A spinorial proof of the rigidity of the Riemannian Schwarzschild manifold
aa r X i v : . [ m a t h . DG ] S e p A SPINORIAL PROOF OF THE RIGIDITY OF THERIEMANNIAN SCHWARZSCHILD MANIFOLD
SIMON RAULOT
Abstract.
We revisit and generalize a recent result of Cederbaum[C2, C3] concerning the rigidity of the Schwarzschild manifold for spinmanifolds. This includes the classical black hole uniqueness theorems[BM, GIS, Hw] as well as the more recent uniqueness theorems for pho-ton spheres [C1, CG1, CG2]. Introduction
An ( n +1)-dimensional vacuum spacetime is a Lorentzian manifold ( L n +1 , g )satisfying the Einstein field equations Ric = 0, where
Ric is the Ricci tensorof the metric g . The vacuum is said to be static when L n +1 = R × M n , g = − N dt + g, where ( M n , g ) is an n -dimensional connected smooth Riemannian manifold,that we will take to be orientable, standing for the unchanging slices ofconstant time and N ∈ C ∞ ( M n ) is a non-trivial smooth function on M n .To model the exterior of an isolated system, it seems physically natural torequire asymptotic flatness, that is, the Cauchy hypersurface M n is usuallytaken to be asymptotically flat. The vacuum Einstein field equations canbe translated into the following two conditions on ( M n , g ) and the lapsefunction N : ∇ N = N Ric, ∆ N = 0 , (1.1)where Ric , ∇ and ∆ are respectively the Ricci tensor, the covariant deriva-tive and the Laplace operator of the Riemannian manifold ( M n , g ). Takingtraces in the first of these two equations and taking into account the secondone, we conclude immediately that the scalar curvature of ( M n , g ) is zero.It is usual to call the triple ( M n , g, N ) a static vacuum triple .The ( n +1)-dimensional Schwarzschild spacetimes [Sc, T] are a 1-parameterfamily of static, spherically symmetric and asymptotically flat solutions tothe vacuum Einstein field equations. For a parameter m ∈ R , it is given bythe static vacuum triple (cid:0) R n \ B r m (0) , g m , N m (cid:1) where the metric g m and the Date : September 22, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Einstein field equations, Schwarzschild spacetime, Rigidity,Spinors, Dirac operator. lapse function are g m = N − m dr + r g S , N m ( r ) = (cid:16) − mr n − (cid:17) / with g S denoting the standard metric on S n − and r m := (2 m ) / ( n − for m > r m := 0 for m ≤ m >
0, this spacetime represents the exterior of a black hole whoseevent horizon occurs at r = r m . A striking result due to Bunting andMasood-ul-Alam [BM] for n = 3 (generalizing the seminal work by Israel[I]) states that the Schwarzschild spacetime is in fact the only asymptoti-cally flat static vacuum spacetime with nondegenerate horizons. For n ≥ g canbe conformally deformed using the lapse function N to get two Riemannianmetrics, one which can be compactified by adding a point at infinity andanother one which is asymptotically flat with zero ADM mass. Then glu-ing these two manifolds along their boundaries gives an asymptotically flatRiemannian manifold with zero scalar curvature and with zero mass. Therigidity part of the positive mass theorem for non-smooth metrics [B, MS]then applies and allows to conclude that ( M n , g ) is conformally flat. For n = 3, this is enough to conclude while for n ≥ (cid:0) R n \ B r m (0) , g m , N m (cid:1) .More recently, new uniqueness theorems for photons spheres have beenstudied using this approach. A timelike hypersurface P in a static vac-uum spacetime ( L , g ) is called a photon sphere if it is a totally umbilicalhypersurface and if the associated lapse function is constant on each of itsconnected components. In the Schwarzschild spacetime, there is only onephoton sphere given by { r = 3 m } which models photons spiraling aroundthe central black hole “at a fixed distance”. In fact, as shown in [CG1],the Schwarzschild spacetime is the unique asymptotically flat static vacuumspacetime with photon sphere as an inner boundary. The main problem toapply the method of Bunting and Masood-ul-Alam in this situation comesmainly from the fact that the gluing hypersurfaces are not totally geodesic sothat the gluing process does not work directly. To overcome this difficulty,Cederbaum and Galloway begin by gluing in a C , fashion some piecesof Schwarzschild time-slices of well-chosen masses on each photon spheres.The resulting manifold has only totally geodesic inner boundary componentsand then the method in [BM] can be used. A generalization of this resultfor higher dimensional static vacuum triple has recently been addressed in[CG2].In [C2, C3], Cederbaum proves that both static vacuum black hole andphoton sphere uniqueness theorems can be deduced from a more generalrigidity result for the Riemannian time-slice of the Schwarzschild spacetime. IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 3
This statement deals with pseudo-static systems ( M n , g, N ) which generalizethe notion of static vacuum triple since they do not need to satisfy thefull set of the static equations (1.1). On the other hand, since the blackhole as well as the photon sphere boundary conditions arise from Lorentziangeometric considerations, they have to be translated into purely Riemannianassumptions. This is done in [C2, J] and this give rises to the notions of nondegenerate static horizons and quasilocal photon surfaces . With thesedefinitions, Cederbaum is able to prove the following general rigidity result: Theorem 1.
Let ( M n , g, N ) be an asymptotically isotropic pseudo-staticsystem of mass m with n ≥ . Assume M n has a compact inner bound-ary whose components are either nondegenerate static horizons or quasilocalphoton surfaces. Then m > , ( M n , g ) is isometric to a suitable piece of theSchwarzschild manifold of mass m and N coincides with N m . In this paper, we address another approach to this problem using spinors.Although we have to assume that the manifold is spin (which is automati-cally satisfied in the 3-dimensional case), we recover the full result of Ceder-baum and even allow to relax the quasilocal photon surface condition. Thisnew type of boundary condition will be referred to as a generalized quasilocalphoton surface . Moreover, our arguments also avoid all the gluing construc-tions which are the delicate part of her proof and our results also includethe black hole [BM, GIS, Hw] and the photon spheres uniqueness theorems[C1, CG1, CG2]. This is done by using a positive mass theorem for mani-folds with inner boundary due to Herzlich [He1, He2] which can be applied ifthe first eigenvalue of the boundary Dirac operator satisfies a certain lowerbound. As we shall see one can check that this lower bound is fulfilled usingboth the Friedrich inequality [F] and a generalization of an inequality ofHijazi-Montiel-Zhang [HMZ]. We then get:
Theorem 2.
Let ( M n , g, N ) be a spin asymptotically isotropic pseudo-staticsystem of mass m with n ≥ . Assume M n has a compact inner boundarywhose components are either nondegenerate static horizons or generalizedquasilocal photon surfaces. Then m > , ( M n , g ) is isometric to a suitablepiece of the Schwarzschild manifold of mass m and N coincides with N m . The idea to use the positive mass theorem of Herzlich for the black holeuniqueness problem was suggested by Walter Simon in [Si1] and I am verygrateful to him for allowing us to reproduce his unpublished alternative proofof the generalization of Israel’s theorem [I] by [MHRS, R] (see Appendix A).It is also a pleasure to thank him for his careful reading as well as forhis valuable comments of a previous version of this paper. We end thiswork by noticing that the use of the positive mass theorem can be droppedfrom the Simon’s approach in the context of 3-dimensional static vacuumtriples (see Appendix B.1). However, since it is essential to deal with thegeneral assumptions of Theorem 2, it is important for the author to includethis proof here. In Appendix B.2, we see that this method is suitable to
SIMON RAULOT get a new proof of the uniqueness of a connected photon surface in thissetting. Carla Cederbaum informed me that a work [CF] with her studentAxel Fehrenbach, in which they get similar results using arguments `a laRobinson [R], is in progress and I would like to thank her for this.Finally, it is a pleasure to thank Piotr Chru´sciel for his invitation to theseminar of the Gravitational Physics team of the University of Vienna aswell as for his hospitality and where this work began.2.
