On the asymptotic behavior of Einstein manifolds with an integral bound on the Weyl curvature
aa r X i v : . [ m a t h . DG ] O c t ON THE ASYMPTOTIC BEHAVIOR OF EINSTEIN MANIFOLDSWITH AN INTEGRAL BOUND ON THE WEYL CURVATURE
ROMAIN GICQUAUD, DANDAN JI † AND YUGUANG SHI † Abstract.
In this paper we consider the geometric behavior near infinity of some Ein-stein manifolds ( X n , g ) with Weyl curvature belonging to a certain L p space. Namely,we show that if ( X n , g ), n ≥
7, admits an essential set and has its Weyl curvature in L p for some 1 < p < n − , then ( X n , g ) must be asymptotically locally hyperbolic. Oneinteresting application of this theorem is to show a rigidity result for the hyperbolicspace under an integral condition for the curvature. Contents
1. Introduction 12. Basic Estimates 33. Pointwise estimate for the Weyl tensor 174. Applications 21References 241.
Introduction
During the last three decades there were lots of interesting works on the asymptoticbehavior of Ricci flat metrics with integral bounds on curvature. See e.g. [4] and [8].These works gave an nice intrinsic characterization of asymptotic locally Euclidean (ALE)manifolds. Inspired by these works, we want to study a similar problem in the contextof asymptotic locally hyperbolic (ALH) manifolds. The ALH case appears much morecomplicated than the ALE case in both geometric and analytic parts. As an example, therescaling argument which is very efficient in the analysis of the asymptotic behavior ofALE metrics does not work in the ALH case because the model is the hyperbolic metricwhich is not scale invariant. Another complication arises from the M¨obius group of S n which allows cuspidal ends. To rule out such ends, we need to assume that the manifold( X n , g ) admits an essential set (see [3] for more details): Definition 1.1.
A non empty compact subset D of a complete noncompact Riemannianmanifold ( X n , g ) is called an essential set if1. D is a compact domain of X n with smooth and strictly convex boundary B := ∂ D ,i.e. its second fundamental forms with respect to the outward unit normal vectorfield is positive definite, Date : September 2, 2018.2000
Mathematics Subject Classification.
Primary 53C25; Secondary 58J05.
Key words and phrases. conformally compact manifold, asymptotically hyperbolic, rigidity. † Research partially supported by NSF grant of China 10990013. D is totally convex: there is no geodesic γ : [ a ; b ] → X such that γ ( a ) , γ ( b ) ∈ D and γ ( c ) D for some c ∈ [ a ; b ],3. the sectional curvature of ( X n , g ) is negative outside D .Assuming that ( X n , g ) is hyperbolic the existence of an essential set is equivalent tothe requirement that ( X n , g ) is convex and co-compact. More generally, it can be shownthat conformally compact and Cartan-Hadamard manifolds admit essential sets, see [11].Together with assumptions on the rate of convergence of the sectional curvature to − X n , g ) in [1, 2, 13, 16].In what follows we define ρ : X → R as the distance function from D : ρ := d g ( D , · ) . In [3], it has been proven that, if D ⊂ X is an essential set, ρ is smooth function andhas no critical point. This implies that the region X n which is outside the essential set D is diffeomorphic to [0 , ∞ ) × B .In this article we want to investigate the behavior at infinity of some Einstein manifoldswith Weyl curvature belonging to a certain L p space. In particular, we show that theyare asymptotically locally hyperbolic Einstein (ALHE) metric outside the essential set D ,meaning that sec g +1 = O ( e − aρ ) for some a >
0. In contrast with the ALE case, themajor difficulty in the ALH setting is the lack of sharp global Sobolev inequalities whichare crucial in applying Moser iterations in the ALE case (see e.g. [4], [8]).However, we observed a nice L -estimate for the Laplace operator acting on 4-tensorssatisfying properties analogous to those of the Weyl tensor for manifolds of dimensiongreater than 5, see Lemma 2.12 below. Thanks to this lemma and combining other tech-niques, we were able to obtain the following result (See also Theorems 3.4): Theorem 1.2.
Let ( X n , g ) , n ≥ , be a complete noncompact Einstein manifold with Ric = − ( n − g. Assume that X n contains an essential set D . We denote W the Weyl tensor of the metric g . If k W k L p ( X n ,g ) < ∞ for some p ∈ (cid:0) n − (cid:1) , then there exists a constant C such that | Rm − K | ≤ Ce − ( n +1) ρ . (1.1) Here Rm is the curvature tensor of the metric g and K the constant curvature tensorwith sectional curvature − with respect to metric g , i.e. K ijkl = − ( g ik g jl − g ij g kl ) . Since ( X n , g ) is Einstein and has a lower bound on its injectivity radius, it will becomeapparent that W ∈ L ∞ . As a consequence if W ∈ L p for some p ∈ (1 , ∞ ), W ∈ L q for all q ≥ p : the smaller p is, the more stringent the assumption.This result turns out to be very useful to prove rigidity theorems. In particular, assum-ing further that the manifold X is simply connected at infinity forces ( X, g ) to be isometricto the hyperbolic space (see Theorem 4.1). We also give a variant of this theorem for staticspacetimes together with a rigidity result in Section 4.We are interested in this article only in complete noncompact manifolds whose curvaturewill be shown to tend to − Einsteinmanifold ” to denote manifolds ( X n , g ) satisfyingRic g = − ( n − g. INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p Einstein metrics constructed in [14], [19] and [5] satisfy | W | ≤ Ce − ρ . In particularW ∈ L p for any p > n − . The case p = n − is more delicate and we plan to address it ina future work. Nevertheless Theorem 3.3 shows that Theorem 1.2 remains true for n ≥ p ≤ n − in the important case of conformally compact metrics.In the ALE case, the curvature behavior at infinity which is the analog of (1.1) isobtained by a Moser iteration where a global Sobolev inequality is involved. However, asmentioned above, in the ALH case such a kind of global Sobolev inequality is not true (see[12] for an illustration of this fact). Hence, we use a variant of the maximum principle toget Estimate (1.1). This is where the assumption n ≥ X n , g ) is Einstein, its Weyl tensor satisfies the following well known equation: △ W + 2( n − W + 2 Q ( W ) = 0 , (1.2)where △ is the Laplace operator acting on tensors and Q is a quadratic expression in theWeyl curvature tensor. See e.g. [2] for a derivation of this formula. Setting B αβγδ := W µ να β W µγνδ , (1.3) Q can be written as follows: Q αβγδ := B αβγδ + B αγβδ − B βαγδ − B βδαγ . (1.4)Note that we are using the Einstein summation convention. Due to the L p -bound of W ,we see that W is small near infinity. Hence, intuitively Equation (1.2) is almost equivalentto the following linear equation: △ W + 2( n − W = 0 . (1.5)By some careful analysis, we are able to show an L spectral estimate of the Laplaceoperator acting on Weyl-type tensors (see Lemma 2.12). Together with a refined Katoinequality and other some other techniques we achieve the proof of Estimate (1.1).Some applications of Theorem 1.2 are considered in this paper. Namely, by Theorem1.2 we are able to show a rigidity theorem for ALHE manifolds with Weyl tensor belongingto L p . We also get the curvature behavior of vacuum static spacetimes with a negativecosmological constant. See Theorems 4.1 and 4.2 for more details.The rest of the paper goes as follows. In § L -estimates for the Weyltensor. Then we show how these estimates can be converted to pointwise estimates in § §
4, we discuss some applications of Theorem 1.2.
Acknowledgements
The authors are grateful to Professor Jie Qing, Dr. Jie Wu andDr. Xue Hu for their interest in this work and for many enlightening discussions.2.
