κ (cid:18) − ρ + 12 κκ (cid:19) ∇ α − κ ∇ κ ∇ α − κ ∇ κ ∇ α − κ ∇ (cid:18) − ρ + 12 κκ (cid:19) α − κ (cid:18) ∇ ∇ α + 52 κ ∇ α + 12 κ ∇ α + (cid:18) − ρ + 12 κκ (cid:19) α (cid:19) . Grouping terms we rewrite in the form, J = − κ ∇ ∇ ∇ α − κκ ∇ ∇ α + J ∇ ∇ α + J ∇ α + J ∇ α + J α. APPENDIX A. APPENDIX TO CHAPTER 2 We calculate the coefficients J , J , J , J as follows. J = − κ − κ = − κ ,J = − κ ∇ κ − κ = − κ (cid:18) − κ + r − d / Γ b (cid:19) − κ = − κ + r − d / Γ b ,J = − κρ − κ (cid:18) − ρ + 12 κκ (cid:19) − κ ∇ κ − κ = 4 κρ − κ − κ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) = − κρ − κ + r − d / Γ g ,J = − ρκ − κ ∇ (cid:18) − ρ + 12 κκ (cid:19) − κ (cid:18) − ρ + 12 κκ (cid:19) = 6 ρκ − κκ + 8 κ ∇ ρ − κ ( κ ∇ κ + κ ∇ κ )= 6 ρκ − κκ + 8 κ (cid:18) − κρ + r − d / Γ g (cid:19) − κ (cid:18) κ (cid:18) − κκ + 2 ρ + r − Γ g (cid:19) + κ (cid:18) − κ + r − d / Γ b (cid:19)(cid:19) = − κ ρ − κκ + κκ + r − d / Γ b + r − Γ g . Hence J ∇ α = − κ + Good ,J ∇ α = (cid:18) − κρ − κ (cid:19) ∇ α + Good ,J α = − κ ρ − κκ + Good . We finally derive, J = − κ ∇ ∇ ∇ α − κκ ∇ ∇ α − κ ∇ ∇ α − κ ∇ α − (cid:18) κρ + 194 κκ (cid:19) ∇ α − (cid:18) κ ρ + 14 κκ (cid:19) α + Good . (A.14.22) .14. PROOF OF THEOREM 2.4.7 Expression for K Also, using Lemma A.14.19 and Lemma A.14.16 (according to which we have the identity[ Q, ∇ ] α = (cid:0) ρ + κκ (cid:1) ∇ α + κκ α + Good ) K = − (cid:104) Q, κ ∇ (cid:105) α = − (cid:18) Q ( κ ) ∇ α + κ [ Q, ∇ ] α + 2 ∇ κ ∇ ∇ α − κ ∇ α (cid:19) = − (cid:18) Q ( κ ) − κ (cid:19) ∇ α − κ (cid:18) (2 ρ + κκ ) ∇ α + 12 κκ α (cid:19) − ∇ κ ∇ ∇ α + Good . Hence, K = −∇ κ ∇ ∇ α − (cid:18) Q ( κ ) − κ (cid:19) ∇ α − κ (cid:0) ρ + κκ (cid:1) ∇ α − κκ α + Good . We calculate the expression, Q ( κ ) − κ = ∇ ∇ κ + 2 κ ∇ κ = ∇ (cid:18) − κ + r − d / Γ b (cid:19) + 2 κ (cid:18) − κ + r − d / Γ b (cid:19) = − κ (cid:0) ∇ κ + κ (cid:1) + ∇ (cid:0) r − d / Γ b (cid:1) + r − d / Γ b = − κ + ∇ (cid:0) r − d / Γ b (cid:1) + r − d / Γ b . Hence, K = −∇ κ ∇ ∇ α + 14 κ ∇ α − κ (cid:0) ρ + κκ (cid:1) ∇ α − κκ α + ∇ (cid:0) r − d / Γ b (cid:1) ∇ α + Good . We note that, ∇ (cid:0) r − d / Γ b (cid:1) ∇ α = ∇ (cid:0) r − d / Γ b ∇ α (cid:1) − r − d / Γ b ∇ ∇ α = ∇ (cid:0) r − d / Γ g d ≤ α (cid:1) − r − d / Γ g d ≤ α = Good . We deduce, K = −∇ κ ∇ ∇ α + 14 κ ∇ α − κ (cid:0) ρ + κκ (cid:1) ∇ α − κκ α + Good= − (cid:18) − κ + r − d / Γ b (cid:19) ∇ ∇ α + 14 κ ∇ α − κ (cid:0) ρ + κκ (cid:1) ∇ α − κκ α + Good= 12 κ ∇ ∇ α + 14 κ ∇ α − κ (cid:0) ρ + κκ (cid:1) ∇ α − κκ α + Good . APPENDIX A. APPENDIX TO CHAPTER 2 In view of Lemma A.14.14 [ ∇ , ∇ ] α = 4 ρα + r − Γ g d / ≤ α . Hence K = 12 κ (cid:0) ∇ ∇ α + 4 ρα + r − Γ g d / ≤ α (cid:1) + 14 κ ∇ α − κ (cid:0) ρ + κκ (cid:1) ∇ α − κκ α + Good= 12 κ ∇ ∇ α + 14 κ ∇ α − κ (2 ρ + κκ ) ∇ α + κ (cid:18) ρ − κκ (cid:19) α + Good . We have thus derived K = 12 κ ∇ ∇ α + 14 κ ∇ α − κ (cid:0) ρ + κκ (cid:1) ∇ α + κ (cid:18) ρ − κκ (cid:19) α + Good . (A.14.23) Expression for L According to Lemma A.14.19 and Lemma A.14.16 (according to which we have the identity[ Q, ∇ ] α = κ ∇ α + κ α + Good ) L = − (cid:104) Q, κe (cid:105) α = − (cid:18) Q ( κ ) ∇ α + κ [ Q, ∇ ] α + 2 ∇ κ ∇ ∇ α − κκ ∇ α (cid:19) = − (cid:18) Q ( κ ) ∇ α + κ (cid:18) κ ∇ α + 12 κ α (cid:19) + 2 ∇ κ ∇ ∇ α − κκ ∇ α (cid:19) + Good= − ∇ κ ∇ ∇ α − (cid:18) Q ( κ ) + 12 κκ (cid:19) ∇ α − κκ α + Good . Note that Q ( κ ) = ∇ ∇ κ + 2 κ ∇ κ + 12 κκ = ∇ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) + 2 κ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) + 12 κκ = − (cid:0) κ ∇ κ + κ ∇ κ (cid:1) + 2 (cid:18) − κρ + r − d / Γ g (cid:19) − κκ + 4 ρκ + 12 κκ + e (cid:0) r − d / Γ g (cid:1) + r − d / Γ g = − (cid:0) κ ∇ κ + κ ∇ κ (cid:1) + ρκ − κκ. Therefore, Q ( κ ) = − κ (cid:18) − κ + r − d / Γ b (cid:19) − κ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) + κρ − κκ + r − d ≤ Γ g = r − d ≤ Γ g . .14. PROOF OF THEOREM 2.4.7 L = − ∇ κ ∇ ∇ α − κκ ∇ α + 54 κκ α + Good= − (cid:18) − κκ + 2 ρ (cid:19) ∇ ∇ α − κκ ∇ α + 54 κκ α + Good . Therefore, L = − (cid:18) − κκ + 2 ρ (cid:19) ∇ ∇ α − κκ ∇ α + 54 κκ α + Good . (A.14.24) Expression for M Similarly, according to Lemma A.14.19, M = (cid:20) Q, ρ − κκ (cid:21) α = Q (cid:18) ρ − κκ (cid:19) α + 2 ∇ (cid:18) ρ − κκ (cid:19) ∇ α − κ (cid:18) ρ − κκ (cid:19) α i.e., M = Q (cid:18) ρ − κκ (cid:19) α + 2 ∇ (cid:18) ρ − κκ (cid:19) ∇ α − κ (cid:18) ρ − κκ (cid:19) α. We calculate, ∇ (cid:18) ρ − κκ (cid:19) = 4 ∇ ρ − κ ∇ κ − κ ∇ κ = 4 (cid:18) − κρ + r − d / Γ g (cid:19) − κ (cid:18) − κ + r − d / Γ b (cid:19) − κ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) = − κρ + 12 κκ + r − d / Γ g . We deduce, M = (cid:18) Q (cid:18) ρ − κκ (cid:19) − κ (cid:18) ρ − κκ (cid:19)(cid:19) α + (cid:18) − κρ + 12 κκ (cid:19) ∇ α + Good . APPENDIX A. APPENDIX TO CHAPTER 2 It remains to calculate M = Q (cid:18) ρ − κκ (cid:19) − κ (cid:18) ρ − κκ (cid:19) = ∇ ∇ (cid:18) ρ − κκ (cid:19) + 2 κ ∇ (cid:18) ρ − κκ (cid:19) = ∇ (cid:18) − κρ + 12 κκ + r − d / Γ g (cid:19) + 2 κ (cid:18) − κρ + 12 κκ + r − d / Γ g (cid:19) = − ρ ∇ κ − κ ∇ ρ + 12 κ ∇ κ + κκ ∇ κ + 2 κ (cid:18) − κρ + 12 κκ (cid:19) + ∇ ( r − d / Γ g ) + r − d / Γ g . Hence, M = − ρ (cid:18) − κ + r − d / Γ b (cid:19) − κ (cid:18) − κρ + r − d / Γ g (cid:19) + 12 κ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) + κκ (cid:18) − κ + r − d / Γ b (cid:19) + 2 κ (cid:18) − κρ + 12 κκ (cid:19) + ∇ ( r − d / Γ g ) + r − d / Γ g = κ ρ + 14 κκ + ∇ ( r − d / Γ g ) + r − d / Γ g . We conclude, M = (cid:18) κ ρ + 14 κκ (cid:19) α + 2 (cid:18) − κρ + 12 κκ (cid:19) ∇ α + Good . (A.14.25)Indeed note that ∇ ( r − d / Γ g ) α = ∇ (cid:0) r − d / Γ g α (cid:1) − r − d / Γ g ∇ α = Good . .14. PROOF OF THEOREM 2.4.7 End of the proof of Proposition A.14.20 Using the equations (A.14.21)–(A.14.25) we deduce, back to (A.14.20),[ Q, L ] α = I + J + K + L + M = (cid:0) ρ − κκ (cid:1) ∇ ∇ α − κ ∇ ∇ α − κ ∇ α − (cid:18) − κκ + 3 ρκ (cid:19) ∇ α − κ (cid:18) − κκ + 2 ρ (cid:19) α − κ ∇ ∇ ∇ α − κκ ∇ ∇ α − κ ∇ ∇ α − κ ∇ α − (cid:18) κρ + 194 κκ (cid:19) ∇ α − (cid:18) κ ρ + 14 κκ (cid:19) α + 12 κ ∇ ∇ α + 14 κ ∇ α − κ (2 ρ + κκ (cid:1) ∇ α + κ (cid:18) ρ − κκ (cid:19) α − (cid:18) − κκ + 2 ρ (cid:19) ∇ ∇ α − κκ ∇ α − κκ α + (cid:18) κ ρ + 14 κκ (cid:19) α + 2 (cid:18) − κρ + 12 κκ (cid:19) ∇ α + Good . We deduce,[ Q, L ] α = − κ ∇ ∇ ∇ α + C (cid:48) ∇ ∇ α + C (cid:48) ∇ ∇ α + C (cid:48) ∇ α + C (cid:48) ∇ α + C (cid:48) α with, C (cid:48) = (cid:0) ρ − κκ (cid:1) − κκ − (cid:18) − κκ + 2 ρ (cid:19) = − ρ − κκ,C (cid:48) = − κ − κ + 12 κ = − κ ,C (cid:48) = − κ − κ + 14 κ = − κ ,C (cid:48) = 12 κκ − ρκ − (cid:18) κρ + 194 κκ (cid:19) − κ (cid:0) ρ + κκ (cid:1) − κκ + 2 (cid:18) − κρ + 12 κκ (cid:19) = − κρ − κκ ,C (cid:48) = − κ (cid:18) − κκ + 2 ρ (cid:19) − (cid:18) κ ρ + 14 κκ (cid:19) + κ (cid:18) ρ − κκ (cid:19) − κκ + (cid:18) κ ρ + 14 κκ (cid:19) = − κ ρ − κκ . Finally we write, recalling the definition of Q = ∇ ∇ + 2 κ ∇ + κ , ∇ ∇ α = Q ( α ) − κ ∇ α − κ α APPENDIX A. APPENDIX TO CHAPTER 2 and, ∇ ∇ ∇ α = ∇ Q ( α ) − κ ∇ ∇ α − κ ∇ α − ∇ κ ∇ α − κ ∇ κα. Hence, − κ ∇ ∇ ∇ α + C (cid:48) ∇ ∇ α = − κ ∇ Q ( α ) + 4 κ ∇ ∇ α + κ ∇ α + 4 κ ∇ κ ∇ α + 2 κ ∇ κα + C (cid:48) (cid:18) Q ( α ) − κ ∇ α − κ α (cid:19) . We deduce,[ Q, L ] α = − κ ∇ Q ( α ) + C (cid:48) Q ( α ) + 4 κ ∇ ∇ α + κ ∇ α + (cid:0) κ ∇ κ − κC (cid:48) (cid:1) ∇ α + (cid:18) κ ∇ κ − κ C (cid:48) (cid:19) α + C (cid:48) ∇ ∇ α + C (cid:48) ∇ α + C (cid:48) ∇ α + C (cid:48) α. Thus, setting C Q = C (cid:48) , we deduce,[ Q, L ] α = − κ ∇ Q ( α ) + C Q Q ( α ) + C ∇ ∇ α + C ∇ α + C ∇ α + C α + Goodwhere, C Q = C (cid:48) = − ρ − κκ,C = 4 κ + C (cid:48) = 4 κ − κ = 0 ,C = κ + C (cid:48) = κ − κ = 0 . Also, C = 2 κ (cid:0) ∇ κ − C (cid:48) (cid:1) + C (cid:48) = 2 κ (cid:18) − κκ + 4 ρ + r − d / Γ g + 8 ρ + 72 κκ (cid:19) + C (cid:48) = 2 κ (cid:18) ρ + 52 κκ (cid:19) + (cid:0) − κρ − κκ (cid:1) + r − d / Γ g = r − d / Γ g ,C = 2 κ ∇ κ − κ C (cid:48) + C (cid:48) = 2 κ (cid:18) − κκ + 2 ρ + r − d / Γ g (cid:19) + 12 κ (cid:18) ρ + 72 κκ (cid:19) − κ ρ − κκ = 8 κ ρ + 34 κκ − κ ρ − κκ + r − d / Γ g = r − d / Γ g . .14. PROOF OF THEOREM 2.4.7 Q, L ] α = − κ ∇ Q ( α ) + C Q Q ( α ) + Good , C Q = − ρ − κκ, as stated in Proposition A.14.20. A.14.4 Proof of Theorem 2.4.7 We start with the following, Lemma A.14.21. We have, (cid:3) ( f