The setting
Here we consider a much broader class than the static vacuum triple,namely the pseudo-static system. Such a system is defined by a triple( M n , g, N ) where M n is an n -dimensional smooth manifold endowed witha smooth Riemannian metric g with nonnegative scalar curvature R and N is a nonnegative smooth harmonic function on M n . It is then immediatefrom (1.1) that a static vacuum triple is a pseudo-static system. In the restof this article, we will always assume that N > ∂M n of M n if it exists.On the other hand, we will use the following definition of asymptoticallyisotropic manifolds (see [CG1, J]). An n -dimensional smooth Riemannianmanifold ( M n , g ), n ≥
3, is asymptotically isotropic of mass m if the man-ifold M n is diffeomorphic to the union of a (possibly empty) compact setand an open end E n which is diffeomorphic to R n \ B , where B is an openball in R n , and if there exists a constant m ∈ R such that, with respect tothe coordinates ( y i ) induced by this diffeomorphism, we have g ij = ( g m ) ij + O (cid:0) s − n (cid:1) for i, j = 1 , ..., n on R n \ B as s := p ( y ) + ... + ( y n ) → ∞ . Here g m := (cid:16) m s n − (cid:17) n − δ denotes the spatial Schwarzschild metric in isotropic coordinates with δ theflat metric on R n . Moreover for such a manifold, a smooth function N : M n → R is called an asymptotic isotropic lapse of mass m if it satisfies N = e N m + O ( s − n )on R n \ B as s → ∞ with respect to the same diffeomorphism, coordinatesand mass m described above. Here, e N m denotes the Schwarzschild lapsefunction in isotropic coordinates, given by e N m ( s ) = 1 − m s n − m s n − . A triple ( M n , g, N ) is called an asymptotically isotropic system of mass m if ( M n , g ) is an asymptotically isotropic manifold of mass m and N is anasymptotic isotropic lapse of same mass m . IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 5
Remark 1. A -dimensional asymptotically flat static vacuum triple (in thesense of (A.1)) is automatically an asymptotically isotropic system from thework of Kennefick and ´O Murchadha [K ´OM] . The statement of Theorem 2 assumes boundary conditions which we nowmake more precise. As mentioned in the introduction, although they are hereexpressed only in terms of Riemannian geometry, these boundary conditionsarise from Lorentzian notions of horizons and photon spheres. We refer to[C2, J] where these characterizations are derived. Assume that M n hasa compact inner boundary ∂M n = ` ki =1 Σ i where Σ i denotes one of itsconnected components for 1 ≤ i ≤ k and let ν be the unit normal to ∂M n pointing toward infinity. A boundary component Σ i of ∂M n is said to bea static horizon if it is a totally geodesic component of the zero level set ofthe lapse N . It is called nondegenerate if its normal derivative is a nonzeroconstant. In the following, such a hypersurface will be referred to as a nondegenerate static horizon and in particular, it satisfies: H i = 0 , N i := N | Σ i = 0 , ν i ( N ) := ∂N∂ν (cid:12)(cid:12) Σ i > , (2.1)where H i is the mean curvature of Σ i in ( M n , g ). In our conventions, themean curvature of an ( n − n -dimensional Euclidean ball is n −
1. Onthe other hand, we will say that Σ i is a quasilocal photon surface if it istotally umbilical, if N i as well as H i are positive constants and if there exista constant c i > R/ i = n − n − c i H i (2.2)and 2 ν i ( N ) = n − n − (cid:16) c i − (cid:17) H i N i . (2.3)Here R/ i denotes the scalar curvature of Σ i with respect to the inducedmetric g/ i := g | Σ i which has to be constant because of (2.2). Note that theconstant c i differs from [C2, C3] by a multiplicative constant. Obviously, theintersection of the photon sphere { r = 3 m } with a time-slice { t = const. } in the Schwarzschild spacetime fits into this class of hypersurfaces. In thefollowing, instead of the assumption (2.2), we will only assume that R/ i ≥ n − n − c i H i . (2.4)In particular, the scalar curvature is not assumed to be constant on Σ i .Therefore a quasilocal photon surface for which (2.2) is relaxed to (2.4) willbe referred to as a generalized quasilocal photon surface . SIMON RAULOT The spinorial tools
In this section, we recall results from spin geometry which are neededto prove Theorem 2. For more details on this wide subject we refer to theclassical monographs [BHMM, LM] and the references therein.On a n -dimensional Riemannian spin manifold ( M n , g ) with boundary,there exists a smooth Hermitian vector bundle over M n called the spinorbundle which will be denoted by S . The sections of this bundle are calledspinors. Moreover, the tangent bundle T M acts on S by Clifford multipli-cation X ⊗ ψ γ ( X ) ψ for any tangent vector fields X and any spinor fields ψ . On the other hand, the Riemannian Levi-Civita connection ∇ lifts tothe so-called spin Levi-Civita connection (also denoted by ∇ ) and definesa metric connection on S that preserves the Clifford multiplication. TheDirac operator is then the first order elliptic differential operator acting onthe spinor bundle S given by D := γ ◦ ∇ . The spin structure on M n alsoinduces (via the unit normal field to ∂M n ) a spin structure on