Basic Estimates
The main purpose of this section is to prove Lemma 2.15 which gives a spectral estimateof some (0 , k W k L p ( X n ,g ) < ∞ for some p ∈ (cid:0) , n − (cid:1) , then k W k L ( X n ,g ) < ∞ . Moreover if n ≥
6, we have k e a ρ W k L ( X n ,g ) < ∞ for some positive a . See Proposition 2.17 for more details. R. GICQUAUD, D. JI AND Y. SHI
We choose once and for all a complete noncompact Einstein manifold ( X n , g ) containingan essential subset D . We first introduce some estimates for a Riccati equation that willbe useful for the analysis of the normal curvature equation. Similar results have beenobtained in [23, Lemma 2.3]. See also [1, 2, 13, 16].In all this section, we use Greek letters to denote indices going from 0 to n − n −
1. Unless otherwise stated, we use the Einsteinsummation convention. For any R ≥
0, we denote D R := { x ∈ X, d g ( x, D ) ≤ R } , and Σ R = ρ − ( R )a slice of constant ρ . Lemma 2.1.
Let ε be a positive constant. Assume that f ( ρ ) is a smooth positive functionof ρ > such that | f ( ρ ) − | ≤ ε . Assume further that y is a solution of y ′ + y = f satisfying y (0) > . Then y satisfies | y − | ≤ e − ρ/ | y (0) − | + ε. Proof.
We claim that y >
0. Indeed, if there exists ρ such that y ( ρ ) ≤
0, we can find some ρ satisfying y ( ρ ) = 0 and y ( τ ) > τ ∈ (0 , ρ ). In particular this implies that y ′ ( ρ ) ≤
0. But 0 ≥ y ′ ( ρ ) + y ( ρ ) = f ( ρ ) > , which is a contradiction. Next, we set z = y −
1. The equation satisfied by y implies thefollowing one for z : ( z ) ′ + 2( y + 1) z = 2 z ( f − . In particular, since we noticed that y >
0, we have( z ) ′ + 2 z < z ( f − ≤ z + ( f − . This inequality can be integrated to yield | z | ≤ p z (0) e − ρ + ε ≤ | z | (0) e − ρ/ + ε. (cid:3) We want to find nice coordinate charts to apply Schauder estimates. We choose touse harmonic coordinates. We refer the reader to [15] and references therein for moreinformations. Let
Q > α ∈ (0; 1) be arbitrary. Since the injectivity radius r I of( X n , g ) is strictly positive (see Lemma 2.2 below), there exists a constant r H > x ∈ X n , there exist harmonic coordinates y , ..., y n on the ball B r H ( x ) in which the metric g satisfies ( Q − δ ≤ g ≤ Qδ, k g − δ k C ,α ≤ Q − , where δ = dy ⊗ dy + ... + dy n ⊗ dy n is the flat metric. Lemma 2.2.
Assume that ( X n , g ) satisfies k W k L p ( X n ,g ) < ∞ for some p ∈ (1; ∞ ) . Thenthe injectivity radius r I ( x ) is bounded from below by some positive constant on ( X n , g ) . INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p Proof.
The injectivity radius of r I ( x ) a point x ∈ X is a positive continuous map from X to R ∗ + ∪ {∞} . Hence it is bounded from below on D by some r . Assume that thereexists a point x ∈ X \ D whose injectivity radius is less than r . We can assume that anypoint y with ρ ( y ) < ρ ( x ) has injectivity radius strictly greater than r . Then there existsa geodesic γ : [0; 1] → X of length r such that γ (0) = γ (1) = x .The function ρ ◦ γ is convex and cannot be constant. Indeed, if ρ ( γ ( t )) >
0, ( ρ ◦ γ ) ′′ ( t ) = S ( ˙ γ, ˙ γ ) ≥
0, with equality iff ˙ γ is colinear to ∇ ρ . So ρ ( γ (1 / < ρ ( x ). Consider now thegeodesics γ and γ defined on the interval [0; 1] by γ ( t ) = γ (cid:18) t (cid:19) ,γ ( t ) = γ (cid:18) − t (cid:19) . They are two geodesics starting at γ (1 /
2) and ending at x , both of length r . Thismeans that γ (1 /
2) has injectivity radius less than r and contradicts the definition of x . (cid:3) Lemma 2.3.
If we further assume that k W k L p ( X n ,g ) < ∞ for some p ∈ (1; ∞ ) , then theWeyl tensor W of ( X n , g ) tends uniformly to zero at infinity.Proof. In harmonic coordinates, the metric g satisfies an equation of the formRic ij = − g kl ∂ k ∂ l g ij + Q ( g, ∂g )where Q is an expression which is quadratic in ∂g , see e.g. [21]. Since g is Einstein,Ric ij = − ( n − g ij , we get by standard elliptic regularity that there exists a constant C such that k g k C ,α (cid:18) B rH ( x ) (cid:19) ≤ C . Thus we get a bound k W k L ∞ (cid:18) B rH ( x ) (cid:19) ≤ C . From W ∈ L p ( X n , g ), we get that for any small µ > R > Z X n \ D R − rH | W | p dV g ! ≤ µ. As a consequence, for any x ∈ M \ D R , we have that k W k L p (cid:18) B rH ( x ) (cid:19) ≤ k W k L p ( X n \ D R − rH ) ≤ µ. Select q ∈ ( n , ∞ ), q > p arbitrarly. From Young’s inequality, there exists β ∈ (0 , k W k L q (cid:18) B rH ( x ) (cid:19) ≤ k W k βL p (cid:18) B rH ( x ) (cid:19) k W k − βL ∞ (cid:18) B rH ( x ) (cid:19) ≤ C − β µ β . From [2], the Weyl tensor satisfies an equation of the form
R. GICQUAUD, D. JI AND Y. SHI △ W + 2( n − Q (W) = 0 , where Q was defined in Equation (1.4). Therefore from the interior Schauder estimates,we get k W k W ,q (cid:18) B r H ( x ) (cid:19) ≤ C " k W k L q (cid:18) B rH ( x ) (cid:19) + kQ (W) k L q (cid:18) B rH ( x ) (cid:19) ≤ C µ β , where we used the fact that k W k L ∞ (cid:18) B rH ( x ) (cid:19) ≤ C to estimate the quadratic term. (cid:3) For simplicity we may use Fermi coordinates ( x , . . . , x n − ) on the slices Σ ρ . We denote S ij the components of the second fundamental form of Σ ρ in this coordinate system. It iswell known that the following equation holds: ∂∂ρ S ji + S jk S ki = − Rm j i , (2.1)where the index 0 refers to the unit normal direction of Σ ρ , that is to say ∇ ρ . We definethe mean curvature of Σ ρ by H = g ij S ij = S ii . Since g is Einstein with scalar curvature − n ( n − αβγδ = − ( g αγ g βδ − g αδ g βγ ) + W αβγδ . (2.2)Combining Equations (2.1) and (2.2) with Lemma 2.3, we get the following lemma: Lemma 2.4.