r ) = r (cid:3) f − r (cid:0) κe f + κe f (cid:1) + r (cid:0) − κκ − ρ (cid:1) f + O ( r d ≤ Γ g · f ) . We postpone the proof of the lemma to the end of the section and continue below theproof of the theorem. According to Lemma A.14.13 L ( α ) = Good where L is the operator L α = −∇ ∇ α + ( c ) (cid:52) / α − κ ∇ α − κ ∇ α − (cid:18) − ρ + 12 κκ (cid:19) α. Applying Q and recalling the definition of the error terms Good we derive, L ( Qα ) = − [ Q, L ] α + Good . Thus, in view of Proposition A.14.20,[ Q, L ] α = − κ ∇ Q ( α ) + C Q Q ( α ) , C Q = − ρ − κκ. We deduce, L ( Qα ) = 2 κ ∇ Q ( α ) − C Q Q ( α ) . Therefore, modulo Good terms,2 κ ∇ Q ( α ) − C Q Q ( α ) = −∇ ∇ ( Qα ) + ( c ) (cid:52) / ( Qα ) − κ ∇ Q ( α ) − κ ∇ Q ( α ) − (cid:18) − ρ + 12 κκ (cid:19) Q ( α ) . APPENDIX A. APPENDIX TO CHAPTER 2 We deduce −∇ ∇ ( Qα ) + ( c ) (cid:52) / ( Qα ) − κ ∇ Q ( α ) − κ ∇ Q ( α ) + (cid:18) C Q − (cid:18) − ρ + 12 κκ (cid:19)(cid:19) Q ( α ) + Good . In view of the expression for (cid:3) in the second part of the Lemma A.14.13 we rewrite inthe form (cid:3) Q ( f ) − κ ∇ Q ( α ) − κ ∇ Q ( α ) − (cid:0) ρ + 4 κκ (cid:1) Q ( α ) = Good + r − Γ g · d /Q ( α ) . Finally, making use of Lemma A.14.21 and recalling that q = r Q ( α ), (cid:3) q = r (cid:3) ( Qα ) − r (cid:0) κe ( Qα ) + κe ( Qα ) (cid:1) + r (cid:0) − κκ − ρ (cid:1) Qf + O ( r d ≤ Γ g · Q ( α )= r (cid:16) κ ∇ Q ( α ) + 2 κ ∇ Q ( α ) + (cid:0) ρ + 4 κκ (cid:1) Q ( α ) + Good (cid:17) − r (cid:0) κe ( Qα ) + κe ( Qα ) (cid:1) + r (cid:0) − κκ − ρ (cid:1) Qf + O ( d ≤ Γ g · q )= − κκ q + r Good . This ends the proof of Theorem 2.4.7. Proof of Lemma A.14.21 We have, (cid:3) ( f r ) = D α D α ( f r ) = D α ( D α f r + f D α r )= r (cid:3) f + 2 D α ( r ) D α f + f (cid:3) ( r )= r (cid:3) f − (cid:0) e ( r ) e f + e ( r ) e f (cid:1) + f (cid:3) ( r ) + r Γ g d f = r (cid:3) f − r (cid:0) e ( r ) e f + e ( r ) e f (cid:1) + f (cid:3) ( r ) + r Γ g d f = r (cid:3) f − r (cid:16) ( κ + Γ b ) e f + ( κ + Γ g ) e f (cid:17) + f (cid:3) ( r ) + r Γ g · d f = r (cid:3) f − r (cid:0) κe f + κe f (cid:1) + f (cid:3) ( r ) + r Γ g · f. Also, (cid:3) ( r ) = − e ( e ( r )) − κe ( r ) + (cid:18) − κ + 2 ω (cid:19) e ( r ) + (cid:52) / ( r ) + 2 ηe θ ( r )= − e ( r e ( r )) − r κ r κ + Γ g ) + 4 r (cid:18) − κ + 2 ω (cid:19) r κ + Γ b ) + (cid:52) / ( r ) + 2 ηe θ ( r )= − r ( e r )( e r ) − r e e r − r κκ + 2 r (cid:18) − κ + 2 ω (cid:19) κ + O ( r Γ b )= − r ( κ + Γ b )( κ + Γ g ) − r e (cid:16) r κ + Γ b ) (cid:17) − r κκ + 4 r ωκ + O ( r Γ b ) . .14. PROOF OF THEOREM 2.4.7 (cid:3) ( r ) = − r κκ + 4 r ωκ − r e ( rκ ) + O ( r d ≤ Γ g ) . Note that, e ( rκ ) = re ( κ ) + r κ ( κ + Γ g ) = r (cid:18) − κ κ + 2 ωκ + 2 d/ η + 2 ρ (cid:19) + r κ ( κ + Γ g )= 2 rρ + 2 rωκ + O ( d ≤ Γ g ) . Hence, (cid:3) ( r ) = − r κκ + 4 r ωκ − r (2 rρ + 2 rωκ ) + O ( r d ≤ Γ g )= r (cid:0) − κκ − ρ (cid:1) + O ( r d ≤ Γ g ) . We conclude (cid:3) ( f r ) = r (cid:3) f − r (cid:0) κe f + κe f (cid:1) + r (cid:0) − κκ − ρ (cid:1) f + O ( r d ≤ Γ g )as stated.62 APPENDIX A. APPENDIX TO CHAPTER 2 ppendix BAPPENDIX TO CHAPTER 8 B.1 Proof of Proposition 8.4.1 Proposition B.1.1. The following wave equations hold true.1. The null curvature component ρ verifies the identity (cid:3) g ρ := κe ρ + κe ρ + 32 (cid:16) κ κ + 2 ρ (cid:17) ρ + Err [ (cid:3) g ρ ] , whereErr [ (cid:3) g ρ ] = 32 ρ (cid:18) − ϑ ϑ + 2( ξ ξ + η η ) (cid:19) + (cid:18) κ − ω (cid:19) (cid:18) ϑ α − ζ β − η β + ξ β ) (cid:19) − ϑ d (cid:63) / β + ( ζ − η ) e β − ηe (Φ) β − ξ ( e β + e (Φ) β ) − ββ − e (cid:18) − ϑ α + ζ β + 2( η β + ξ β ) (cid:19) − d (cid:63) / ( κ ) β + 2 d (cid:63) / ( ω ) β + 3 η d (cid:63) / ( ρ ) − d/ (cid:16) − ϑβ + ξα (cid:17) − ηe θ ρ. 2. The small curvature quantity, ˜ ρ := r (cid:18) ρ + 2 mr (cid:19) APPENDIX B. APPENDIX TO CHAPTER 8 verifies the wave equation, (cid:3) g ( ˜ ρ ) + 8 mr ˜ ρ = − m (cid:3) g ( r ) − (cid:0) r − mr (cid:1) r − mr (cid:18) κκ + 4Υ r (cid:19) − mr ( Aκ + Aκ ) + Err [ (cid:3) g ˜ ρ ] , whereErr [ (cid:3) g ˜ ρ ] := − mr AA + 3 r ˜ ρ + 32 (cid:32) A e ( r ) r + 43 A e ( r ) r (cid:33) ˜ ρ + (cid:18) (cid:16) κκ − mr + 23 r (cid:3) g ( r ) (cid:17) + 8 mr (cid:19) ˜ ρ − Ae ( ˜ ρ ) − Ae ( ˜ ρ ) + 2 r Ae ( m ) + 2 r Ae ( m )+4 D a ( m ) D a (cid:18) r (cid:19) + 2 r (cid:3) g ( m ) + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ ) + r Err [ (cid:3) g ρ ] . Proof. We prove the result in the following steps. Step 1. We start by deriving the wave equation for ρ . From Bianchi, ρ satisfies e ρ + 32 κρ = d/ β − ϑ α + ζ β + 2( η β + ξ β ) . Differentiating with respect to e , we obtain e ( e ( ρ )) + 32 κe ( ρ ) + 32 e ( κ ) ρ = e ( d/ β ) + e (cid:18) − ϑ α + ζ β + 2( η β + ξ β ) (cid:19) . Also, β satisfies from Bianchi e β + κβ = − d (cid:63) / ρ + 2 ωβ + 3 ηρ − ϑβ + ξα. Differentiating with respect to d/ , we infer d/ ( e β ) + κ d/ β − d (cid:63) / ( κ ) β = − d/ d (cid:63) / ρ + 2 ω d/ β − d (cid:63) / ( ω ) β + 3 ρ d/ η − η d (cid:63) / ( ρ )+ d/ (cid:16) − ϑβ + ξα (cid:17) and hence d/ d (cid:63) / ρ = − d/ ( e β ) − κ d/ β + 2 ω d/ β + 3 ρ d/ η + d (cid:63) / ( κ ) β − d (cid:63) / ( ω ) β − η d (cid:63) / ( ρ ) + d/ (cid:16) − ϑβ + ξα (cid:17) . .1. PROOF OF PROPOSITION 8.4.1 d/ d (cid:63) / ρ from the one for e ( e ( ρ )). This yields e ( e ( ρ )) + d/ d (cid:63) / ρ + 32 κe ( ρ ) + 32 e ( κ ) ρ = [ e , d/ ] β − κ d/ β + 2 ω d/ β + 3 ρ d/ η + e (cid:18) − ϑ α + ζ β + 2( η β + ξ β ) (cid:19) + d (cid:63) / ( κ ) β − d (cid:63) / ( ω ) β − η d (cid:63) / ( ρ ) + d/ (cid:16) − ϑβ + ξα (cid:17) . Next, we recall the following commutator identity[ e , d/ ] β = − κ d/ β + 12 ϑ d (cid:63) / β − ( ζ − η ) e β + ηe (Φ) β + ξ ( e β + e (Φ) β ) + ββ. We infer e ( e ( ρ )) + d/ d (cid:63) / ρ + 32 κe ( ρ ) + 32 e ( κ ) ρ + (cid:18) κ − ω (cid:19) d/ β − ρ d/ η = 12 ϑ d (cid:63) / β − ( ζ − η ) e β + ηe (Φ) β + ξ ( e β + e (Φ) β ) + ββ + e (cid:18) − ϑ α + ζ β + 2( η β + ξ β ) (cid:19) + d (cid:63) / ( κ ) β − d (cid:63) / ( ω ) β − η d (cid:63) / ( ρ ) + d/ (cid:16) − ϑβ + ξα (cid:17) . Next, we make use of the Bianchi identities and the null structure equations to compute32 e ( κ ) ρ + (cid:18) κ − ω (cid:19) d/ β − ρ d/ η = 32 ρ (cid:18) − κ κ + 2 ωκ + 2 d/ η + 2 ρ − ϑ ϑ + 2( ξ ξ + η η ) (cid:19) + (cid:18) κ − ω (cid:19) (cid:18) e ρ + 32 κρ + 12 ϑ α − ζ β − η β + ξ β ) (cid:19) − ρ d/ η = (cid:18) κ − ω (cid:19) e ρ + 32 ρ (cid:16) κ κ + 2 ρ (cid:17) + 32 ρ (cid:18) − ϑ ϑ + 2( ξ ξ + η η ) (cid:19) + (cid:18) κ − ω (cid:19) (cid:18) ϑ α − ζ β − η β + ξ β ) (cid:19) . APPENDIX B. APPENDIX TO CHAPTER 8 This yields e ( e ( ρ )) − (cid:52) / ρ + 32 κe ( ρ ) + (cid:18) κ − ω (cid:19) e ρ + 32 ρ (cid:16) κ κ + 2 ρ (cid:17) = − ρ (cid:18) − ϑ ϑ + 2( ξ ξ + η η ) (cid:19) − (cid:18) κ − ω (cid:19) (cid:18) ϑ α − ζ β − η β + ξ β ) (cid:19) + 12 ϑ d (cid:63) / β − ( ζ − η ) e β + ηe (Φ) β + ξ ( e β + e (Φ) β ) + ββ + e (cid:18) − ϑ α + ζ β + 2( η β + ξ β ) (cid:19) + d (cid:63) / ( κ ) β − d (cid:63) / ( ω ) β − η d (cid:63) / ( ρ ) + d/ (cid:16) − ϑβ + ξα (cid:17) , where we used the fact that d/ d (cid:63) / = −(cid:52) / .Next, recall the formula for the wave operator acting on a scalar ψ (cid:3) g ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ. We infer e ( e ( ρ )) − (cid:52) / ρ + 32 κe ( ρ ) + (cid:18) κ − ω (cid:19) e ρ + 32 ρ (cid:16) κκ + 2 ρ (cid:17) = − (cid:3) g ρ + (cid:18) ω − κ (cid:19) e ρ − κe ρ + 2 ηe θ ρ + 32 κe ( ρ ) + (cid:18) κ − ω (cid:19) e ρ + 32 ρ (cid:16) κ κ + 2 ρ (cid:17) and hence (cid:3) g ρ = κe ρ + κe ρ + 32 (cid:16) κ κ + 2 ρ (cid:17) ρ + 32 ρ (cid:18) − ϑ ϑ + 2( ξ ξ + η η ) (cid:19) + (cid:18) κ − ω (cid:19) (cid:18) ϑ α − ζ β − η β + ξ β ) (cid:19) − ϑ d (cid:63) / β + ( ζ − η ) e β − ηe (Φ) β − ξ ( e β + e (Φ) β ) − ββ − e (cid:18) − ϑ α + ζ β + 2( η β + ξ β ) (cid:19) − d (cid:63) / ( κ ) β + 2 d (cid:63) / ( ω ) β + 3 η d (cid:63) / ( ρ ) − d/ (cid:16) − ϑβ + ξα (cid:17) − ηe θ ρ. .1. PROOF OF PROPOSITION 8.4.1 Step 2. We derive the following, identity (cid:3) g ( r ρ ) = − Ae ( r ρ ) − Ae ( r ρ )+ 32 (cid:32) A e ( r ) r + 43 A e ( r ) r + κκ + 2 ρ + 23 r (cid:3) g ( r ) (cid:33) r ρ + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ ) + r Err[ (cid:3) g ρ ] . (B.1.1) Proof. r ρ satisfies the following wave equation (cid:3) g ( r ρ ) = r (cid:3) g ρ + 2 D a ( r ) D a ( ρ ) + ρ (cid:3) g ( r ) . On the other hand, recall that we have (cid:3) g ρ = κe ρ + κe ρ + 32 (cid:16) κ κ + 2 ρ (cid:17) ρ + Err[ (cid:3) g ρ ] . We deduce (cid:3) g ( r ρ ) = (cid:0) r κ − e ( r ) (cid:1) e ρ + (cid:0) r κ − e ( r ) (cid:1) e ρ + 32 (cid:18) κκ + 2 ρ + 23 r (cid:3) g ( r ) (cid:19) r ρ + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ ) + r Err[ (cid:3) g ρ ]= − Ae ( r ρ ) − Ae ( r ρ )+ 32 (cid:32) A e ( r ) r + 43 A e ( r ) r + κκ + 2 ρ + 23 r (cid:3) g ( r ) (cid:33) r ρ + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ )+ r Err[ (cid:3) g ρ ]as desired. Step 3. We now derive the desired formula for (cid:3) g ˜ ρ . In view of the definition of ˜ ρ , wehave (cid:3) g ( ˜ ρ ) = (cid:3) g ( r ρ ) + (cid:3) g (cid:18) mr (cid:19) = (cid:3) g ( r ρ ) + 2 m (cid:3) g (cid:18) r (cid:19) + 4 D a ( m ) D a (cid:18) r (cid:19) + 2 r (cid:3) g ( m ) . Together with B.1.1 we deduce, (cid:3) g ( ˜ ρ ) = − Ae ( r ρ ) − Ae ( r ρ )+ 32 (cid:32) A e ( r ) r + 43 A e ( r ) r + κκ + 2 ρ + 23 r (cid:3) g ( r ) (cid:33) r ρ + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ )+2 m (cid:3) g (cid:18) r (cid:19) + 4 D a ( m ) D a (cid:18) r (cid:19) + 2 r (cid:3) g ( m ) + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ ) + r Err[ (cid:3) g ρ ] . APPENDIX B. APPENDIX TO CHAPTER 8 Next, we use r ρ = ˜ ρ − mr − . This yields (cid:3) g ( ˜ ρ ) − (cid:32) κκ − mr + 23 r (cid:3) g ( r ) (cid:33) ˜ ρ = 2 m (cid:3) g (cid:18) r (cid:19) − mr κκ + 12 m r − mr (cid:3) g ( r ) − mA e ( r ) r − mA e ( r ) r + 3 r ˜ ρ + 32 (cid:32) A e ( r ) r + 43 A e ( r ) r (cid:33) ˜ ρ − Ae ( ˜ ρ ) − Ae ( ˜ ρ ) + 2 r Ae ( m ) + 2 r Ae ( m )+4 D a ( m ) D a (cid:18) r (cid:19) + 2 r (cid:3) g ( m ) + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ ) + r Err[ (cid:3) g ρ ] . Note that in Schwarzschild,32 (cid:18) κκ − mr + 23 r (cid:3) g ( r ) (cid:19) = − mr and hence (cid:3) g ( ˜ ρ ) + 8 mr ˜ ρ = 2 m (cid:3) g (cid:18) r (cid:19) − mr κκ + 12 m r − mr (cid:3) g ( r ) − m e ( r ) r − mA e ( r ) r + 3 r ˜ ρ + 32 (cid:32) A e ( r ) r + 43 A e ( r ) r (cid:33) ˜ ρ + (cid:32) (cid:32) κκ − mr + 23 r (cid:3) g ( r ) (cid:33) + 8 mr (cid:33) ˜ ρ − Ae ( ˜ ρ ) − Ae ( ˜ ρ ) + 2 r Ae ( m ) + 2 r Ae ( m )+4 D a ( m ) D a (cid:18) r (cid:19) + 2 r (cid:3) g ( m ) + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ ) + r Err[ (cid:3) g ρ ] . Also, we have (cid:3) g (cid:18) r (cid:19) − r (cid:3) g ( r ) = − (cid:3) g ( r ) r + 2 D α ( r ) D α ( r ) r − (cid:3) g ( r ) r − D α ( r ) D α ( r ) r = − (cid:3) g ( r ) r .1. PROOF OF PROPOSITION 8.4.1 (cid:3) g ( ˜ ρ ) + 8 mr ˜ ρ − m (cid:3) g ( r ) r − mr κκ + 12 m r − m e ( r ) r − mA e ( r ) r + 3 r ˜ ρ + 32 (cid:32) A e ( r ) r + 43 A e ( r ) r (cid:33) ˜ ρ + (cid:32) (cid:32) κκ − mr + 23 r (cid:3) g ( r ) (cid:33) + 8 mr (cid:33) ˜ ρ − Ae ( ˜ ρ ) − Ae ( ˜ ρ ) + 2 r Ae ( m ) + 2 r Ae ( m )+4 D a ( m ) D a (cid:18) r (cid:19) + 2 r (cid:3) g ( m ) + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ ) + r Err[ (cid:3) g ρ ] . Finally, since − m e ( r ) r − mA e ( r ) r = − mA κr − mA κr − mr AA and − m (cid:3) g ( r ) r − mr κκ + 12 m r = − m (cid:3) g ( r ) − (cid:0) r − mr (cid:1) r − mr (cid:18) κκ + 4Υ r (cid:19) , we obtain (cid:3) g ( ˜ ρ ) + 8 mr ˜ ρ = − m (cid:3) g ( r ) − (cid:0) r − mr (cid:1) r − mr (cid:18) κκ + 4Υ r (cid:19) − mA κr − mA κr − mr AA + 3 r ˜ ρ + 32 (cid:32) A e ( r ) r + 43 A e ( r ) r (cid:33) ˜ ρ + (cid:32) (cid:32) κκ − mr + 23 r (cid:3) g ( r ) (cid:33) + 8 mr (cid:33) ˜ ρ − Ae ( ˜ ρ ) − Ae ( ˜ ρ ) + 2 r Ae ( m ) + 2 r Ae ( m )+4 D a ( m ) D a (cid:18) r (cid:19) + 2 r (cid:3) g ( m ) + 4 r d (cid:63) / ( r ) d (cid:63) / ( ρ ) + r Err[ (cid:3) g ρ ] . APPENDIX B. APPENDIX TO CHAPTER 8 ppendix CAPPENDIX TO CHAPTER 9 C.1 Proof of Lemma 9.2.6 We start with the following Lemma C.1.1. Let k ≥ an integer and let f ∈ s k ( S ) . Then, we have ( d/ S k f ) = √ γ (cid:112) γ S (cid:40) ◦ d/ k ( f ) + (cid:32) k U (cid:90) (cid:16) √ γ ( e θ ( κ ) − e θ ( ϑ )) (cid:17) λ dλ + k S (cid:90) (cid:0) √ γe θ (cid:0) κ − ϑ − Ω( κ − ϑ ) − bγ / e θ Φ (cid:1)(cid:1) λ dλ + k (cid:16) κ − ϑ − Ω( κ − ϑ ) − bγ / e θ Φ (cid:17) U (cid:48) + k κ + ϑ ) S (cid:48) (cid:33) f (cid:41) where for ≤ λ ≤ , λ denotes the pull back by ψ λ ( ◦ u, ◦ s, θ ) := ( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) . Proof. For p ∈ ◦ S and f a Z -invariant scalar function on S , we have by definition of thepush forward of a vectorfield[Ψ ( ∂ θ ) f ] Ψ( p ) = [ ∂ θ ( f ◦ Ψ)] p . We infer ( d/ S k f ) = 1 (cid:112) γ S (cid:16) ∂ θ ( f ) + k∂ θ (Φ ) f (cid:17) APPENDIX C. APPENDIX TO CHAPTER 9 and hence ( d/ S k f ) = √ γ (cid:112) γ S (cid:16) e θ ( f ) + ke θ (Φ) f + k ( e θ (Φ ) − e θ (Φ)) f (cid:17) = √ γ (cid:112) γ S (cid:16) ◦ d/ k ( f ) + k ( e θ (Φ ) − e θ (Φ)) f (cid:17) . Next, we have e θ (Φ ) − e θ (Φ) = √ γ − (cid:16) ∂ θ (Φ ) − ∂ θ Φ (cid:17) and (cid:16) ∂ θ (Φ ) − ∂ θ Φ (cid:17) ( ◦ u, ◦ s, θ )= ∂ θ [Φ( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ )] − ∂ θ Φ( ◦ u, ◦ s, θ )= ( ∂ θ Φ)( ◦ u + U ( θ ) , ◦ s + S ( θ ) , θ ) − ∂ θ Φ( ◦ u, ◦ s, θ ) + (cid:104) ( ∂ u Φ) U (cid:48) + ( ∂ s Φ) S (cid:48) (cid:105) ( ◦ u, ◦ s, θ )= (cid:90) ddλ (cid:104) ( ∂ θ Φ)( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) (cid:105) dλ + (cid:104) ( ∂ u Φ) U (cid:48) + ( ∂ s Φ) S (cid:48) (cid:105) ( ◦ u, ◦ s, θ )= U ( θ ) (cid:90) ( ∂ u ∂ θ Φ)( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) dλ + S ( θ ) (cid:90) ( ∂ s ∂ θ Φ)( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) dλ + (cid:104) ( ∂ u Φ) U (cid:48) + ( ∂ s Φ) S (cid:48) (cid:105) ( ◦ u, ◦ s, θ )which we rewrite ∂ θ (Φ ) − ∂ θ Φ = U (cid:90) ( ∂ u ∂ θ Φ) λ dλ + S (cid:90) ( ∂ s ∂ θ Φ) λ dλ + ( ∂ u Φ) U (cid:48) + ( ∂ s Φ) S (cid:48) where λ denotes the pull back by the map ψ λ ( ◦ u, ◦ s, θ ) = ( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) . Next, recall that, ∂ s = e , ∂ u = 12 (cid:0) e − Ω e − bγ / e θ (cid:1) , ∂ θ = √ γe θ . Hence, ∂ θ ∂ s Φ = √ γe θ e (Φ) ∂ θ ∂ u Φ = 12 √ γe θ (cid:0) e Φ − Ω e Φ − bγ / e θ Φ (cid:1) .1. PROOF OF LEMMA 9.2.6 ∂ θ (Φ ) − ∂ θ Φ = U (cid:90) (cid:16) √ γe θ e (Φ) (cid:17) λ dλ + S (cid:90) (cid:18) √ γe θ (cid:0) e Φ − Ω e Φ − bγ / e θ Φ (cid:1)(cid:19) λ dλ + 12 (cid:16) e Φ − Ω e Φ − bγ / e θ Φ (cid:17) U (cid:48) + ( e Φ) S (cid:48) = 12 U (cid:90) (cid:16) √ γ ( e θ ( κ ) − e θ ( ϑ )) (cid:17) λ dλ + 14 S (cid:90) (cid:0) √ γe θ (cid:0) κ − ϑ − Ω( κ − ϑ ) − bγ / e θ Φ (cid:1)(cid:1) λ dλ + 14 (cid:16) κ − ϑ − Ω( κ − ϑ ) − bγ / e θ Φ (cid:17) U (cid:48) + 12 ( κ + ϑ ) S (cid:48) . We deduce( d/ S k f ) = √ γ (cid:112) γ S (cid:40) ◦ d/ k ( f ) + (cid:32) k U (cid:90) (cid:16) √ γ ( e θ ( κ ) − e θ ( ϑ )) (cid:17) λ dλ + k S (cid:90) (cid:0) √ γe θ (cid:0) κ − ϑ − Ω( κ − ϑ ) − bγ / e θ Φ (cid:1)(cid:1) λ dλ + k (cid:16) κ − ϑ − Ω( κ − ϑ ) − bγ / e θ Φ (cid:17) U (cid:48) + k κ + ϑ ) S (cid:48) (cid:33) f (cid:41) . This concludes the proof of the lemma.We are ready to prove the higher derivative comparison Lemma 9.2.6 which we recallbelow. Lemma C.1.2. Let ◦ S ⊂ R = R ( ◦ (cid:15), ◦ δ ) as in Definition 9.1.1 verifying the assumptions A1-A3 . Let Ψ : ◦ S −→ S be Z -invariant deformation. Assume the bound (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) L ∞ ( ◦ S ) + ◦ r − max ≤ s ≤ s max (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:46) ◦ δ. (C.1.1) Then, we have for any reduced scalar h defined on R(cid:107) h (cid:107) h s ( S ) (cid:46) sup R | d ≤ k h | for ≤ s ≤ s max . Also, if f ∈ h s ( S ) and f is its pull-back by ψ , we have (cid:107) f (cid:107) h s ( S ) = (cid:107) f (cid:107) h s ( ◦ S, g/ S , ) = (cid:107) f (cid:107) h s ( ◦ S, ◦ g/ ) (1 + O ( ◦ (cid:15) )) for ≤ s ≤ s max . APPENDIX C. APPENDIX TO CHAPTER 9 Remark C.1.3. Note that the estimates of the lemma are independent of the size ◦ r of thesphere ◦ S = S ( ◦ u, ◦ s ) ⊂ R , see Definition 9.1.1. To simplify the argument below we assume ◦ r ≈ . The general case can be easily deduced by a simple scaling argument or makingobvious adjustments in the inequalities below.Proof. We argue by iteration. We consider the following iteration assumptionsIf (9.2.14) holds, then we have (cid:107) h (cid:107) h s ( S ) (cid:46) sup R | d ≤ s h | , (C.1.2)and If (9.2.14) holds, then we have (cid:107) f (cid:107) h s ( ◦ S, g/ S , ) = (cid:107) f (cid:107) h s ( ◦ S, ◦ g/ ) (1 + O ( ◦ δ )) . (C.1.3)First, note that (C.1.2) holds trivially for s = 0 and (C.1.3) holds for s = 0 by Lemma9.2.3. Thus, from now on, we assume that (C.1.2) holds for some s with 0 ≤ s ≤ s max − s with 0 ≤ s ≤ s max − 3, and our goal is to prove that italso holds for s replaced by s + 1.We start with (C.1.2). We have d/ S k h = e S θ h + e S θ (Φ) h. Now, recall that we have e S θ = 1 (cid:112) γ S ∂ S θ , ∂ S θ | Ψ( p ) = (cid:18)(cid:18) S (cid:48) − 12 Ω U (cid:48) (cid:19) e + 12 U (cid:48) e + √ γ (cid:18) − bU (cid:48) (cid:19) e θ (cid:19) (cid:12)(cid:12)(cid:12) Ψ( p ) . This yields( d/ S k h ) | Ψ( p ) = (cid:40) (cid:112) γ S (cid:32) (cid:18) S (cid:48) − 12 Ω U (cid:48) (cid:19) e ( h ) + (cid:18) S (cid:48) − 12 Ω U (cid:48) (cid:19) e (Φ) h + 12 U (cid:48) e ( h ) + 12 U (cid:48) e (Φ) h + √ γ (cid:18) − bU (cid:48) (cid:19) d/ k ( h ) (cid:33)(cid:41) | Ψ( p ) . Together with the iteration assumption (C.1.3), we infer (cid:107) d/ S k h (cid:107) h s ( S ) = (cid:107) ( d/ S k h ) (cid:107) h s ( ◦ S, ◦ g/ ) (1 + O ( ◦ δ )) (cid:46) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:32) (cid:18) S (cid:48) − 12 Ω U (cid:48) (cid:19) ( e ( h )) + (cid:18) S (cid:48) − 12 Ω U (cid:48) (cid:19) ( e (Φ) h ) + 12 U (cid:48) ( e ( h )) + 12 U (cid:48) ( e (Φ) h ) + (cid:112) γ (cid:18) − b U (cid:48) (cid:19) ( d/ k ( h )) (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) .1. PROOF OF LEMMA 9.2.6 (cid:107) d/ S k h (cid:107) h s ( S ) (cid:46) (cid:13)(cid:13) ( d/ k ( h )) (cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:32) (cid:112) γ (cid:112) γ S , − (cid:33) ( d/ k ( h )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:32) (cid:18) S (cid:48) − 12 Ω U (cid:48) (cid:19) ( e ( h )) + (cid:18) S (cid:48) − 12 Ω U (cid:48) (cid:19) ( e (Φ) h ) + 12 U (cid:48) ( e ( h )) + 12 U (cid:48) ( e (Φ) h ) − (cid:112) γ b U (cid:48) ( d/ k ( h )) (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) . Together with a non sharp product rule in h s ( ◦ S, ◦ g/ ) and the repeated use of the iterationassumptions (C.1.2) (C.1.3), we can bound the right hand side of the above inequality by (cid:46) (cid:18) (cid:107)√ γ (cid:107) h s ( S ) + (cid:13)(cid:13)(cid:13) ( √ γ ) (cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( S ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:107) d/ k ( h ) (cid:107) h s ( S ) + (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( S ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) × (cid:18) (cid:107) (Ω , b √ γ ) (cid:107) h s ( S ) + (cid:13)(cid:13) (Ω , b √ γ ) (cid:13)(cid:13) h ∞ ( ◦ S ) (cid:19) (cid:13)(cid:13)(cid:13)(cid:16) ( e , e , d/ k ) h, e (Φ) h, e (Φ) h (cid:17)(cid:13)(cid:13)(cid:13) h s ( S ) Therefore (cid:107) d/ S k h (cid:107) h s ( S ) can be bounded by (cid:46) (cid:18) (cid:107)√ γ (cid:107) h s ( S ) + (cid:13)(cid:13)(cid:13) ( √ γ ) (cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( S ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) sup R (cid:12)(cid:12) d ≤ s d/ k h (cid:12)(cid:12) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( S ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:18) (cid:107) (Ω , b √ γ ) (cid:107) h s ( S ) + (cid:13)(cid:13) (Ω , b √ γ ) (cid:13)(cid:13) h ∞ ( ◦ S ) (cid:19) × sup R (cid:12)(cid:12) d ≤ s ( d h, e (Φ) h, e (Φ) h ) (cid:12)(cid:12) , where we used in the last inequality the assumption (9.2.14) on ( U (cid:48) , S (cid:48) ). Together with76 APPENDIX C. APPENDIX TO CHAPTER 9 (9.1.12) and (9.1.15), we infer (cid:107) d/ S k h (cid:107) h s ( S ) (cid:46) (cid:40) (cid:18) (cid:13)(cid:13)(cid:13) ( √ γ ) (cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( S ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( S ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:18) (cid:13)(cid:13) (Ω , b √ γ ) (cid:13)(cid:13) h ∞ ( ◦ S ) (cid:19) (cid:41) × sup R (cid:12)(cid:12) d ≤ s +1 h (cid:12)(cid:12) . Also, for a reduced scalar v defined on R , we have in view of the assumption (9.2.14) on( U (cid:48) , S (cid:48) ) (cid:107) v (cid:107) h ∞ ( ◦ S ) = (cid:107) v ◦ ψ (cid:107) h ∞ ( ◦ S ) (cid:46) (cid:18) ≤ θ ≤ π | ψ (cid:48) ( θ ) | (cid:19) sup R | d ≤ v | (cid:46) (cid:18) (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h ∞ ( ◦ S ) (cid:19) sup R | d ≤ v | (cid:46) (1 + ◦ δ ) sup R | d ≤ v | . (C.1.4)Together with (9.1.12) and (9.1.15), we infer (cid:107) d/ S k h (cid:107) h s ( S ) (cid:46) (cid:40) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( S ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:41) sup R (cid:12)(cid:12) d ≤ s +1 h (cid:12)(cid:12) . Now, recall that γ S ( ψ ( θ )) = γ ( ψ ( θ )) + (cid:18) Ω( ψ ( θ )) + 14 ( b ( ψ ( θ ))) γ ( ψ ( θ )) (cid:19) ( U (cid:48) ( θ )) − U (cid:48) ( θ ) S (cid:48) ( θ ) − γ ( ψ ( θ )) b ( ψ ( θ )) U (cid:48) ( θ ) . Together with a repeated application of the iteration assumptions and a non sharp productrule in h s ( ◦ S, ◦ g/ ) and (C.1.4), this yields (cid:13)(cid:13) γ S (cid:13)(cid:13) h s ( S ) + (cid:13)(cid:13) γ S , (cid:13)(cid:13) h ∞ ( ◦ S ) (cid:46) (cid:18) R | d ≤ ( γ, Ω , b γ, bγ ) | + sup R | d ≤ s ( γ, Ω , b γ, bγ ) | (cid:19) × (cid:18) (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h ∞ ( ◦ S ) + (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h s ( ◦ S, ◦ g/ ) (cid:19) (cid:46) .1. PROOF OF LEMMA 9.2.6 U (cid:48) , S (cid:48) ) and (9.1.15). Weinfer (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( S ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:112) γ S , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:46) (cid:107) d/ S k h (cid:107) h s ( S ) (cid:46) sup R (cid:12)(cid:12) d ≤ s +1 h (cid:12)(cid:12) which corresponds to the first of our iteration assumption (C.1.2) with s replaced with s + 1 for s ≤ s max − s replaced with s + 1 for s ≤ s max − 3. Recall from Lemma C.1.1 that we have for f ∈ s k ( S )( d/ S k f ) = √ γ (cid:112) γ S (cid:40) ◦ d/ k ( f ) + (cid:32) k U (cid:90) (cid:16) √ γ ( e θ ( κ ) − e θ ( ϑ )) (cid:17) λ dλ + k S (cid:90) (cid:0) √ γe θ (cid:0) κ − ϑ − Ω( κ − ϑ ) − bγ / e θ Φ (cid:1)(cid:1) λ dλ + k (cid:16) κ − ϑ − Ω( κ − ϑ ) − bγ / e θ Φ (cid:17) U (cid:48) + k κ + ϑ ) S (cid:48) (cid:33) f (cid:41) where for 0 ≤ λ ≤ 1, λ denotes the pull back by ψ λ ( ◦ u, ◦ s, θ ) = ( ◦ u + λU ( θ ) , ◦ s + λS ( θ ) , θ ) . For convenience, we rewrite some of the terms as follows e θ ( κ ) − e θ ( ϑ ) = − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) ,bγ / e θ Φ = 12 γ / ( d/ b + d (cid:63) / b ) , APPENDIX C. APPENDIX TO CHAPTER 9 and e θ (cid:0) κ − ϑ − Ω( κ − ϑ ) − bγ / e θ Φ (cid:1) = − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / (Ω κ ) − d (cid:63) / (Ω) ϑ + 12 Ω( d/ ϑ − d (cid:63) / ϑ )+ d (cid:63) / ( γ / )( d/ ϑ + d (cid:63) / ϑ ) b − γ / e θ ( be θ Φ)= − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / (Ω κ ) − d (cid:63) / (Ω) ϑ + 12 Ω( d/ ϑ − d (cid:63) / ϑ )+ d (cid:63) / ( γ / )( d/ ϑ + d (cid:63) / ϑ ) b − γ / ( − e θ (Φ) d (cid:63) / b − Kb )= − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / (Ω κ ) − d (cid:63) / (Ω) ϑ + 12 Ω( d/ ϑ − d (cid:63) / ϑ )+ d (cid:63) / ( γ / )( d/ ϑ + d (cid:63) / ϑ ) b + 12 γ / ( d/ d (cid:63) / b + d (cid:63) / d (cid:63) / b ) + 2 γ / Kb where we used the identities e θ ( e θ (Φ)) = − ( e θ (Φ)) − K, γ / e θ Φ d (cid:63) / b = 12 γ / ( d/ d (cid:63) / b + d (cid:63) / d (cid:63) / b ) . This yields( d/ S k f ) = √ γ (cid:112) γ S (cid:40) ◦ d/ k ( f ) + (cid:32) k U (cid:90) (cid:18) √ γ (cid:18) − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) (cid:19)(cid:19) λ dλ + k S (cid:90) (cid:32) √ γ (cid:32) − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / (Ω κ ) − d (cid:63) / (Ω) ϑ + 12 Ω( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / ( γ / )( d/ ϑ + d (cid:63) / ϑ ) b + 12 γ / ( d/ d (cid:63) / b + d (cid:63) / d (cid:63) / b ) + 2 γ / Kb (cid:33)(cid:33) λ dλ + k (cid:16) κ − ϑ − Ω( κ − ϑ ) − γ / ( d/ ϑ + d (cid:63) / ϑ ) b (cid:17) U (cid:48) + k κ + ϑ ) S (cid:48) (cid:33) f (cid:41) . Next, we take the h s ( ◦ S, ◦ g/ )-norm of this identity, and we use the iteration assumption to .1. PROOF OF LEMMA 9.2.6 h s ( ◦ S, g/ S , )-norm. We infer (cid:107) ( d/ S k f ) (cid:107) h s ( ◦ S, g/ S , ) (1 + O ( ◦ δ ))= (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ γ (cid:112) γ S (cid:40) ◦ d/ k ( f ) + (cid:32) k U (cid:90) (cid:18) √ γ (cid:18) − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) (cid:19)(cid:19) λ dλ + k S (cid:90) (cid:32) √ γ (cid:32) − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / (Ω κ ) − d (cid:63) / (Ω) ϑ + 12 Ω( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / ( γ / )( d/ ϑ + d (cid:63) / ϑ ) b + 12 γ / ( d/ d (cid:63) / b + d (cid:63) / d (cid:63) / b ) + 2 γ / Kb (cid:33)(cid:33) λ dλ + k (cid:16) κ − ϑ − Ω( κ − ϑ ) − γ / ( d/ ϑ + d (cid:63) / ϑ ) b (cid:17) U (cid:48) + k κ + ϑ ) S (cid:48) (cid:33) f (cid:41)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) . Next, we use a non sharp product rule in h s ( ◦ S, ◦ g/ ) to infer (cid:107) ( d/ S k f ) (cid:107) h s ( ◦ S, g/ S , ) (1 + O ( ◦ δ ))= O (1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ γ (cid:112) γ S − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:40) (cid:13)(cid:13)(cid:13)(cid:13) ◦ d/ k ( f ) (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) + O (1) (cid:32) (cid:107) U (cid:107) h s +1 ( ◦ S, ◦ g/ ) (cid:90) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) √ γ (cid:18) − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) (cid:19)(cid:19) λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) dλ + (cid:107) S (cid:107) h s +1 ( ◦ S, ◦ g/ ) (cid:90) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:32) √ γ (cid:32) − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / (Ω κ ) − d (cid:63) / (Ω) ϑ + 12 Ω( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / ( γ / )( d/ ϑ + d (cid:63) / ϑ ) b + 12 γ / ( d/ d (cid:63) / b + d (cid:63) / d (cid:63) / b ) + 2 γ / Kb (cid:33)(cid:33) λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) dλ + (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) κ − ϑ − Ω( κ − ϑ ) − γ / ( d/ ϑ + d (cid:63) / ϑ ) b (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:107) U (cid:48) (cid:107) h s +1 ( ◦ S, ◦ g/ ) + (cid:13)(cid:13) ( κ + ϑ ) (cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:107) S (cid:48) (cid:107) h s +1 ( ◦ S, ◦ g/ ) (cid:33) (cid:13)(cid:13) f (cid:13)(cid:13) h s +1 ( ◦ S, ◦ g/ ) (cid:41) . APPENDIX C. APPENDIX TO CHAPTER 9 Since s + 1 ≤ s max − 2, we infer in view of (9.2.14) and the fact that U (0) = S (0) = 0, (cid:107) ( d/ S k f ) (cid:107) h s ( ◦ S, g/ S , ) (1 + O ( ◦ δ ))= O (1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ γ (cid:112) γ S − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:40) (cid:13)(cid:13)(cid:13)(cid:13) ◦ d/ k ( f ) (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) + O ( ◦ δ ) (cid:32) (cid:90) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) √ γ (cid:18) − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) (cid:19)(cid:19) λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) dλ + (cid:90) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:32) √ γ (cid:32) − d (cid:63) / ( κ ) − 12 ( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / (Ω κ ) − d (cid:63) / (Ω) ϑ + 12 Ω( d/ ϑ − d (cid:63) / ϑ ) + d (cid:63) / ( γ / )( d/ ϑ + d (cid:63) / ϑ ) b + 12 γ / ( d/ d (cid:63) / b + d (cid:63) / d (cid:63) / b ) + 2 γ / Kb (cid:33)(cid:33) λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) dλ + (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) κ − ϑ − Ω( κ − ϑ ) − γ / ( d/ ϑ + d (cid:63) / ϑ ) b (cid:17) (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) + (cid:13)(cid:13) ( κ + ϑ ) (cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:33) (cid:13)(cid:13) f (cid:13)(cid:13) h s +1 ( ◦ S, ◦ g/ ) (cid:41) . Next, we have by the iteration assumption (C.1.3) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) + sup ≤ λ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) λ (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) (cid:46) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, g/ S , ) + sup ≤ λ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) λ (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, g/ S , λ ) (cid:46) (cid:13)(cid:13)(cid:13) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:13)(cid:13)(cid:13) h s ( S ) + sup ≤ λ ≤ (cid:13)(cid:13)(cid:13) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:13)(cid:13)(cid:13) h s ( S λ ) where the surface S λ is the image of ◦ S by ψ λ . Since s ≤ s max − 3, we infer in view of ouriteration assumption (C.1.2) and our assumptions (9.1.12) (9.1.15) on the ( u, s )-foliation (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) + sup ≤ λ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) λ (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) (cid:46) sup R (cid:12)(cid:12)(cid:12) d ≤ s d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17) ˇΓ (cid:12)(cid:12)(cid:12) (cid:46) sup R (cid:12)(cid:12)(cid:12) d ≤ s +2 (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17) ˇΓ (cid:12)(cid:12)(cid:12) (cid:46) ◦ δ. (C.1.5) .1. PROOF OF LEMMA 9.2.6 (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) (cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) + sup ≤ λ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) λ (cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) = (cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) ◦ ψ (cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) + sup ≤ λ ≤ (cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) ◦ ψ λ (cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:46) (cid:18) sup R (cid:12)(cid:12)(cid:12) d ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:12)(cid:12)(cid:12)(cid:19) (cid:18) ≤ θ ≤ π | ψ (cid:48) ( θ ) | (cid:19) (cid:46) (cid:18) sup R (cid:12)(cid:12)(cid:12) d ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:12)(cid:12)(cid:12)(cid:19) (cid:18) (cid:107) ( U (cid:48) , S (cid:48) ) (cid:107) h ∞ ( ◦ S ) (cid:19) (cid:46) ◦ (cid:15) where we used our assumptions (9.1.12) (9.1.15) on the ( u, s )-foliation and our assumption(9.2.14) on ( U (cid:48) , S (cid:48) ). Therefore, (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) (cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:46) ◦ δ sup ≤ λ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) ˇΓ , r − γ − , b, Ω + Υ (cid:17)(cid:17) λ (cid:13)(cid:13)(cid:13)(cid:13) h ∞ ( ◦ S ) (cid:46) ◦ δ. (C.1.6)We deduce (cid:107) ( d/ S k f ) (cid:107) h s ( ◦ S, g/ S , ) (1 + O ( ◦ δ ))= O (1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ γ (cid:112) γ S − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:40) (cid:13)(cid:13)(cid:13)(cid:13) ◦ d/ k ( f ) (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) + O ( ◦ δ ) (cid:13)(cid:13) f (cid:13)(cid:13) h s +1 ( ◦ S, ◦ g/ ) (cid:41) . Next, we estimate the term in the RHS involving γ and γ S . From the proof of Lemma9.2.3, we have γ S , − γ = 12 U (cid:90) (cid:0)(cid:0) e − Ω e − bγ / e θ (cid:1) γ (cid:1) λ dλ + S (cid:90) ( e γ ) λ dλ + (cid:18) Ω + 14 b γ (cid:19) ( U (cid:48) ) − U (cid:48) S (cid:48) − ( γb ) U (cid:48) . APPENDIX C. APPENDIX TO CHAPTER 9 Using a non sharp product rule, we infer (cid:13)(cid:13) γ S , − γ (cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:46) (cid:107) U (cid:107) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:90) (cid:13)(cid:13)(cid:13)(cid:0)(cid:0) e − Ω e − bγ / e θ (cid:1) γ (cid:1) λ (cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) dλ (cid:107) S (cid:107) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:90) (cid:13)(cid:13)(cid:13) ( e γ ) λ (cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) dλ + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) Ω + 14 b γ (cid:19) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:107) U (cid:48) (cid:107) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) + (cid:107) U (cid:48) (cid:107) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:107) S (cid:48) (cid:107) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) + (cid:13)(cid:13)(cid:13) ( γb ) (cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:107) U (cid:48) (cid:107) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:46) ◦ δ (cid:90) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) d/ ≤ (cid:16) r − γ − , b, Ω + Υ (cid:17)(cid:17) λ (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) dλ + ◦ δ (cid:13)(cid:13)(cid:13)(cid:0) r − γ − , b, Ω + Υ (cid:1) (cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) + ◦ δ where we used our assumption (9.2.14) on ( U (cid:48) , S (cid:48) ) and the fact that U (0) = S (0) = 0.Using the estimates (C.1.5) (C.1.6) for ( r − γ − , b, Ω + Υ), we infer (cid:13)(cid:13) γ S , − γ (cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:46) ◦ δ. Together with (9.1.15) for γ , we infer (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ γ (cid:112) γ S − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) ∩ h ∞ ( ◦ S ) (cid:46) ◦ δ and hence (cid:107) ( d/ S k f ) (cid:107) h s ( ◦ S, g/ S , ) (1 + O ( ◦ δ )) = (cid:18) O ( ◦ δ ) (cid:19) (cid:40)(cid:13)(cid:13)(cid:13)(cid:13) ◦ d/ k ( f ) (cid:13)(cid:13)(cid:13)(cid:13) h s ( ◦ S, ◦ g/ ) + O ( ◦ δ ) (cid:13)(cid:13) f (cid:13)(cid:13) h s +1 ( ◦ S, ◦ g/ ) (cid:41) . Now, we have (cid:107) f (cid:107) h s +1 ( ◦ S, g/ S , ) = (cid:107) f (cid:107) L ( ◦ S, g/ S , ) + (cid:107) ( d/ S k f ) (cid:107) h s ( ◦ S, g/ S , ) , (cid:107) f (cid:107) h s +1 ( ◦ S, ◦ g/ ) = (cid:107) f (cid:107) L ( ◦ S, ◦ g/ ) + (cid:107) ◦ d/ k ( f ) (cid:107) h s ( ◦ S, ◦ g/ ) . Together with Lemma 9.2.3, this yields (cid:107) f (cid:107) h s +1 ( ◦ S, g/ S , ) = (cid:107) f (cid:107) h s +1 ( ◦ S, ◦ g/ ) (1 + O ( ◦ δ )) . .1. PROOF OF LEMMA 9.2.6 s replaced with s + 1 for s ≤ s max − 3. Thus, we have finally derived both iteration assumption (C.1.3) and (C.1.2)with s replaced with s + 1 respectively for s ≤ s max − s ≤ s max − 3. Hence, wededuce that they hold respectively for 0 ≤ s ≤ s max and 0 ≤ s ≤ s max − 2, i.e. (cid:107) h (cid:107) h k ( S ) (cid:46) sup R | d ≤ k h | for 0 ≤ s ≤ s max and (cid:107) f (cid:107) h s ( ◦ S, g/ S , ) = (cid:107) f (cid:107) h s ( ◦ S, ◦ g/ ) (1 + O ( ◦ δ )) for 0 ≤ s ≤ s max − . Together with Lemma 9.2.2, we deduce (cid:107) f (cid:107) h s ( S ) = (cid:107) f (cid:107) h s +1 ( ◦ S, g/ S , ) = (cid:107) f (cid:107) h s +1 ( ◦ S, ◦ g/ ) (1 + O ( ◦ δ )) for all 0 ≤ s ≤ s max − . This concludes the proof of the lemma.84 APPENDIX C. APPENDIX TO CHAPTER 9 ppendix DAPPENDIX TO CHAPTER 10 D.1 Horizontal S -tensors Consider a null pair e , e on ( M , g ) and, at every point p ∈ M the horizontal space S = { e , e } ⊥ . Let γ the metric induced on S . By definition, for all X, Y ∈ T S M , i.e.vectors in M tangent to S , h ( X, Y ) = g ( X, Y )For any Y ∈ T ( M ) we define its horizontal projection, Y ⊥ = Y + 12 g ( Y, e ) e + 12 g ( Y, e ) e (D.1.1) Definition D.1.1. A k -covariant tensor-field U is said to be S -horizontal, U ∈ T kS ( M ) ,if for any X , . . . X k we have, U ( Y , . . . Y k ) = U ( Y ⊥ , . . . Y ⊥ k )We define the projection operator,Π νµ := δ νµ − 12 ( e ) µ ( e ) ν − 12 ( e ) µ ( e ) ν Clearly Π µα Π βµ = Π βα . An arbitrary tensor U α ...α m is said to an S - horizontal tensor, orsimply S -tensor, if Π β α . . . Π β m α m U β ...β m = U α ...α m . APPENDIX D. APPENDIX TO CHAPTER 10 Definition D.1.2. Given X ∈ T ( M ) and Y ∈ T S ( M ) we define, ˙ D X Y := ( D X Y ) ⊥ Remark D.1.3. In the particular case when S is integrable and both X, Y ∈ T S M then ˙ D X Y is the standard induced covariant differentiation on S . Definition D.1.4. Given a general, covariant, S - horizontal tensor-field U we define itshorizontal covariant derivative according to the formula, ˙ D X U ( Y , . . . Y k ) = X ( U ( Y , . . . Y k )) − U ( ˙ D X Y , . . . Y k ) − . . . − U ( Y , . . . ˙ D X Y k ) . (D.1.2) where X ∈ TM and Y , . . . Y k ∈ T S M . Proposition D.1.5. For all X ∈ TM and Y , Y ∈ T S M , Xh ( Y , Y ) = h ( ˙ D X Y , Y ) + h ( Y , ˙ D X Y ) . Proof. Indeed, Xh ( Y , Y ) = X g ( Y , Y ) = g ( D X Y , Y ) + g ( Y , D X Y ) = g ( ˙ D X Y , Y ) + g ( Y , ˙ D X Y )= h ( ˙ D X Y , Y ) + h ( Y , ˙ D X Y )Given an orthonormal frame e , e on S we have,˙ D µ e A = (cid:88) B =1 , (Λ µ ) AB e B A, B = 1 , µ ) αβ := g ( D µ e β , e α ) D.1.1 Mixed tensors We consider tensors T k M ⊗ T lS M , i.e. tensors of the form, U µ ...µ k ,A ...A L for which we define,˙ D µ U ν ...ν k ,A ...A L = e µ U ν ...ν k ,A ...A l − U D µ ν ...ν k ,A ...A l − . . . − U ν ... D µ ν k ,A ...A l − U ν ...ν k , ˙ D µ A ...A l − U ν ...ν k ,A ... ˙ D µ A l We are now ready to prove the following, .1. HORIZONTAL S -TENSORS Proposition D.1.6. We have the curvature formula ( ˙ D µ ˙ D ν − ˙ D ν ˙ D µ )Ψ A = R A B µν Ψ B More generally, ( ˙ D µ ˙ D ν − ˙ D ν ˙ D µ )Ψ λA = R λ σ µν Ψ σA + R A B µν Ψ λB Proof. Straightforward verification. D.1.2 Invariant Lagrangian We introduce, L = g µν h AB ˙ D µ Ψ A ˙ D µ Ψ B + V h AB Ψ A Ψ B Proposition D.1.7. The Euler Lagrange equations are given by: ˙ (cid:3) Ψ A = V Ψ A where ˙ (cid:3) Ψ A := g µν ˙ D µ ˙ D ν Ψ A . Proof. The variation of the action is given by,0 = 2 (cid:90) M h AB (cid:16) g µν ˙ D µ Ψ A ˙ D ν ( δ Ψ) B + V Ψ A δ Ψ B (cid:17) dv g = 2 (cid:90) M D ν (cid:16) g µν h AB ˙ D µ Ψ A ( δ Ψ) B (cid:17) dv g − (cid:90) M h AB (cid:16) g µν ˙ D ν ˙ D µ Ψ A ( δ Ψ) B − V Ψ A δ Ψ B (cid:17) dv g = − (cid:90) M h AB (cid:16) g µν ˙ D ν ˙ D µ Ψ A ( δ Ψ) B − V Ψ A δ Ψ B (cid:17) dv g from which the proposition follows. D.1.3 Comparison of the Lagrangians Let Ψ ∈ S ( M ) and ψ ∈ s its reduced form. Note that the Lagrangian of the scalarequation (cid:3) g ψ = V ψ + 4( e θ Φ) ψ APPENDIX D. APPENDIX TO CHAPTER 10 is given by, L ( ψ ) : = g µν ∂ µ ψ∂ ν ψ + ( V + 4( e θ Φ) ) ψ while the Lagrangian for, ˙ (cid:3) g Ψ = V Ψis given by L (Ψ) = g µν ˙ D µ Ψ · ˙ D ν Ψ + V Ψ · Ψ Proposition D.1.8. We have, L (Ψ) = 2 L ( ψ ) (D.1.3) Proof. Observe that, g µν ˙ D µ Ψ ˙ D ν Ψ = − ˙ D Ψ · ˙ D Ψ + ˙ D θ Ψ · ˙ D θ Ψ + ˙ D ϕ Ψ · ˙ D ϕ ΨNow, recalling that, ∇ / ϕ e ϕ = − e θ Φ e θ , ∇ / ϕ e θ = e θ (Φ) e ϕ ∇ / θ e θ = 0 ∇ / θ e ϕ = 0we deduce˙ D Ψ · ˙ D Ψ = e Ψ · e Ψ = 2 e ψe ψ ˙ D θ Ψ · ˙ D θ Ψ = ˙ D θ Ψ θθ ˙ D θ Ψ θθ + 2 ˙ D θ Ψ θϕ ˙ D θ Ψ θϕ + ˙ D θ Ψ ϕϕ ˙ D θ Ψ ϕϕ = 2( e θ ψ ) ˙ D ϕ Ψ · ˙ D ϕ Ψ = ˙ D ϕ Ψ θθ ˙ D ϕ Ψ θθ + 2 ˙ D ϕ Ψ θϕ ˙ D ϕ Ψ θϕ + ˙ D ϕ Ψ ϕϕ ˙ D ϕ Ψ ϕϕ = 2( e ϕ ψ ) + 2( − Ψ ˙ D ϕ θϕ − Ψ θ ˙ D ϕ ϕ ) · ( − Ψ ˙ D ϕ θϕ − Ψ θ ˙ D ϕ ϕ )= 2( e ϕ ψ ) + 2( − e θ (Φ)Ψ ϕϕ + e θ (Φ)Ψ θθ ) · ( − e θ (Φ)Ψ ϕϕ + e θ (Φ)Ψ θθ )= 2( e ϕ ψ ) + 8( e θ Φ) ψ Hence, g µν ˙ D µ Ψ ˙ D ν Ψ = − e ψe ψ + 2( e θ ψ ) + 2( e ϕ ψ ) + 4( e θ Φ) ψ and L (Ψ) = − e Ψ e ψ + 2( e θ ψ ) + 2( e ϕ ψ ) + 8( e θ Φ) ψ + 2 V ψ .1. HORIZONTAL S -TENSORS D.1.4 Energy-Momentum tensor Consider the energy-momentum tensor, Q µν := ˙ D µ Ψ · ˙ D ν Ψ − g µν (cid:16) ˙ D λ Ψ · ˙ D λ Ψ + V Ψ · Ψ (cid:17) Lemma D.1.9. We have, D ν Q µν = ˙ D µ Ψ · (cid:0) ˙ (cid:3) Ψ − V ψ (cid:1) + ˙ D ν Ψ A R ABνµ Ψ B − D µ V Ψ · Ψ Proof. We have, D ν Q µν = ˙ D ν ˙ D ν Ψ · ˙ D µ Ψ + ˙ D ν Ψ · (cid:16) ˙ D ν ˙ D µ − ˙ D µ ˙ D ν (cid:17) Ψ − V D µ Ψ · Ψ − D µ V Ψ · Ψ= ˙ D µ Ψ · ˙ D ν ˙ D ν Ψ + ˙ D ν Ψ A R ABνµ Ψ B − V D µ ΨΨ − D µ V Ψ · Ψ= ˙ D µ Ψ (cid:0) ˙ (cid:3) Ψ − V Ψ (cid:1) + ˙ D ν Ψ A R ABνµ Ψ B − D µ V Ψ · Ψ Lemma D.1.10. Relative to an arbitrary Z -polarized frame e , e , e θ , e ϕ we have, Q = | e Ψ | , Q = | e Ψ | , Q = |∇ / Ψ | + V | Ψ | . If ψ is the reduced form of Ψ , Q = 2( e ψ ) , Q = 2( e ψ ) , Q = 2( e θ ψ ) + 2( e ϕ ψ ) + 2 V | ψ | + 8( e θ Φ) ψ . Also, g µν Q µν = −L (Ψ) − V | Ψ | , |L (Ψ) | (cid:46) | e Ψ | | e Ψ | + |∇ / Ψ | + V | Ψ | , and |Q AB | ≤ | e Ψ || e Ψ | + |∇ / Ψ | + | V || Ψ | , |Q A | ≤ | e Ψ ||∇ / Ψ | , |Q A | ≤ | e Ψ ||∇ / Ψ | . APPENDIX D. APPENDIX TO CHAPTER 10 D.2 Standard Calculation Proposition D.2.1. Consider an admissible spacetime M and Ψ ∈ S ( M ) and X avectorfield of the form, X = ae + be , 1. The -form P µ = Q µν X ν verifies, D µ P µ = X µ ˙ D µ Ψ · (cid:0) ˙ (cid:3) Ψ − V Ψ (cid:1) − X ( V )Ψ · Ψ 2. Let X as above, w a scalar and M a one form. Define, P µ = P µ [ X, w, M ] = Q µν X ν + 12 w Ψ · ˙ D µ Ψ − | Ψ | ∂ µ w + 14 | Ψ | M µ Then, with | Ψ | := Ψ · Ψ , D µ P µ [ X, w, M ] = 12 Q · ( X ) π − X ( V )Ψ · Ψ + 12 w L [Ψ] − | Ψ | (cid:3) g w + 14 | Ψ | Div M + 12 Ψ · ˙ D µ Ψ M µ + (cid:18) X (Ψ) + 12 w Ψ (cid:19) · (cid:0) ˙ (cid:3) Ψ − V Ψ (cid:1) Proof. Let P µ [ X, , 0] = Q µν X ν , Then, D µ P µ = X µ ˙ D µ Ψ · (cid:16) ˙ D ν ˙ D ν Ψ − V Ψ (cid:17) + X µ ˙ D ν Ψ A R ABνµ Ψ B − X µ D µ V Ψ · Ψ= X µ ˙ D µ Ψ · (cid:0) ˙ (cid:3) Ψ − V Ψ (cid:1) − X ( V ) | Ψ | Assume X = ae + be . Then, since