its boundary.This allows to define the extrinsic spinor bundle S / := S | ∂M n over ∂M n onwhich there exists a Clifford multiplication γ/ and a metric connection ∇ / .Similarly, the extrinsic Dirac operator is defined by taking the Clifford traceof the covariant derivative ∇ / that is D/ := γ/ ◦ ∇ / . From the spin structureon ∂M n , one can also construct an intrinsic spinor bundle for the inducedmetric g/ , denoted by S ∂ , and endowed with a Clifford multiplication γ ∂ anda spin Levi-Civita connection ∇ ∂ . Note that the (intrinsic) Dirac operatoron ( ∂M n , g/ ) is obviously defined by D ∂ = γ ∂ ◦ ∇ ∂ . In fact, we have anisomorphism (cid:0) S / , ∇ / , γ/ (cid:1) ≃ (cid:26) (cid:0) S ∂ , ∇ ∂ , γ ∂ (cid:1) if n is odd (cid:0) S ∂ , ∇ ∂ , γ ∂ (cid:1) ⊕ (cid:0) S ∂ , ∇ ∂ , − γ ∂ (cid:1) if n is evenso that the restriction of a spinor field on M n to ∂M n and the extensionof a spinor field on ∂M n to M n are well-defined. These identifications alsoimply in particular that the spectrum of the extrinsic Dirac operator is anintrinsic invariant of the boundary: it only depends on the spin and Rie-mannian structures of ∂M n and not on how it is embedded in M n . The firstnonnegative eigenvalue of the extrinsic Dirac operator, which correspondsto the lowest eigenvalue (in absolute value) of D ∂ , will be denoted by λ ( D/ ).3.1. Herzlich’s positive mass theorem for spin manifolds with bound-ary.
One of the main result needed in our approach is a sharp version of thepositive mass theorem for asymptotically flat spin manifold with boundarydue to Herzlich [He1, He2]. It is important to note that the spin assumptionis not only assumed just to adapt the Witten approach [W] to this setting.As we shall briefly recall below, the choice of the boundary condition underwhich the Dirac operator is studied is crucial to get rigidity. Note that apositive mass theorem for asymptotically flat manifolds with compact innerboundary has recently been obtained without the spin assumption by Hirschand Miao [HM].
IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 7
Recall that a Riemannian manifold ( M n , g ) is said to be asymptoticallyflat if the complement of some compact set is diffeomorphic to the com-plement of a ball in R n and the difference between the metric g and theEuclidean metric δ in this chart behaves like s − τ , its first derivatives like s − τ − and its second derivative like s − τ − where τ > ( n − /
2. If moreoverthe scalar curvature is integrable, its ADM mass, defined by m ADM ( g ) := 12( n − ω n − lim r →∞ Z S r (cid:16) div δ g − d (cid:0) tr δ g (cid:1)(cid:17) ( ν r ) , is a geometric invariant of ( M n , g ) by Bartnik [B] and Chru´sciel [Ch] inde-pendently. Here S r is a coordinate sphere of radius r with ν r as its unitnormal vector field pointing towards infinity, δ is the Euclidean metric and ω n − is the volume of the standard ( n − R n . Moreover, thevolume integral is with respect to the Euclidean metric and div δ (resp. tr δ )is the divergence (resp. the trace) with respect to this metric. Obviously,an asymptotically isotropic manifold of mass m as defined in Section 2 isan asymptotically flat manifold with ADM mass equals to m . The positivemass theorem asserts that if in addition to all the previous assumptions,the scalar curvature is nonnegative then the ADM mass is also nonnegativeand if it is zero, ( M n , g ) must be isometric to the Euclidean space. Thisresult was first proved by Schoen and Yau [SY1, SY2] for 3-dimensionalmanifolds and thereafter, they showed how there method can be used for di-mensions less than eight. In a recent preprint [SY3], the higher-dimensionalcases have been treated. On the other hand, Witten [W] discovered a proofwith a completely different method relying on spin geometry which we nowdiscuss.In the spin setting and if the manifold has a compact inner boundary, theproof of the positive mass theorem relies on the existence of ψ ∈ Γ( S ) (insome weighted Sobolev or H¨older spaces) such that Dψ = 0 and ψ → ψ where ψ ∈ Γ( S ) is constant near infinity and where the boundary conditionon ∂M n has to be well-chosen. Then, integrating by parts the famous Schr¨o-dinger-Lichnerowicz formula on large domains Ω r := { s ≤ r } with r > r → ∞ leads to12 ( n − ω n − m ADM ( g ) = Z M (cid:16) |∇ ψ | + R | ψ | (cid:17) − k X i =1 Z Σ i h D/ i ψ i + H i ψ i , ψ i i where D/ i is the restriction of D/ to S / i := S / | Σ i and ψ i = ψ | Σ i for all i ∈ { , ..., k } . Since the scalar curvature is assumed to be nonnegative,it turns out that the right-hand side of this expression is nonnegative if onecan ensure that each boundary term is nonpositive. This can be done byimposing the Atiyah-Patodi-Singer boundary condition on ψ | ∂M n as well asan additional assumption on the first eigenvalue of the boundary Dirac op-erator. We refer to the original papers of Herzlich (and to [BC2] for a more SIMON RAULOT detailed treatment of the analytic part) for a rigorous proof of this result. Astraightforward adaptation of these arguments allows to obtain the followingversion of Herzlich’s positive mass theorem (compare with [He1, Proposition2.1] and [He2, Proposition 2.1]):
Theorem 3.
Let ( M n , g ) be a n -dimensional Riemannian spin asymptoti-cally flat manifold with integrable scalar curvature R ≥ and with a compactinner boundary ∂M n := ` ki =1 Σ i such that λ ( D/ i ) ≥ H i > for all i = 1 , ..., k . Then the mass is nonnegative and if the mass is zero, ( M n , g ) is flat, the mean curvature H i is constant and (3.1) is an equalityfor all i = 1 , ..., k . Here λ ( D/ i ) denotes the first nonzero eigenvalue of the extrinsic Diracoperator on Σ i endowed with the metric g/ i := g | Σ i . It is important to pointout that this result is sharp since the exterior of round balls in Euclideanspace are flat manifolds with zero mass for which (3.1) is an equality.3.2. First eigenvalue of the Dirac operator.
Our approach (following[Si1], see Appendix A) consists to apply the previous positive mass theoremto a certain conformal deformation of the Riemannian manifold ( M n , g ).In particular, we have to check that the assumption (3.1) holds and so wehave to deal with estimates for the first eigenvalue of the Dirac operator oncompact Riemannian spin manifolds. This is a vast subject on which theinterested reader may consult [BHMM, G].3.2.1. The Friedrich inequality.
The first sharp inequality concerning eigen-values of the Dirac operator on compact Riemannian spin manifolds is dueto Friedrich [F] and is now known as the Friedrich inequality. It asserts that,if (Σ n − , g/ ) is such a manifold and if λ ( D/ ) denotes it first Dirac eigenvalue,then λ ( D/ ) ≥ n − n −
2) inf Σ n − R/ (3.2)where R/ is the scalar curvature of Σ n − with respect to the metric g/ . More-over, equality occurs if and only if the manifold carries a real Killing spinor.In particular, it is an Einstein manifold with positive scalar curvature. Notethat if Σ n − is disconnected, this inequality holds on each of its connectedcomponents.3.2.2. A Hijazi-Montiel-Zhang-like inequality.
If now we assume that Σ n − is the boundary of an n -dimensional compact Riemannian spin manifold( M n , g ), one can relate this first eigenvalue with extrinsic geometric invari-ants. In [HMZ], Hijazi, Montiel and Zhang prove that if the scalar curvature IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 9 of ( M n , g ) and the mean curvature of Σ n − = ∂M n are both nonnegative,it holds that λ ( D/ i ) ≥
12 inf Σ i H i (3.3)for all i = 1 , ..., k . Here we let Σ n − := ` ki =1 Σ i where for i = 1 , ..., k , Σ i is a connected component of Σ n − with metric g/ i := g | Σ i and first Diraceigenvalue λ ( D/ i ). It turns out that, in our situation, this inequality doesnot apply directly since one cannot ensure, a priori, that the mean curvatureis nonnegative. This is the first thing which has to be taken into accountin our approach. The second one is that the metric fails to be smooth atan interior point (where it is in fact C , ). We shall see that the works ofBartnik and Chru´sciel [BC1, BC2] allow to deal with this problem. So weend this section by recalling some of their results and we refer to their papersfor the definition of the Sobolev spaces which appear below.Consider M n a smooth, compact, spin manifold with boundary Σ n − endowed with a Riemannian metric g which is smooth on M n \ { p } and W ,qloc at p ∈ M n \ Σ n − with q > n/