Let H be the mean curvature of the hypersurfaces of constant ρ . If k W k L p ( X n ,g ) < ∞ for some p ∈ (1; ∞ ) , then H = ( n −
1) + o (1) .Proof. We fix an arbitrary ε >
0. From Equation (2.2), the Riccati equation for theWeingarten operator (2.1) can be rewritten as follows: ∂∂ρ S ji + S jk S ki = δ ji − W j i . From Lemma 2.3, there exists ρ > | W | < ε on X \ D ρ . It follows fromstandard methods (see e.g. [21, Chapter 6]) together with Lemma 2.1 that S satisfies | S − δ | ≤ sup Σ ρ | S − δ | ! e − ( ρ − ρ ) / + ε on X n \ D ρ . In particular, H = tr( S ) is controlled at infinity: | H − ( n − | ≤ ( n − sup Σ ρ | S − δ | ! e − ( ρ − ρ ) / + ( n − ε. Since ε was arbitrary, this proves the lemma. (cid:3) As a consequence of this lemma, we get the following L -estimate: Lemma 2.5 (Cheng-Yau estimate) . Assume that k W k L p ( X n ,g ) < ∞ for some p ∈ (1; ∞ ) .For every ε > , there exists a compact subset K ε ⊃ D such that for any u ∈ C ∞ c ( X \ K ε ) , − Z X u △ u dV g ≥ (cid:20) ( n − − ε (cid:21) Z X u dV g . INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p Proof.
We set ϕ = e − n − ρ . Remark that if ρ is large enough, H ≥ ( n − − εn − on X \ D ρ : −△ ϕ = − ( n − ϕ + n − Hϕ ≥ (cid:18) ( n − − ε (cid:19) ϕ. We rewrite △ u = △ (cid:18) ϕ uϕ (cid:19) = △ ϕϕ u + 2 (cid:28) dϕ, d uϕ (cid:29) + ϕ △ uϕ . So, − Z X \ D ρ u △ u dV g = − Z X \ D ρ △ ϕϕ u dV g − Z X \ D ρ u (cid:28) dϕ, d uϕ (cid:29) dV g − Z X \ D ρ uϕ △ uϕ dV g ≥ (cid:18) ( n − − ε (cid:19) Z X \ D ρ u dV g − Z X \ D ρ u (cid:28) dϕ, d uϕ (cid:29) dV g + Z X \ D ρ (cid:28) d ( uϕ ) , d uϕ (cid:29) dV g ≥ (cid:18) ( n − − ε (cid:19) Z X \ D ρ u dV g − Z X \ D ρ u (cid:28) dϕ, d uϕ (cid:29) dV g + Z X \ D ρ ϕ (cid:28) du, d uϕ (cid:29) dV g ≥ (cid:18) ( n − − ε (cid:19) Z X \ D ρ u dV g + Z X \ D ρ ϕ (cid:12)(cid:12)(cid:12)(cid:12) d uϕ (cid:12)(cid:12)(cid:12)(cid:12) dV g ≥ (cid:18) ( n − − ε (cid:19) Z X \ D ρ u dV g . (cid:3) As noted in [14] and [19], this simple estimate immediately yields an estimate for thecovariant Laplacian acting on tensor fields by making use of Kato’s inequality. Unfortu-nately, this estimate is not sharp enough to get useful estimates. In [19], Lee mainly dealswith symmetric 2-tensors. In order to get sharp estimates he considers r -tensor fields as( r − X -valued 2-forms. Some new observations are needed. Let us begin with the followingdefinition: Definition 2.6.
We say that a (0 , p + 2)-tensor ω belongs to Λ p T ∗ X , if it satisfies ω ( Y , Y ; Z , · · · , Z s , · · · , Z l , · · · , Z p )= − ω ( Y , Y ; Z , · · · , Z s − , Z l , Z s +1 , · · · , Z l − , Z s , Z l +1 , · · · , Z p ) , for every Y , Y , Z , · · · , Z p ∈ T X and any pair s, l with 1 ≤ s < l ≤ p . R. GICQUAUD, D. JI AND Y. SHI
It can be easily shown that in local coordinates ( x µ ) a (0 , p + 2) − tensor ω ∈ Λ p T ∗ X can be written as ω = 1 p ! ω µνα ··· α p dx µ ⊗ dx ν ⊗ ( dx α ∧ · · · ∧ dx α p ) , where the coefficients ω µνα ··· α p = ω ( ∂∂x µ , ∂∂x ν ; ∂∂x α , · · · , ∂∂x α p )satisfy ω µνα ··· α l ··· α s ··· α p = − ω µνα ··· α s ··· α l ··· α p (1 ≤ s < l ≤ p ) . For any local orthogonal frame { e µ } and dual coframe { e µ } , the exterior derivative D : C ∞ ( X ; Λ p T ∗ X ) → C ∞ ( X ; Λ p +1 T ∗ X )on T ∗ X -valued p -forms is given by Dω := e µ ∧ ∇ e µ ω for every ω ∈ Λ p T ∗ X . It is standard matter to check that D does not depend on thechoice of the frame { e µ } , see e.g. [6]. This can be seen as a consequence of the followingproposition which gives an intrinsic definition of D : Proposition 2.7. If ω ∈ Λ p T ∗ X , then Dω ( X , X ; Y , · · · , Y p ) = p X m =0 ( − m ( ∇ Y m ω )( X , X ; Y , · · · , b Y m , · · · Y p ) , for any X , X , Y , · · · , Y p ∈ T X.
Proof.
Choose a point x ∈ X and an orthonormal frame { e µ } such that ∇ e µ = ∇ e µ = 0at x , where { e µ } is the coframe dual to { e µ } . For a ω ∈ Λ p T ∗ X , ω can be written as ω = 1 p ! ω µν ; α α ··· α p e µ ⊗ e ν ⊗ ( e α ∧ e α ∧ · · · e α p ) . Then computing at x , Dω = e σ ∧ ∇ e σ ω = 1 p ! e σ ∧ ∇ e σ ( ω µν ; α α ··· α p e µ ⊗ e ν ⊗ ( e α ∧ e α ∧ · · · e α p ))= 1 p ! e σ ∧ ( ∇ e σ ω µν ; α α ··· α p ) e µ ⊗ e ν ⊗ ( e α ∧ e α ∧ · · · e α p )= 1 p ! ( ∇ e σ ω µν ; α α ··· α p ) e µ ⊗ e ν ⊗ ( e σ ∧ e α ∧ e α ∧ · · · e α p ) . Hence
INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p Dω ( Y , · · · , Y p ) = 1 p ! ( ∇ e σ ω µν ; α α ··· α p ) e µ ⊗ e ν ⊗ ( e σ ∧ e α ∧ · · · ∧ e α p )( Y , · · · , Y p )= p ! p X m =0 ( − m e σ ( Y m )( ∇ e σ ω µν ; α α ··· α p ) e µ ⊗ e ν ⊗ ( e α ∧ · · · ∧ e α p ) ! · ( Y , · · · , b Y m , · · · Y p )= p X m =0 ( − m e σ ( Y m )( ∇ e σ ω )( Y , · · · , b Y m , · · · Y p )= p X m =0 ( − m ( ∇ Y m ω )( Y , · · · , b Y m , · · · Y p ) . (cid:3) Let D ∗ be the formal L -adjoint of D . If ω ∈ Λ p T ∗ X , we define the divergence of ω ,div ω ∈ Λ p − T ∗ X , as follows:div ω ( X , X ; Y , · · · , Y p − ) := n X m =1 ( ∇ e m ω )( X , X ; e m , Y , · · · , Y p − ) . In local coordinates, that is(div ω ) µν ; α α ··· α p − = g γδ ∇ γ ω µν ; δα ··· α p − . Proposition 2.8. On Λ p T ∗ X , D ∗ = − div . Proof.