only the middle components of R are relevant, andrecalling that R AB = − (cid:63) ρ ∈ AB = 0, we derive, X µ ˙ D ν Ψ A R ABν Ψ B = a ˙ D Ψ A R AB Ψ B + b ˙ D Ψ A R AB Ψ B = 0To prove the second part of the proposition we write with N [Ψ] := ˙ (cid:3) Ψ − V Ψ, D µ P µ [ X, w, M ] = 12 Q · ( X ) π + X (Ψ) · N [Ψ] − X ( V )Ψ · Ψ + 12 D µ w Ψ · ˙ D µ Ψ+ 12 w ˙ D µ Ψ · ˙ D µ Ψ + 12 w Ψ ˙ (cid:3) g Ψ − 12 Ψ · ˙ D µ Ψ ∂ µ w − | Ψ | (cid:3) g w + 14 | Ψ | Div M + 12 Ψ · ˙ D µ Ψ M µ = 12 Q · ( X ) π − X ( V )Ψ · Ψ + 12 w ˙ D µ Ψ · ˙ D µ Ψ + 12 w Ψ ( V Ψ + N [Ψ]) − | Ψ | (cid:3) g w + 14 | Ψ | Div M + 12 Ψ · ˙ D µ Ψ M µ + X (Ψ) · Ψ · N [Ψ] .3. VECTORFIELD X F D µ P µ [ X, w, M ] = 12 Q · ( X ) π − X ( V )Ψ · Ψ + 12 w L [Ψ] − | Ψ | (cid:3) g w + 14 | Ψ | Div M + 12 Ψ · ˙ D µ Ψ M µ + (cid:18) X (Ψ) + 12 w Ψ (cid:19) · N [Ψ]as desired. Remark D.2.2. As consequence of the proposition above we deduce that every time weuse vectorfields of the form ae + be as multipliers, the equation (cid:3) Ψ − V Ψ = N is treatedexactly in the same manner as the scalar equation (cid:3) ψ − V ψ = N . Remark D.2.3. Note that in Schwarzschild our potential V = κκ = 4Υ r − verifies, ∂ r V = ∂ r (cid:20) r − (cid:18) − mr (cid:19)(cid:21) = − r − (cid:18) − mr (cid:19) + 2 mr = − r − mr . D.3 Vectorfield X f Lemma D.3.1. Let X f := f e . Then with ( X ) Λ = fr and ( X ) (cid:101) π = ( X ) π − ( X ) Λ g = ( X ) π − fr g , • We have, ( X ) (cid:101) π = 0 , ( X ) π ϕ = 0 , ( X ) π ϕ = 0 , ( X ) (cid:101) π = − e f + 4 f ω + 4 fr = − (cid:18) e ( f ) − fr (cid:19) + 4 f ω, ( X ) (cid:101) π θ = 2 f ξ, ( X ) (cid:101) π AB = 2 f (1+3) χ AB − fr g/ AB = 2 f (cid:18) (1+3) χ AB − r δ AB (cid:19) , ( X ) (cid:101) π θ = 2 f ( η + ζ ) , ( X ) (cid:101) π = − f ω − e ( f ) . (D.3.1)92 APPENDIX D. APPENDIX TO CHAPTER 10 • In particular, we have, ( X ) (cid:101) π = − f (cid:48) + 4 fr + O ( (cid:15) ) min { w , , w , / } ( | f | + r | f (cid:48) | ) , ( X ) (cid:101) π A = (cid:15) min { w , , w , / } , ( X ) (cid:101) π AB = O ( (cid:15) ) min { w , , w , / }| f | , ( X ) (cid:101) π A = O ( (cid:15) ) w , | f | , ( X ) (cid:101) π = 4 f (cid:48) Υ − (cid:48) + O ( (cid:15) ) w , ( | f | + r | f (cid:48) | ) . (D.3.2) • We have, (cid:3) ( X ) Λ = 2 r f (cid:48)(cid:48) + O (cid:16) mr + (cid:15)w , (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) . (D.3.3) Proof. We calculate ( X ) π αβ = g ( D e α X, e β ) + g ( D e β X, e α ), ( X ) π = 0 ( X ) π = − e f + 4 f ω ( X ) π θ = 2 f ξ ( X ) π AB = 2 f (1+3) χ AB ( X ) π θ = 2 f ( η + ζ ) ( X ) π = − f ω − e ( f )We deduce, for ( X ) (cid:101) π = ( X ) π − ( X ) Λ g = ( X ) π − fr g , ( X ) (cid:101) π = 0 ( X ) (cid:101) π = − e f + 4 f ω + 4 fr = − (cid:18) e ( f ) − fr (cid:19) + 4 f ω ( X ) (cid:101) π θ = 2 f ξ ( X ) (cid:101) π AB = 2 f (1+3) χ AB − fr g/ AB = 2 f (cid:18) (1+3) χ AB − r δ AB (cid:19) ( X ) (cid:101) π θ = 2 f ( η + ζ ) ( X ) (cid:101) π = − f ω − e ( f )Under the assumptions (10.2.8)– (10.2.9) on the Ricci coefficients (with respect to the .3. VECTORFIELD X F e (cid:48) , e (cid:48) )), we deduce, ( X ) (cid:101) π = − e f + 4 f ω = − f (cid:48) + 4 fr − f (cid:48) ( e ( r ) − 1) + 4 f ( ω − − f (cid:48) + 4 fr + (cid:15) min { w , , w , / } ( | f | + r | f (cid:48) | ) ( X ) (cid:101) π A = (cid:15) min { w , , w , / } , ( X ) (cid:101) π AB = (cid:15) min { w , , w , / }| f | ( X ) (cid:101) π A = min { w , , w , / }| f | ( X ) (cid:101) π = − f ω − e ( f ) = − f (cid:16) mr + (cid:15)w , (cid:17) − f (cid:48) ( − Υ + (cid:15)w , )= 4 f (cid:48) Υ − (cid:48) + (cid:15)w , ( | f | + r | f (cid:48) | )To prove formula (D.3.3) we make use of the following (see also Lemma 10.1.11), Lemma D.3.2. If h = h ( r ) then (cid:3) h = Υ h (cid:48)(cid:48) ( r ) + (cid:18) r − mr (cid:19) h (cid:48) + O ( (cid:15) ) w , (cid:0) | h | + r | h (cid:48) | + r | h (cid:48)(cid:48) | (cid:1) Proof. For a general scalar h , (cid:3) h = − 12 ( e e + e e ) h + (cid:52) / h + (cid:18) (1+3) ω − (1+3) tr χ (cid:19) e h + ( (1+3) ω − (1+3) tr χ ) e h with (cid:52) / h = e θ e θ h + ( e θ Φ) e θ h = 0 if h is radial. Thus, (cid:3) h = − 12 ( e e + e e ) h + ( (1+3) ω − (1+3) tr χ ) e h + ( (1+3) ω − (1+3) tr χ ) e h = − f (cid:48)(cid:48) ( e r )( e r ) − h (cid:48) ( e e + e e ) r + h (cid:48) (cid:20) ( (1+3) ω − (1+3) tr χ ) e r + ( (1+3) ω − (1+3) tr χ ) e r (cid:21) = − h (cid:48)(cid:48) ( − Υ + O ( (cid:15) ) w , )(1 + O ( (cid:15) ) w , ) + (cid:0) mr + O ( (cid:15) ) w , (cid:1) h (cid:48) + h (cid:48) (cid:20) ( mr + Υ r + O ( (cid:15) ) w , )(1 + O ( (cid:15) ) w , ) + ( − r + O ( (cid:15) ) w , )( − Υ + O ( (cid:15) ) w , (cid:21) = Υ h (cid:48)(cid:48) + (cid:18) r − mr (cid:19) h (cid:48) + O ( (cid:15) ) w , (cid:0) | h | + r | h (cid:48) | + r | h (cid:48)(cid:48) | (cid:1) which concludes the proof of Lemma D.3.2.94 APPENDIX D. APPENDIX TO CHAPTER 10 In view of Lemma D.3.2, (cid:3) ( X ) Λ = (cid:3) (cid:18) fr (cid:19) = Υ (cid:18) fr (cid:19) (cid:48)(cid:48) + (cid:18) r − mr (cid:19) ( 2 fr ) (cid:48) + O ( (cid:15) ) w , (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) Note that,Υ (cid:18) fr (cid:19) (cid:48)(cid:48) + (cid:18) r − mr (cid:19) (cid:18) fr (cid:19) (cid:48) = Υ (cid:18) f (cid:48)(cid:48) r − f (cid:48) r + 4 fr (cid:19) + (cid:18) r − mr (cid:19) (cid:18) f (cid:48) r − fr (cid:19) = 2Υ r f (cid:48)(cid:48) − (Υ − 1) 4 f (cid:48) r + (Υ − 1) 4 fr − mr (cid:18) f (cid:48) r − fr (cid:19) = 2 r + O (cid:16) mr (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) Hence, (cid:3) ( X ) Λ = 2 r f (cid:48)(cid:48) + O (cid:16) mr (cid:15)w , (cid:17) (cid:0) | f | + r | f (cid:48) | + r | f (cid:48)(cid:48) | (cid:1) as desired. This concludes the proof of Lemma D.3.1. D.4 Proof of Proposition 10.3.1 In view of the following Leibniz rule which holds for any scalar f , −(cid:52) / ( f ψ ) = d (cid:63) / d/ ( f ψ ) + 2 Kf ψ = d (cid:63) / ( f d/ ψ + e θ ( f ) ψ ) + 2 Kf ψ = − f (cid:52) / ψ − e θ ( f ) d/ ψ + e θ ( f ) d (cid:63) / ψ − (cid:52) / ( f ) ψ, we have the following computation e ( (cid:3) ( rψ )) = e ( r (cid:3) ψ ) − e ( e ( r ) e ψ ) − e ( e ( r ) e ψ ) − e ( e θ ( r ) d/ ψ )+2 e ( e θ ( r ) d (cid:63) / ψ ) + e ( (cid:3) ( r ) ψ )= e ( r (cid:3) ψ ) − e (cid:16) r κ + A ) e ψ (cid:17) − e (cid:16) r κ + A ) e ψ (cid:17) + e ( (cid:3) ( r ) ψ ) + r − d ≤ (Γ g ) d ≤ ψ = e ( r (cid:3) ψ ) − e ( rκe ψ ) − e ( rκe ψ ) + e ( (cid:3) ( r ) ψ ) + r − Err , where we have introduced the notation, used throughout the proof of Proposition 10.3.1,Err := r Γ g e e ψ + r Γ b e d ψ + d ≤ (Γ b ) d ≤ ψ + r d ≤ (Γ g ) e ψ + d ≤ (Γ g ) d ψ. .4. PROOF OF PROPOSITION 10.3.1 (cid:3) ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ. We infer (cid:3) ( r ) = − e ( e ( r )) + (cid:52) / ( r ) + (cid:18) ω − κ (cid:19) e ( r ) − κe ( r ) + 2 ηe θ ( r )= − e (cid:16) r κ + A ) (cid:17) + (cid:18) ω − κ (cid:19) r κ + A ) − κ r κ + A ) + r − d ≤ Γ g = − e ( rκ ) + 12 (cid:18) ω − κ (cid:19) rκ − rκκ + r d ≤ Γ g = 2 r + O ( r − ) + d ≤ Γ b and hence e ( (cid:3) ( rψ )) = e ( r (cid:3) ψ ) − e ( rκe ψ ) − e ( rκe ψ ) + e ( (cid:3) ( r ) ψ ) + d ≤ (Γ g ) d ≤ ψ = e ( r (cid:3) ψ ) − e ( rκe ψ ) − e ( rκe ψ ) + e (cid:18) r ψ (cid:19) + O ( r − ) d ≤ ψ + r − Errso that e ( r (cid:3) ψ ) = e ( (cid:3) ( rψ )) + 12 e ( rκe ψ ) + 12 e ( rκe ψ ) − e (cid:18) r ψ (cid:19) + O ( r − ) d ≤ ψ + r − Err . We infer (cid:3) ( e ( rψ )) − e ( r (cid:3) ψ ) = [ (cid:3) , e ]( rψ ) − e ( rκe ψ ) − e ( rκe ψ ) + e (cid:18) r ψ (cid:19) + O ( r − ) d ≤ ψ + r − Err . Next, using again (cid:3) ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ, we infer[ (cid:3) , e ] ψ = − e [ e , e ] ψ + [ (cid:52) / , e ] ψ + (cid:18) ω − κ (cid:19) [ e , e ] ψ − e (cid:18) ω − κ (cid:19) e ψ + 12 e ( κ ) e ψ + 2 η [ e θ , e ] ψ − e ( η ) e θ ψ = − e [ e , e ] ψ + [ (cid:52) / , e ] ψ + (cid:18) ω − κ (cid:19) [ e , e ] ψ − (cid:18) e ( ω ) − (cid:18) − κ − ωκ (cid:19)(cid:19) e ψ + 12 (cid:18) − κκ + 2 ωκ + 2 ρ (cid:19) e ψ + 2 η [ e θ , e ] ψ + r − d ≤ (Γ g ) d ψ. APPENDIX D. APPENDIX TO CHAPTER 10 Now, recall [ e , e ] = 2 ωe − ωe + 2( η − η ) e θ , and, in view of Lemma 2.1.51, the following commutation formulae for reduced scalars1. If f ∈ s k ,[ d/ k , e ] = 12 κ d/ k f + Com k ( f ) , Com k ( f ) = − ϑ d (cid:63) / k +1 f − ( ζ + η ) e f − kηe Φ f − ξ ( e f + ke (Φ) f ) − kβf. 2. If f ∈ s k − [ d (cid:63) / k , e ] f = 12 κ d/ k f + Com ∗ k ( f ) , Com ∗ k ( f ) = − ϑ d/ k − f + ( ζ + η ) e f − ( k − ηe Φ f + ξ ( e f − ( k − e (Φ) f ) − ( k − βf. We infer[ (cid:3) , e ] ψ = − e (cid:16) (2 ωe − ωe + 2 ηe θ ) ψ (cid:17) + κ (cid:52) / ψ + (cid:18) ω − κ (cid:19) (cid:16) ωe − ωe + 2 ηe θ (cid:17) ψ − (cid:18) e ( ω ) + 14 κ + ωκ (cid:19) e ψ + 12 (cid:18) − κκ + 2 ωκ + 2 ρ (cid:19) e ψ + r − d ≤ (Γ g ) d ≤ ψ = 2 e ( ωe ψ ) + κ (cid:52) / ψ − (cid:18) e ( ω ) + 14 κ (cid:19) e ψ − κκe ψ + O ( r − ) d ≤ ψ + r − Err= 2 ωe ( e ψ ) + κ (cid:52) / ψ − κ e ψ − κκe ψ + O ( r − ) d ≤ ψ + r − Err . This implies[ (cid:3) , e ]( rψ ) = 2 ωe ( e ( rψ )) + κ (cid:52) / ( rψ ) − κ e ( rψ ) − κκe ( rψ ) + O ( r − ) d ≤ ψ + r − Err= 2 ωe ( e ( rψ )) + 2 ω [ e , e ] rψ + κ (cid:52) / ( rψ ) − κ e ( rψ ) − κκe ( rψ )+ O ( r − ) d ≤ ψ + r − Err= 2 ωe ( e ( rψ )) + κ (cid:52) / ( rψ ) − κ e ( rψ ) − κκe ( rψ ) + O ( r − ) d ≤ ψ + r − Err .4. PROOF OF PROPOSITION 10.3.1 (cid:3) ( e ( rψ )) − e ( r (cid:3) ψ )= [ (cid:3) , e ]( rψ ) − e ( rκe ψ ) − e ( rκe ψ ) + e (cid:18) r ψ (cid:19) + O ( r − ) d ≤ ψ + r − Err= 2 ωe ( e ( rψ )) − e ( rκe ψ ) − e ( rκe ψ ) + κ (cid:52) / ( rψ ) − κ e ( rψ ) − κκe ( rψ ) + e (cid:18) r ψ (cid:19) + O ( r − ) d ≤ ψ + r − Err . Next, we compute − e ( rκe ψ ) − e ( rκe ψ ) + κ (cid:52) / ( rψ ) − κ e ( rψ ) + e (cid:18) r ψ (cid:19) = − e ( κ ( e ( rψ ) − e ( r ) ψ )) − e ( e ( rκψ )) − 12 [ e , e ]( rκψ ) + 12 e ( e ( rκ ) ψ )+ rκ (cid:52) / ψ − rκ e ψ − κ e ( r ) ψ + 2 r e ( rψ ) + e (cid:18) r (cid:19) rψ + r − d ≤ (Γ g ) d ≤ ψ = − e ( κ ( e ( rψ ))) + 12 e (cid:16) κ r κ + A ) ψ (cid:17) − e ( κe ( rψ )) − e ( e ( κ ) rψ ) − (cid:16) − ωe + 2 ωe − η − η ) e θ (cid:17) ( rκψ ) + 12 e ( e ( rκ ) ψ )+ rκ (cid:52) / ψ − rκ e ψ − κ r κ + A ) ψ + 2 r e ( rψ ) − e ( r ) r ψ + r − d ≤ (Γ g ) d ≤ ψ i.e. − e ( rκe ψ ) − e ( rκe ψ ) + κ (cid:52) / ( rψ ) − κ e ( rψ ) + e (cid:18) r ψ (cid:19) = − e ( κ ( e ( rψ ))) + 14 e ( rκκψ ) − e ( κe ( rψ )) − e (cid:18)(cid:18) − κ − ωκ (cid:19) rψ (cid:19) − ωe ( rκψ ) + 12 e ( e ( rκ ) ψ ) + rκ (cid:52) / ψ − rκ e ψ − rκ κψ + 2 r e ( rψ ) − κr ψ + O ( r − ) d ≤ ψ + r − Err= − e ( κ ( e ( rψ ))) + 14 κκe ( rψ ) + 14 e ( κκ ) rψ − e ( κe ( rψ )) − e (cid:18)(cid:18) − κ − ωκ (cid:19) rψ (cid:19) − ωe ( rκψ ) + 12 r − e ( rκ ) e ( rψ ) + 12 e ( r − e ( rκ )) rψ + rκ (cid:52) / ψ − rκ e ψ − rκ κψ + 2 r e ( rψ ) − κr ψ + O ( r − ) d ≤ ψ + r − Err . APPENDIX D. APPENDIX TO CHAPTER 10 We infer (cid:3) ( e ( rψ )) − e ( r (cid:3) ψ )= 2 ωe ( e ( rψ )) − e ( κ ( e ( rψ ))) + 14 e ( κκ ) rψ − e ( κe ( rψ )) − e (cid:18)(cid:18) − κ − ωκ (cid:19) rψ (cid:19) − ωe ( rκψ ) + 12 r − e ( rκ ) e ( rψ ) + 12 e ( r − e ( rκ )) rψ + rκ (cid:52) / ψ − rκ e ψ − rκ κψ + 2 r e ( rψ ) − κr ψ + O ( r − ) d ≤ ψ + r − Err . Since e ( rψ ) = r Υˇ e ψ , this may be rewritten as (cid:3) ( r Υˇ e ψ ) − e ( r (cid:3) ψ )= 2 ωe ( r Υˇ e ψ ) − e ( r Υ κ ˇ e ψ ) + 14 e ( κκ ) rψ − e ( r Υ κ ˇ e ψ ) − e (cid:18)(cid:18) − κ − ωκ (cid:19) rψ (cid:19) − ωe ( rκψ ) + 12 e ( rκ )Υˇ e ψ + 12 e ( r − e ( rκ )) rψ + rκ (cid:52) / ψ − rκ e ψ − rκ κψ + 2 r Υˇ e ψ − κr ψ + O ( r − ) d ≤ ψ + r − Err . Now, since (cid:3) ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ, we have rκ (cid:52) / ψ = rκ (cid:3) ψ + rκe e ψ − rκ (cid:18) ω − κ (cid:19) e ψ + 12 rκ e ψ + r − d ≤ (Γ b ) d ≤ ψ = rκ (cid:3) ψ + rκe ( r − e ( rψ )) − rκe ( r − e ( r ) ψ ) + 12 κκe ( rψ ) − κκe ( r ) ψ + 12 rκ e ψ + r − d ≤ (Γ b ) d ≤ ψ = rκ (cid:3) ψ + rκe (Υˇ e ψ ) − rκe ( κ ) ψ + 12 κκr Υˇ e ψ − r κ κψ + r − d ≤ (Γ b ) d ≤ ψ and hence (cid:3) ( r Υˇ e ψ ) − e ( r (cid:3) ψ )= 2 ωe ( r Υˇ e ψ ) − e ( r Υ κ ˇ e ψ ) + 14 e ( κκ ) rψ − e ( r Υ κ ˇ e ψ ) − e (cid:18)(cid:18) − κ − ωκ (cid:19) rψ (cid:19) − ωe ( rκψ ) + 12 e ( rκ )Υˇ e ψ + 12 e ( r − e ( rκ )) rψ + rκ (cid:3) ψ + rκe (Υˇ e ψ ) − rκe ( κ ) ψ − r κ κψ − rκ e ψ − rκ κψ − κr ψ + O ( r − ) d ≤ ψ + r − Err . .4. PROOF OF PROPOSITION 10.3.1 e ( κκ ) rψ − e (cid:18)(cid:18) − κ − ωκ (cid:19) rψ (cid:19) − ωe ( rκψ ) + 12 e ( r − e ( rκ )) rψ − rκe ( κ ) ψ − r κ κψ − rκ e ψ − rκ κψ − κr ψ = r κρψ + r − d ≤ (Γ b ) ψ = O ( r − ) ψ + r − d ≤ (Γ b ) ψ so that (cid:3) ( r Υˇ e ψ ) = e ( r (cid:3) ψ ) + rκ (cid:3) ψ + 2 ωe ( r Υˇ e ψ ) − e ( r Υ κ ˇ e ψ ) − e ( r Υ κ ˇ e ψ )+ 12 e ( rκ )Υˇ e ψ + rκe (Υˇ e ψ ) + O ( r − ) d ≤ ψ + r − Err . Since (cid:3) ( r ˇ e ψ ) = r Υ − (cid:3) ( r Υˇ e ψ ) − e ( r Υ − ) e ( r Υˇ e ψ ) − e ( r Υ − ) e ( r Υˇ e ψ )+ (cid:3) ( r Υ − ) r Υˇ e ψ + d ≤ (Γ g ) d ≤ ψ, we infer (cid:3) ( r ˇ e ψ ) = r Υ − e ( r (cid:3) ψ ) + r Υ − κ (cid:3) ψ + 2 r Υ − ωe ( r Υˇ e ψ ) − r Υ − e ( r Υ κ ˇ e ψ ) − r Υ − e ( r Υ κ ˇ e ψ ) + 12 re ( rκ )ˇ e ψ + r Υ − κe (Υˇ e ψ ) − e ( r Υ − ) e ( r Υˇ e ψ ) − e ( r Υ − ) e ( r Υˇ e ψ )+ (cid:3) ( r Υ − ) r Υˇ e ψ + O ( r − ) d ≤ ψ + Err . Now, we have2 r Υ − ωe ( r Υˇ e ψ ) − r Υ − e ( r Υ κ ˇ e ψ ) − r Υ − e ( r Υ κ ˇ e ψ ) + 12 re ( rκ )ˇ e ψ + r Υ − κe (Υˇ e ψ ) − e ( r Υ − ) e ( r Υˇ e ψ ) − e ( r Υ − ) e ( r Υˇ e ψ ) + (cid:3) ( r Υ − ) r Υˇ e ψ = 2 r − mr Υ e (ˇ e ψ )+ (cid:40) r Υ − ωe ( r Υ) − r Υ − e ( r Υ κ ) − r Υ − e ( r Υ κ ) + 12 re ( rκ ) + r Υ − κe (Υ) − e ( r Υ − ) e ( r Υ) − e ( r Υ − ) e ( r Υ) + (cid:3) ( r Υ − ) r Υ (cid:41) ˇ e ψ + Err . APPENDIX D. APPENDIX TO CHAPTER 10 Also, we have2 r Υ − ωe ( r Υ) − r Υ − e ( r Υ κ ) − r Υ − e ( r Υ κ ) + 12 re ( rκ ) + r Υ − κe (Υ) − e ( r Υ − ) e ( r Υ) − e ( r Υ − ) e ( r Υ) + (cid:3) ( r Υ − ) r Υ= 4 + O ( r − ) + r Γ b = − r κκ + O ( r − ) + r Γ b . We infer ( (cid:3) + κκ )( r ˇ e ψ ) = r Υ − e ( r (cid:3) ψ ) + r Υ − κ (cid:3) ψ + 2 r − mr Υ e (ˇ e ψ )+ O ( r − ) d ≤ ψ + Err . In view of the wee equation satisfied by ψ , i.e. (cid:3) ψ + κκψ = N, we have r Υ − e ( r (cid:3) ψ ) + r Υ − κ (cid:3) ψ + 2 r − mr Υ e (ˇ e ψ )= r Υ − e ( r ( N − κκψ )) + r Υ − κ ( N − κκψ ) + 2 r − mr Υ e ( r ˇ e ψ ) − − mr Υ e ( r )ˇ e ψ = r Υ − e ( rN ) + r Υ − κN + 2 r − mr Υ e ( r ˇ e ψ ) + 4 mr ˇ e ψ − r Υ − κρψ + d ≤ (Γ b ) d ≤ ψ = r (cid:18) Υ − e ( N ) + 3 r N (cid:19) + 2 r − mr Υ e ( r ˇ e ψ ) + O ( r − ) d ≤ ψ + d ≤ (Γ b ) d ≤ ψ, from which we deduce( (cid:3) + κκ )( r ˇ e ψ ) = r (cid:18) Υ − e ( N ) + 3 r N (cid:19) + 2 r − mr Υ e ( r ˇ e ψ ) + O ( r − ) d ≤ ψ + Err . Since ˇ ψ = f ˇ e ψ = f r r ˇ e ψ, we infer( (cid:3) + κκ ) ˇ ψ = f r ( (cid:3) + κκ )( r ˇ e ψ ) − e (cid:18) f r (cid:19) e ( r ˇ e ψ ) − e (cid:18) f r (cid:19) e ( r ˇ e ψ )+ e θ (cid:18) f r (cid:19) d/ ( r ˇ e ψ ) − e θ (cid:18) f r (cid:19) d (cid:63) / ( r ˇ e ψ ) + (cid:3) (cid:18) f r (cid:19) r ˇ e ψ .4. PROOF OF PROPOSITION 10.3.1 (cid:3) + κκ ) ˇ ψ = f (cid:18) Υ − e ( N ) + 3 r N (cid:19) + f r (cid:40) r − mr Υ e ( r ˇ e ψ ) + O ( r − ) d ≤ ψ + Err (cid:41) − e (cid:18) f r (cid:19) e ( r ˇ e ψ ) − e (cid:18) f r (cid:19) e ( r ˇ e ψ )+ e θ (cid:18) f r (cid:19) d/ ( r ˇ e ψ ) − e θ (cid:18) f r (cid:19) d (cid:63) / ( r ˇ e ψ ) + (cid:3) (cid:18) f r (cid:19) r ˇ e ψ. Now, recall that Err is defined byErr = r Γ g e e ψ + r Γ b e d ψ + d ≤ (Γ b ) d ≤ ψ + r d ≤ (Γ g ) e ψ + d ≤ (Γ g ) d ψ. so that( (cid:3) + κκ ) ˇ ψ = f (cid:18) Υ − e ( N ) + 3 r N (cid:19) + f r (cid:40) r − mr Υ e ( r ˇ e ψ ) + O ( r − ) d ≤ ψ + r Γ g e e ψ + r Γ b e d ψ + d ≤ (Γ b ) d ≤ ψ + r d ≤ (Γ g ) e ψ + d ≤ (Γ g ) d ψ (cid:41) − e (cid:18) f r (cid:19) e ( r ˇ e ψ ) − e (cid:18) f r (cid:19) e ( r ˇ e ψ )+ e θ (cid:18) f r (cid:19) d/ ( r ˇ e ψ ) − e θ (cid:18) f r (cid:19) d (cid:63) / ( r ˇ e ψ ) + (cid:3) (cid:18) f r (cid:19) r ˇ e ψ. In view of (cid:3) ψ = − e e ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ, we have r Γ g e e ψ = r Γ g (cid:18) − (cid:3) ψ + (cid:52) / ψ + (cid:18) ω − κ (cid:19) e ψ − κe ψ + 2 ηe θ ψ (cid:19) = − r Γ g N + r Γ g e ψ + Γ g d ≤ ψ APPENDIX D. APPENDIX TO CHAPTER 10 and hence( (cid:3) + κκ ) ˇ ψ = f (cid:18) Υ − e ( N ) + 3 r N (cid:19) + f r (cid:40) r − mr Υ e ( r ˇ e ψ ) + O ( r − ) d ≤ ψ + r Γ b e d ψ + d ≤ (Γ b ) d ≤ ψ + r d ≤ (Γ g ) e ψ + d ≤ (Γ g ) d ψ (cid:41) − e (cid:18) f r (cid:19) e ( r ˇ e ψ ) − e (cid:18) f r (cid:19) e ( r ˇ e ψ )+ e θ (cid:18) f r (cid:19) d/ ( r ˇ e ψ ) − e θ (cid:18) f r (cid:19) d (cid:63) / ( r ˇ e ψ ) + (cid:3) (cid:18) f r (cid:19) r ˇ e ψ. 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