Select arbitrary θ ∈ Λ p T ∗ X and ω ∈ Λ p − T ∗ X with compact support. Then itfollows from Proposition 2.7 that Z X n h θ, Dω i dV g = 1 p ! Z X n θ µν ; α ··· α p − p − X m =0 ( − m ∇ e ασ ω µν ; α ··· b α m ··· α p − ! dV g = 1 p ! Z X n p − X m =0 θ µν ; α m α ··· b α m ··· α p − ∇ e km ω µν ; α ··· b α m ··· α p − dV g = 1( p − Z X n p − X m =0 θ µν ; α m α ··· b α m ··· α p − ∇ e αm ω µν ; α ··· b α m ··· α p − dV g = 1( p − Z X n p − X m =0 ( − div θ ) µν ; α ··· b α m ··· α p − ω µν ; α ··· b α m ··· α p − dV g = Z X n h− div θ, ω i dV g . (cid:3) We define the Hodge Laplacian on T ∗ X -valued p -forms Λ p T ∗ X as follows e △ := DD ∗ + D ∗ D, and the covariant Laplace operator on ω ∈ Λ p T ∗ X by △ ω = tr( ∇ ω ) , where the trace is taken with respect to the two indices of the Hessian. Proposition 2.9. If ω ∈ T ∗ X , then e △ ω = −△ ω . For a 1-form θ ∈ T ∗ X , we let θ ∨ : Λ p T ∗ X → Λ p − T ∗ X denote the adjoint of themap θ ∧ : Λ p − T ∗ X → Λ p T ∗ X with respect to g , so that h θ ∧ ω, η i = h ω, θ ∨ η i for ω ∈ Λ p T ∗ X and η ∈ Λ p +1 T ∗ X . In coordinates,( θ ∨ ω ) µν ; α ··· α p − = g γδ θ γ ω µν ; δα ··· α p − . For any ξ ∈ Λ p T ∗ X and any function u , we define H ( u ) ξ as H ( u ) ξ := ( ∇ e µ ,e ν u ) e µ ∧ ( e ν ∨ ξ ) . (2.3) Proposition 2.10.
Let ω ∈ Λ p T X and f be a function. We have1. D ( f ω ) = f Dω + df ∧ ω ; D ∗ ( f ω ) = f D ∗ ω − df ∨ ω ; D ∗ ( df ∧ ω ) = − ( △ f ) ω − ∇ ∇ f ω − df ∧ D ∗ ω + H ( f ) ω ; | df ∧ ω | + | df ∨ ω | = | df | | ω | . Proof.
1. According to the definition, D ( f ω ) = e µ ∧ ∇ e µ ( f ω )= e µ ∧ ( e µ ( f ) ω + f ∇ e µ ω )= f Dω + df ∧ ω ;2. In local coordinates, D ∗ ( f ω ) µν ; α ··· α p − = − div( f ω ) µν ; α ··· α p − = − g γδ ∇ γ ( f ω ) µν ; α ··· α p − = − g γδ ( f ∇ γ ω µν ; δα ··· α p − + ∇ γ f ω µν ; δα ··· α p − )= − f div ω − g γδ ∇ a f ω µν ; δα ··· α p − = f D ∗ ω µν ; α ··· α p − − ( df ∨ ω ) µν ; α ··· α p − ;3. ( df ∧ ω ) µν ; δα ··· α p = ( ∇ b f ) ω µν ; α ··· α p + p X m =1 ( − m ( ∇ α m f ) ω µν ; δα ··· d α m ··· α p ,D ∗ ( df ∧ ω ) µν ; α ··· α p = − div( df ∧ ω ) µν ; α ··· α p = − g γδ ∇ γ ( df ∧ ω ) µν ; δα ··· α p = − g γδ ( ∇ γ,δ f ω µν ; α ··· α p + ( ∇ δ f ) ∇ γ ω µν ; α ··· α p ) − p X m =1 g γδ (( − m ( ∇ α m ,γ f ) ω µν ; δα ··· d α m ··· α p +( − m ( ∇ α m f ) ∇ γ ω µν ; δα ··· d α m ··· α p )= (( −△ f ) ω − ∇ ∇ f ω + H ( f ) ω − df ∧ D ∗ ω ) µν ; α ··· α p . INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p | df ∧ ω | = 1( p + 1)! p X m =1 ( − m ( ∇ α m f ) ω µν ; α ··· d α m ··· α p ! p X s =1 ( − s ( ∇ α s f ) ω µν ; α ··· c α s ··· α p ! = 1( p + 1)! p X m =1 ( − m ( ∇ α m f ) ω µν ; α ··· d α m ··· α p p X s =1 ,s = m ( − s ( ∇ α s f ) ω µν ; α ··· c α s ··· α p +( ∇ α m f ) ω µν ; α ··· d α m ··· α p ( ∇ α m f ) ω µν ; α ··· d α m ··· α p (cid:17) = | df | | ω | + 1( p + 1)! p X m =1 ( − m ( ∇ α m f ) ω µν ; α ··· d α m ··· α p p X s =1 ,s = m ( − s ( ∇ α s f ) ω µν ; α ··· c α s ··· α p = | df | | ω | − p + 1)! p X m =1 p X s =1 ,s = m ( ∇ α m f ) ω µν ; α s α ··· d α m ··· c α s ··· α p ( ∇ α s f ) ω µν ; α m α ··· d α m ··· c α s ··· α p = | df | | ω | − | df ∨ ω | . (cid:3) The following lemma is taken from [19, Lemma 7.9]:
Lemma 2.11.
For any smooth compactly supported section ξ of Λ q T ∗ X , and any positive C function ϕ on X , the following integral formula holds ( ξ, e △ ξ ) ≥ Z X (cid:10) ξ, ( − ϕ − △ ϕ + 2 H (log ϕ ) ξ ) (cid:11) dV g . Here h· , ·i is the induced inner product of tensor bundles and ( · , · ) is R X n h· , ·i dV g . As in [19, Lemma 7.10] and [20, Lemma 2.2], we also have the following result:
Lemma 2.12.
Let ( X n , g ) be a complete non-compact Einstein manifold of dimension n ≥ . Then for every small ε > there exists a compact set K ( ε ) such that the followingestimate holds for any smooth section ξ of Λ T ∗ X compactly supported in X n \ K ( ε ) : ( ξ, e △ ξ ) ≥ (cid:20) ( n − − C ( n, ε ) (cid:21) Z X n | ξ | dV g . Proof.
We let { e µ } , 0 ≤ µ ≤ n be a local orthonormal coframe of g such that e = dρ .This implies that { e i } , 1 ≤ i ≤ n − ρ . For convenience, we also denote g = dρ + g ij ( ρ, θ ) dx i dx j . We denote ε ′ := ε n − . We set ϕ ( x ) = e − n − ρ . Arguing as in the proof of Lemma 2.4, we get that if ρ is large enough, | S − δ | ≤ ε ′ , on X \ D ρ . Restricting ourselves to X \ D ρ , this implies that − ϕ − △ ϕ = − ( n − n − △ ρ ≥ − ( n − n − n − − ( n − ε = ( n − n + 3)4 − ( n − ε ′ , (2.4)and ∇ i,j log ϕ = − n − ∇ i,j ρ ≥ − n − g ij − ε ′ g ij , ∇ ,j log ϕ = 0 , ∇ , (log ϕ ) = 0 . From these estimates, we get that, for any ξ which is compactly supported in X \ D ρ , h H (log ϕ ) ξ, ξ i = (cid:10) ∇ µ,ν (log ϕ ) e µ ∧ ( e ν ∨ ξ ) , ξ (cid:11) = 2 ∇ µ,ν (log ϕ ) h e µ ∨ ξ, e ν ∨ ξ i = 2 ∇ e i ,e j (log ϕ ) (cid:10) e i ∨ ξ, e j ∨ ξ (cid:11) ≥ − (cid:18) n −
52 + n − ε ′ (cid:19) δ ij (cid:10) e i ∨ ξ, e j ∨ ξ (cid:11) ≥ − (2( n −
5) + ( n − ε ) h ξ, ξ i . (2.5)Here we have used the fact that n − X i =1 | e i ∨ ξ | ≤ n − X µ =0 | e µ ∨ ξ | = 2 | ξ | , for ξ ∈ Λ T X . Combining equation (2.5) and (2.4) and Lemma 2.11, we have (cid:16) ξ, e △ ξ (cid:17) ≥ Z X n (cid:18) ( n − n + 3)4 − n − − (3 n − ε ′ (cid:19) h ξ, ξ i dV g ≥ (cid:18) ( n − − ε (cid:19) Z X n | ξ | dV g . This proves the lemma with K ( ε ) = D ρ . (cid:3) Note that a (0,4)-tensor ω such that ω ( · , · ; Y , Y ) = − ω ( · , · ; Y , Y ) for any Y , Y ∈ T X ,can be considered as a Λ X -valued 2-form, i.e. ω ∈ Λ ( X, Λ X ). In the remaining of thissection we will consider such (0,4)-tensors. The following lemma gives a Weitzenb¨ockformula relating the covariant Laplacian on such tensors to e △ : Lemma 2.13.
For a section ω of Λ ( X, Λ X ) , INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p e △ ω αβγδ = −△ ω αβγδ + Ric νγ ω αβνδ − Ric νδ ω αβνγ − Rm ν µδ γ ω αβµν + Rm ν µαγ ω νβµδ + Rm ν µβγ ω ανµδ − Rm νµγ δ ω αβµν + Rm ν µα δ ω νβµγ + Rm ν µβ δ ω ανµγ . Proof.
Note that the last two indices of ω are considered to be the 2-form indices. ByProposition 2.8, Proposition 2.7 and some direct computations, we get( D ∗ Dω ) αβγδ = −∇ µ ( Dω ) αβµγδ = −∇ µ ∇ µ ω αβγδ + ∇ µ ∇ γ ω αβµδ − ∇ µ ∇ δ ω αβµγ ;( DD ∗ ω ) αβγδ = ( D ∗ ω αβδ ) γ − ( D ∗ ω αβγ ) δ = −∇ γ ∇ µ ω αβµδ + ∇ δ ∇ µ ω αβµγ . Then, applying the Ricci identity, ∇ δ ∇ γ ω α ··· α − ∇ γ ∇ δ ω α ··· α = X s =1 ω α ··· α s − να s +1 ··· α Rm να s γδ , we finally get e △ ω αβγδ = ( DD ∗ + D ∗ D ) ω αβγδ = −∇ µ ∇ µ ω αβγσ + ∇ µ ∇ γ ω αβµσ − ∇ γ ∇ µ ω αβµσ , µ + ∇ σ ∇ µ ω αβµγ − ∇ µ ∇ σ ω αβµγ = −△ ω αβγσ + Ric νγ ω αβνσ + Rm ν µδγ ω αβµν − Rm ν µα γ ω νβµσ − Rm ν µβ γ ω ανµσ − Ric νδ ω αβνγ + Rm ν µγ δ ω αβµν − Rm νµα δ ω νβµγ − Rm νµβ δ ω ανµγ . (cid:3) Definition 2.14.
We say that a 4-tensor ω belongs to e Σ if it satisfies the following threeassumptions:1. ω αβγδ = − ω βαγδ ,2. ω αβγδ + ω αγδβ + ω αδβγ = 0 , ω αβγδ = ω γδαβ . Furthermore, if ω is trace-free, meaning that g ik ω ijkl = 0, we say that ω ∈ e Σ .Note that any element of e Σ belongs to Λ ( X, Λ X ). Combining Lemmas 2.12 and2.13, we obtain the following estimate: Lemma 2.15.
Let ( X n , g ) be an n -dimensional Einstein manifold containing an essentialset D with n > . Then for every ε > there exists a compact set K ( ε ) ⊃ D suchthat the following estimate holds for any smooth -tensor ω ∈ e Σ compactly supported in X n \ K ( ε ) : Z X n |∇ ω | dV g ≥ (cid:18) ( n − − C ( n, ε ) (cid:19) Z X n | ω | dV g . Proof.
From Lemma 2.3, there exists a compact set K ( ε ) ⊃ D such that k Rm − K k L ∞ ( X n \ K ( ε )) = k W k L ∞ ( X n \ K ( ε )) ≤ ε. By a direct computation, we have
Ric νγ ω αβνδ − Ric νδ ω αβνγ = − n − ω αβγδ ; − Rm ν µδ γ ω αβµν + Rm ν µαγ ω νβµδ + Rm ν µβγ ω ανµδ = − ω αβδγ − ω γβαδ − ω αγβδ + O ( εω )= O ( εω ); − Rm νµγ δ ω αβµν + Rm ν µα δ ω νβµγ + Rm ν µβ δ ω ανµγ = ω αβγδ + ω δβαγ + ω αδβγ + O ( εω )= O ( εω ) . Using Lemma 2.13 together with Lemma 2.12, we get: Z X n |∇ ω | dV g = ( ω, −△ ω )= ( ω, e △ ω αβγδ − Ric νγ ω αβνδ + Ric νδ ω αβνγ + Rm ν µδ γ ω αβµν − Rm ν µαγ ω νβµδ − Rm ν µβγ ω ανµδ + Rm νµγ δ ω αβµν − Rm ν µα δ ω νβµγ − Rm ν µβ δ ω ανµγ ) ≥ ( n − Z X n | ω | dV g + 2( n − Z X n | ω | dV g − C ( n, ε ) Z X n | ω | dV g ≥ (cid:18) ( n − − C ( n, ε ) (cid:19) Z X n | ω | dV g . (cid:3) Remark . By a density argument, it is not difficult to see that Lemmas 2.5 and 2.15 arestill true if we replace the condition that u or ω has compact support by u ∈ W , ( X n \ K )(resp. ω ∈ W , ( X n \ K )). Here the subscript 0 means that u (resp. ω ) has vanishingtrace on ∂K (resp. ∂K ).Our next goal is to make use of the above estimates to get weighted L -estimate for theWeyl tensor. More precisely, we have: Proposition 2.17.
Suppose that ( X n , g ) , n ≥ , is a complete noncompact Einsteinmanifold with an essential set D . If k W k L p ( X n ,g ) < ∞ , with < p < n − , then k W k L ( X n ,g ) < ∞ . Furthermore if n ≥ , we have k e a ρ W k W , ( X n ,g ) < ∞ for any a ∈ [0; n − . Before giving the proof of this proposition, we need to make a preliminary definition.Formula (1.4) together with Equation (1.3) define a quadratic map from e Σ to itself. Wedefine the associated symmetric bilinear map as follows: Q ( ξ, ω ) αβγδ := ξ µ να β ω µγνδ + ξ µ να γ ω µβνδ − ξ µ νβ α ω µγνδ − ξ µ νβ δ ω µανγ . This map enjoys the following nice property:
Claim . For every ω, ξ ∈ e Σ , we have hQ ( ω, W) , ξ i = h ω, Q ( ξ, W) i . Equivalently, the map ω
7→ Q (W , ω ) is symmetric. INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p Proof.
The proof is a straightforward calculation: hQ ( ω, W) , ξ i = (cid:16) ω µ να β W µγνδ + ω µ να γ W µβνδ − ω µ νβ α W µγνδ − ω µ νβ δ W µανγ (cid:17) ξ αβγδ = 2 (cid:16) ω µ να β W µγνδ + ω µ να γ W µβνδ (cid:17) ξ αβγδ = 2 ω µ να β W µγνδ (cid:0) ξ αβγδ + ξ αγβδ (cid:1) = − ω µ να β W µγνδ (cid:0) ξ αγδβ + ξ αδβγ (cid:1) = 4 ω αµβν ξ αγβδ W µ νγ δ − ω αµβν ξ αδβγ W µ νγ δ . where we used the first Bianchi identity, Property 2 of Definition 2.14, to get the fourthline. Under this form, the claim becomes clear by swapping γ (resp. δ ) and µ (resp. ν ). (cid:3) Having made this definition, we can give a proof of Proposition 2.17:
Proof of Proposition 2.17.
We remark that if n < p < n − ≤
2. Hence, from thefact that W ∈ L ∞ , which was proven in Lemma 2.3, we conclude that W ∈ L . As aconsequence, we now restrict our attention to the case n ≥
6. We also assume that p > b ∈ R and using H¨older inequality, we get Z X n e − bρ | W | g dV g ≤ (cid:18)Z X n | W | pg dV g (cid:19) p (cid:18)Z X n e − bpp − ρ dV g (cid:19) p − p . (2.6)Note that the second integral appearing in the right-hand side can be rewritten as follows: (cid:18)Z X n e − bpp − ρ dV g (cid:19) p = Vol( D ) + Z ∞ e − bpp − ρ | Σ ρ | dρ, (2.7)where | Σ ρ | denotes the area of Σ ρ . Using Lemma 2.4, we get: ddρ | Σ ρ | = Z Σ ρ HdV g = ( n − o (1)) | Σ ρ | . Integrating this differential estimate, we obtain | Σ ρ | = | Σ | e ( n − ρ + o ( ρ ) . In particular, Integral (2.7) converges if and only if n − − bpp − <
0, that is to say b > n − − n − p . From Equation (2.6), we conclude that Z X n e − bρ | W | g dV g < ∞ (2.8)for any b > n − − n − p .We now select ε > χ which vanishes on K = K ( ε ) and which equals one outside a larger compact subset K ′ ⊃ K ( ε ). We set f W := χW . We remark that f W satisfies the following equation: △ f W + 2( n − f W + 2 Q (W , f W ) = θ, (2.9)where θ is a tensor belonging to e Σ and whose support is contained in supp( ∇ χ ) ⊂ K ′ \ K . For any compactly supported Lipschitz function f , we have Z X (cid:12)(cid:12)(cid:12) ∇ ( f f W ) (cid:12)(cid:12)(cid:12) dV g = Z X f (cid:12)(cid:12)(cid:12) ∇ f W (cid:12)(cid:12)(cid:12) dV g + 2 Z X f D ∇ f ⊗ f W , ∇ f W E dV g + Z X |∇ f | (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g = Z X D ∇ ( f f W ) , f W E dV g + Z X |∇ f | (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g = − Z X f Df W , △ f W E dV g + Z X |∇ f | (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g = 2( n − Z X f (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g + 2 Z X f Df W , Q (W , f W ) E dV g − Z X f Df W , θ E dV g + Z X |∇ f | (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g . Since f f W is compactly supported in X \ K , we conclude from Lemma 2.15 that (cid:20) ( n − − ε (cid:21) Z X f (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g ≤ n − Z X f (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g + 2 Z X f Df W , Q (W , f W ) E dV g − Z X f Df W , θ E dV g + Z X |∇ f | (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g . From the fact that | W | < ε on X n \ K , we conclude that (cid:20) ( n − − C ( n, ε ) (cid:21) Z X f (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g ≤ − Z X f Df W , θ E dV g + Z X |∇ f | (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g . (2.10)By a simple density argument using Inequality (2.8), it can be shown that Estimate(2.10) still holds for any function f such that f, |∇ f | = O ( e − bρ ) for some b > n − − n − p .We choose f = f R ( ρ ) where f R is a 1-parameter family of functions defined as follows: f R ( ρ ) := (cid:26) e aρ if ρ ≤ R,e aR − b ( ρ − R ) if ρ ≥ R. It is easy to see that these functions are Lipschitz continuous and satisfy f, |∇ f | = O ( e − bρ ). From the fact that |∇ f | = a f if ρ < R and |∇ f | = b f if ρ > R , we finallyget: (cid:20) ( n − − a − C ( n, ε ) (cid:21) Z D R f (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g + (cid:20) ( n − − b − C ( n, ε ) (cid:21) Z X \ D R f (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g ≤ − Z X f Df W , θ E dV g . (2.11)Choosing b < n − , which is possible since n − − n − p < n − , and ε so small that ( n − − b − C ( n, ε ) ≥
0, we finally get (cid:20) ( n − − a − C ( n, ε ) (cid:21) Z D R e aρ (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g ≤ − Z X e aρ Df W , θ E dV g . (2.12) INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p Letting R tend to infinity, and upon reducing the value of ε so that ( n − − a − C ( n, ε ) >
0, we finally get Z D R e aρ (cid:12)(cid:12)(cid:12)f W (cid:12)(cid:12)(cid:12) dV g < ∞ . This ends the proof of Proposition 2.17. (cid:3) Pointwise estimate for the Weyl tensor
In this section, we assume that ( X n , g ) is an AHE manifold with Weyl tensor satisfying k W k L p ( X,g ) < ∞ for some p ≤ n − . Note that on Einstein manifolds we always haveW = Rm − K . The main purpose of this section is to give a pointwise decay estimatefor W. We achieve this by two steps: first, we get the estimate by assuming ( X n , g ) is a C ,µ -conformally compact Einstein manifold. Obviously, even in this case the result hasits own interests. Later, we remove the condition of C ,µ -regularity and try to obtain thepointwise estimate of | W | in more general situations. Unfortunately, due to some technicalreasons mentioned in the introduction, we have to assume n ≥ Definition 3.1.
We say that (
X, g ) is a C k,α -conformally compact manifold if • there exists a smooth manifold X with boundary ∂X whose interior is X : X = X \ ∂X • and for some defining function x , g = x g extends to a C k,α metric on X ,where a defining function x is a smooth function x : X → bR + such that x − (0) = ∂X with dx = 0 at every point of ∂X .Furthermore, assuming that sec g → − x satisfies | dx | g ≡ ∂X . C k,α -conformally compact manifolds whose curvature tends to − asymptotically hyperbolic . We refer the reader to [19] and references therein for moredetails on these manifolds.In order to get the pointwise decay of W which is mentioned above, we need the followinglemma, which was observed in [24]. Lemma 3.2.
Suppose that ( X n , g ) is a conformally compact Einstein manifold of regu-larity C . If its conformal infinity is conformally flat, then | W | = O ( r n +1 ) where r is the defining function determined by some conformal infinity. Here is the outline of the proof of the above lemma. We refer the reader to [24] fordetails. Straightforward calculations yield that if an Einstein metric g is at least C conformally compact, then the sectional curvature in X satisfiessec g = − O ( r ) . (3.1)The most basic and important fact about asymptotically hyperbolic manifolds is that aconformal infinity ( ∂X, g ) determines a unique defining function r in a collar neighborhoodof ∂X such that g = r − ( dr + g r ) , where g r is an r -dependent family of metrics on ∂X with g r | r =0 = g . See e.g. [18]. Itfollows from the work of Fefferman and Graham [10] that the Einstein equation impliesthe following asymptotic expansion for the metric g . For n even, g r = g + g (2) r + (even powers) + g ( n − r n − + g ( n − r n − + ..., where the g ( j ) are tensors on ∂X and g ( n − is trace-free with respect to g . The tensors g ( j ) for j ≤ n − g , but g ( n − is formallyundetermined. For n odd the analogous expansion is g r = g + g (2) r + (even powers) + kr n − log r + g ( n − r n − + ..., where the g ( j ) ’s are locally determined for j ≤ n − k is locally determined and trace-free,but g ( n − is formally undetermined.Due to Theorem A in [9], we know that if ( X n , g ) is a conformally compact of regularity C and its conformal infinity is smooth, then in fact ( X n , g ) is conformally compact oforder C ∞ if n is even or if n is odd and k ≡
0, where k is a conformally covariant tensor.Therefore according Fefferman-Graham expansion, we can get | W | g = O ( r n +1 ) if theconformal infinity is locally conformally flat. Theorem 3.3.
Suppose that ( X n , g ) is a conformally compact Einstein manifold of di-mension n ≥ and of regularity C ,µ for some µ ∈ (0; 1) . If we further suppose that thereexists p ∈ (cid:0) n − (cid:3) such that k W k L p ( X n ,g ) < ∞ , then | W | = O ( r n +1 ) , where r is some special defining function.Proof. Let r : X → R be an arbitrary defining function for the conformal infinity ∂X of X and let g = ρ g be the compactified metric. We denote with a bar quantities associatedto the metric g , e.g. W denotes its Weyl tensor. By assumption, g is a C ,µ metric on X = X ∪ ∂X .We first note that | W | g = ρ (cid:12)(cid:12) W (cid:12)(cid:12) g . As a consequence, Z X | W | pg dV g = Z X ρ p − n (cid:12)(cid:12) W (cid:12)(cid:12) pg dV g . Since g is C ,µ , W is a continuous 4-tensor. From the fact that p ≤ n − , the functionof ρ appearing in the integral on the right-hand side blows up faster than ρ − whenapproaching ∂X . As a consequence, if this quantity is to be finite, this impose that W ≡ ∂X .From the Fefferman-Graham expansion of the metric g , we immediately see that thesecond fundamental form of ∂X in the manifold X vanishes.If b g denotes the metric induced on the conformal infinity ∂X , it follows from the Gauss-Codazzi equations that the Riemann tensor d Rm of b g is equal to the restriction of Rm to T ( ∂X ). We denote P and b P the Schouten tensors of the metrics g and b g . From thedecomposition of the Riemann tensors ( d Rm = b g ? b P + c W , Rm = g ? P + Wit follows that c W = W + (cid:16) P − b P (cid:17) ? b g, INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p Since W ≡ ∂X , we conclude that c W = (cid:16) P − b P (cid:17) ? b g, which implies c W ≡ e Σ ( ∂X ).As a consequence, we have proven that ∂X is locally conformally flat. The theoremfollows from Lemma 3.2. (cid:3) Finally we remove the condition C ,µ -regularity to give the pointwise estimate of | W | .According to Proposition 2.17 we get weighted L -estimate for the Weyl tensor. UsingLemma 2.5, we are able to show the following theorem: Theorem 3.4.
Suppose that ( X n , g ) , n ≥ is a complete noncompact Einstein manifoldwith an essential set D . If k W k L p ( X n ,g ) < ∞ for some p ∈ (cid:2) n − (cid:1) , then | W | ≤ Ce − ( n +1) ρ . An essential element in the proof of this theorem is [16, Theorem 1.2] which we recallhere for the sake of completeness:
Proposition 3.5.
Suppose that ( X n , g ) is a complete Riemannian manifold with an es-sential set D . If the Riemann tensor satisfies the following assumptions: | Rm − K | = O ( e − aρ ) , |∇ Rm | = O ( e − aρ ) for some constant a > , then there is a smooth closed manifold ∂X and a smooth structureon X = X ∪ ∂X , such that setting x = e − ρ and extending it by zero on ∂X , x is a definingfunction for ∂X and the metric g = x g extends to a C ,µ metric on the manifold X forsome µ ∈ (0; 1) . That is to say ( X, g ) is C ,µ -conformally compact.Proof of Theorem 3.4. We will assume that n ≥ n = 7.The first step is to obtain an exponential (pointwise) decay of | W | at infinity. To this end,we set W := e aρ W for some a > satisfies △ W = ( △ e aρ )W + 2 h∇ e aρ , ∇ W i + e aρ △ W= (cid:2) a + ( n − a + o (1) (cid:3) W + 2 ae aρ ∇ ∇ ρ W + e aρ △ W= (cid:2) a + ( n − a −
2) + o (1) (cid:3) W + 2 ae aρ ∇ ∇ ρ W − Q (W , W ) , (3.2)where we used △ ρ = H = n − o (1) (see Lemma 2.4). Next, we compute △| W | intwo different ways at any point where | W | 6 = 0: △| W | = 2 (cid:16) |∇| W || + | W |△| W | (cid:17) = 2 (cid:16) |∇ W | + h W , △ W i (cid:17) = 2 (cid:16) |∇ W | + (cid:2) a + ( n − a −
2) + o (1) (cid:3) | W | +2 ae aρ h∇ ∇ ρ W , W i − h W , Q (W , W ) i ) . As a consequence, we get the following equation for | W | : | W |△| W |− (cid:2) a + ( n − a −
2) + o (1) (cid:3) | W | = |∇ W | −|∇| W || +2 ae aρ h∇ ∇ ρ W , W i , (3.3) where we used Lemma 2.3 to get h W , Q (W , W ) i = o (cid:0) | W | (cid:1) .The following refined Kato inequality holds for the Weyl tensor of any Einstein manifold(see e.g. [7]): |∇| W || ≤ n − n + 1 |∇ W | . (3.4)We are going to take advantage of it to estimate the right-hand side of Equation (3.3).We first remark that |∇ W | = e aρ |∇ W | + 2 ae aρ h∇ ∇ ρ W , W i + a e aρ | W | , |∇| W || = e aρ |∇| W || + 2 ae aρ h∇ ∇ ρ | W | , | W |i + a e aρ | W | = e aρ |∇| W || + ae aρ ∇ ∇ ρ | W | + a e aρ | W | = e aρ |∇| W || + 2 ae aρ h∇ ∇ ρ W , W i + a e aρ | W | . Therefore, using Inequality (3.4), we get: |∇ W | − |∇| W || ≥ n − e aρ |∇ W | (3.5)Next, using Young’s inequality, we remark that2 ae aρ h∇ ∇ ρ W , W i ≥ − n − e aρ |∇ W | − n − a e aρ | W | . Thus Equation (3.3) yields the following differential inequality: | W |△| W | − (cid:2) a + ( n − a −
2) + o (1) (cid:3) | W | ≥ − n − a | W | . (3.6)We select a = n − n − . The previous inequality becomes △| W | ≥ (cid:20) −
12 ( n − n − n − o (1) (cid:21) | W | , (3.7)at any point where | W | >
0. Note that when n > a < n − so from Proposition 2.17, W ∈ L . We claim that | W | ≤ Ce − n − ρ . Indeed, set b := n − + δ for some small δ > (cid:18) △ + 12 ( n − n − n − (cid:19) e − bρ = (cid:18) δ −
14 ( n − n − n − o (1) (cid:19) e − bρ . Select ε > ε <
14 ( n − n − n − . Provided that δ < ε , there exists a compact set K ⊃ D such that △| W | ≥ − (cid:20)
12 ( n − n − n − ε (cid:21) | W | , △ e − bρ ≤ (cid:20)
12 ( n − n − n − ε (cid:21) e − bρ , and such that for any W , -function ϕ supported in K , the following L -estimate holds(Lemma 2.5): Z X n |∇ ϕ | dV g ≥ (cid:20) ( n − − ε (cid:21) Z ϕ dV g . INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p We set ψ = | W | − Ce − bρ where C is chosen so large that ψ < K . Then ψ satisfies △ ψ ≥ − (cid:20)
12 ( n − n − n − ε (cid:21) ψ. (3.8)We also define ψ + = max { ψ, } and note that ψ + ∈ W , and supp ψ + ⊂ X \ K . FromInequality (3.8), we get (cid:20) ( n − − ε (cid:21) Z X | ψ + | dV g ≤ Z X n |∇ ψ + | dV g ≤ − Z X n ψ + △ ψdV g ≤ (cid:20)
12 ( n − n − n − ε (cid:21) Z X n ( ψ + ) dV g , (cid:20)
14 ( n − n − n − − ε (cid:21) Z X n ( ψ + ) dV g ≤ . From our assumption on ε , this immediately implies that ψ + ≡
0, that is to say | W | ≤ Ce − bρ , or equivalently, | W | ≤ Ce − (cid:18)
12 ( n − n − + δ (cid:19) ρ . Since n ≥
12 ( n − n − >
2. In particular, from Proposition 3.5, we conclude that themanifold ( X n , g ) is C ,µ -conformally compact for some µ ∈ (0; 1). So it falls into theassumptions of Theorem 3.3. This concludes the proof of Theorem 3.4 for n ≥ n = 7, then n − n − = > n − so we can no longer apply Proposition 2.17 for thisvalue of a . Instead we choose a = n − − . Inequality (3.6) becomes △| W | ≥ (cid:20) o (1) (cid:21) | W | . Setting b := 3 + δ , it can be checked that e − bρ satisfies △ e − bρ ≤ (cid:20) o (1) (cid:21) e − bρ , outside some compact subset. We can then rephrase the previous proof, the only point tonote is that ( n − > . This is what allows the use of the asymptotic L -estimate (Lemma 2.5). (cid:3) Applications
Together with Theorem 1.2, the rigidity result [16, Theorem 1.6] implies
Theorem 4.1.
Suppose that ( X n , g ) , n ≥ is a complete noncompact Einstein manifoldwith an essential set D and that X n is simply connected at infinity. If we further assume k W k L p ( X n ,g ) < ∞ for some p satisfying < p < n − , then ( X n , g ) is isometric to H n . Proof.
By Theorem 3.4, we know that there exits a constant
C > | W | ≤ Ce − ( n +1) ρ . On the other hand, by a direct refinenemt of the proof of Lemma 2.3, we also have |∇ W | ≤ C ′ e − ( n +1) ρ for some constant C ′ >
0. See also [2, Theorem 4.3]. The theorem then follows from [16,Theorem 5.1]. (cid:3)
As another application, we consider a similar question for static vacuum spacetimes. Werecall that an ( n + 1)-dimensional static spacetime ( N n +1 , g ) is a solution of the vacuumEinstein equations Ric − Scal2 g + Λ g = 0of the form N n +1 = R × M n ,g = − V dt + h where ( M n , h ) is a Riemannian manifold, V is a positive function on M n and Λ is theso-called cosmological constant which we choose equal to − n ( n − . The vacuum Einsteinequations can be written in terms of h and V asRic h + nh = Hess( V ) V , (4.1)and △ h V = nV. (4.2)Computing the trace of these two equations, we see that h has constant scalar curvatureScal = − n ( n − M n , h, V ) a static vacuum. We setE := Rm h − K , T := Ric h + ( n − h = Hess( V ) V − h. As another application of Theorem 1.2, we state the following theorem:
Theorem 4.2.
Suppose given ( M n , h, V ) a static vacuum with n ≥ such that ( M, h ) has an essential set D and R M n V | E | ph dV h < ∞ , for some p ∈ (cid:0) n (cid:1) . If we further assumethat h dV, dρ i > outside of D , then there exists a constant C > such that | E | ≤ Ce − ( n +2) ρ and (cid:12)(cid:12)(cid:12)(cid:12) Hess( V ) V − h (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − ( n +2) ρ , where ρ is the distance to the essential set D .Remark . The assumption h dV, dρ i > D is natural and reasonable. Indeed,it is expected that V grows as e ρ at infinity. More precisely, V is expected to have thefollowing expansion at infinity: V = ve ρ + O (1), where v ( x ) = v ( x i ) is a positive C function depending only on the projection of a point on the boundary of the essential set.The assumption h dV, dρ i > INSTEIN MANIFOLDS WITH WEYL CURVATURE IN L p of the fact that V = ve ρ is a nice approximate solution of the equation △ V = nV . Werefer the reader to [17] and [13] for more details.Notice that for static vacuum ( M n , h, V ), the Riemannian metric g = V dθ + h is anEinstein metric on S × M . Hence we consider the Einstein manifold ( S × M, V dθ + h ).For convenience, in the following, the index 0 refers to the direction ∂ θ . Latin indices takevalues 1 to n and refer to coordinates on M .In order to prove Theorem 4.2, we need the following two lemmas: Lemma 4.4.
Let ( M n , h, V ) be a static vacuum, if R M V | E | ph dV h < ∞ , then Z S × M | W g | pg dV g < ∞ . Proof.
By a direct computation, using Equations (4.1) and (4.2), we get W ijkl ( g ) = E ijkl ,W jkl ( g ) = 0 ,W j l ( g ) = − V ( V − ∇ h V − h ) = − V T, and | W( g ) | g = | E | h + 4 | T | h . Note that T ik = h jl E ijkl , thus there is a constant C = C ( n ) such that | E | h ≤ | W g | g ≤ C | E | h . Therefore the assumption Z M n V | E | ph dV h < ∞ is equivalent to Z S × M n | W g | pg dV g < ∞ . (cid:3) Lemma 4.5.
Let ( M n , h, V ) be a static vacuum. If ( M n , h ) has an essential set D , ρ isthe distance to D and h dρ, dV i > outside D , then the manifold ( S × M n , g ) admits anessential set.Proof. From [11, Lemma 2.5.11], the existence of an essential set is a consequence of thefollowing two facts:1. sec g < K ⊂ S × M ,2. there exists a proper smooth function f whose Hessian is positive definite outside K .We note that the assumption h dρ, dV i > D implies in particular that inf V =inf D V ( V grows along the gradient lines of ρ ). This implies that the metric g has injectivityradius bounded from below. Then, mimicking the proof of Lemma 2.3 and using Lemma4.4, we get that | W g | → Next we extend ρ to S × M by making it constant along the circles S . The Hessianof ρ can be computed explicitely: ∇ ( g ) ij ρ = ∇ ( h ) ij ρ, ∇ ( g )0 i ρ = 0 , ∇ ( g )00 ρ = V h dρ, dV i . It is then straightforward to see from the assumptions that Hess g ( ρ ) is positive definiteoutside D . This proves the lemma. (cid:3) Theorem 4.2 is then a consequence of Theorem 3.4 applied to the metric g = V dθ + h .If we further assume that M is spin, then we fall into the assumptions of [25, Theorem1] (See also Theorem 1.2 in [22]) so we get the following theorem: Theorem 4.6.
Suppose that ( M n , h, V ) is a static vacuum with n ≥ . Assume furtherthat1. M is spin,2. ( M, h ) has an essential set D ,3. h dρ, dV i > outside D ,4. and R M n V | E | ph dV h < ∞ for some p ∈ (cid:0) n (cid:1) then ( M n , h ) is the hyperbolic space and V = cosh( r ) , where r is the distance function to acertain point x ∈ M . Equivalently, the spacetime ( R × M, − V dt + h ) is the anti-deSitterspace. References
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Romain Gicquaud, Laboratoire de Math´ematiques et de Physique Th´eorique, UFR Scienceset Technologie, Universit´e Franc¸ois Rabelais, Parc de Grandmont, 37300 Tours, France
E-mail address : [email protected] Dandan Ji, Key Laboratory of Pure and Applied mathematics, School of Mathematics Sci-ence, Peking University, Beijing, 100871, P.R. China.
E-mail address : [email protected] Yuguang Shi, Key Laboratory of Pure and Applied mathematics, School of MathematicsScience, Peking University, Beijing, 100871, P.R. China.
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