2. The choice of the boundary conditionfor the Dirac operator in our approach is crucial and the motivations for thischoice are similar to those of Herzlich (as explained in the previous section).The Atiyah-Patodi-Singer boundary condition P > is defined as the L -orthogonal projection on the positive eigenspaces of the Dirac operator D/ .It is shown in [BC1, Corollary 7.4] that the boundary problem (cid:26) Dψ = η on M n P > ψ | Σ n − = ζ along Σ n − (3.4)for ( η, ζ ) ∈ L × P > H / ∗ has a solution ψ ∈ H if and only if Ker ( D ∗ , P ∗ > )is reduced to zero. Here D ∗ denotes the L -formal adjoint of D and P ∗ > theadjoint boundary condition of P > . It is in fact straightforward to see that D ∗ = D and P ∗ > = P ≥ , the L -orthogonal projection on the nonnegativeeigenspaces of D/ , so that the condition on the existence of a solution to (3.4)can be expressed as Ker ( D, P ≥ ) = { } . (3.5)On the other hand, if the data Ψ and Φ are smooth, the interior [BC1,Theorem 3.8] and boundary [BC1, Theorem 6.6] regularity results apply on M n \ { p } , since the metric is smooth there. Therefore, the spinor field ψ is smooth on M n \ { p } . As we will see in Section 5, these facts imply inparticular that the Hijazi-Montiel-Zhang inequality (3.3) holds under theseweaker assumptions on ( M n , g ).4. Proof of Theorem 2
Let ( M n , g, N ) be a spin asymptotically isotropic pseudo-static system ofmass m with n ≥
3. In the following, we write ∂M n = ` ki =1 Σ i and weassume that Σ i is • a nondegenerate static horizon for 1 ≤ i ≤ i , • a generalized quasilocal photon surface for i + 1 ≤ i ≤ k .Then consider the metric conformally related to g defined by g + = Φ n − + g with Φ + = 1 + N . (4.1)From the well-known transformation of the scalar curvature under a confor-mal change of the metric, the scalar curvature R + of g + is easily computedto be R + = Φ − n +2 n − + (cid:16) − n − n − + + R Φ + (cid:17) = R Φ − n − + ≥ N is harmonic with respect to g and R ≥
0. On the other hand, adirect computation using the asymptotically isotropy of the metric g as wellas the one of the lapse N allows to prove that ( M n , g + ) is asymptotically flatwith zero ADM mass. It is then enough to check that the condition (3.1) inTheorem 3 is fulfilled to conclude that g + is flat. For this reason, one has tocompute the mean curvature of each boundary components with respect tothe metric g + . This is achieved by using the classical formula which relatesthe mean curvature of the boundary of two conformally related metrics,namely H + i = Φ − nn − + (cid:16) n − n − ∂ Φ + ∂ν + H i Φ + (cid:17) for all i ∈ { , ..., k } . Then if Σ i is a nondegenerate static horizon, it followsfrom (2.1) and the previous formula that its mean curvature is given by thepositive constant H + i = 2 nn − n − n − κ i (4.2)where κ i := ν i ( N ) > i for i ∈ { , ..., i } . Onthe other hand, it follows from (2.3) that if i ∈ { i + 1 , ..., k } , the meancurvature of Σ i is also a positive constant whose value is H + i = 12 H i Φ − nn − + (cid:16) c i N i + 1 (cid:17) . (4.3)A first attempt to prove that the condition (3.1) holds is to apply theFriedrich inequality (3.2) on each component of the boundary. It is eas-ily seen to be unfruitful for the nondegenerate static horizon components.For the generalized quasilocal photon surfaces, it is useful to consider theinteger j ∈ { i + 1 , ..., k } for which N i ≥ c − / i if i + 1 ≤ i ≤ j and N i ≤ c − / i IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 11 if j + 1 ≤ i ≤ k . Then we compute that the scalar curvature R/ + i of (Σ i , g/ + i )with g/ + i := g + | Σ i satisfies R/ + i = Φ − n − + R/ i ≥ n − n − − n − + cH i > i ∈ { i +1 , ..., k } . The previous inequality is a direct consequence of (2.4).Combining the Friedrich inequality (3.2) for the first eigenvalue λ ( D/ + i ) ofthe Dirac operator D/ + i on (Σ i , g/ + i ) with (4.4) yields λ ( D/ + i ) ≥ n − n −
2) inf Σ i R/ + i ≥
14 Φ − n − + c i H i . The last inequality is deduced from the facts that the function Φ + and themean curvature H i are constant on the generalized quasilocal photon surfaceΣ i . Now from (4.3), we have that14 Φ − n − + c i H i ≥ (cid:0) H + i (cid:1) (4.5)if and only if 14 Φ − n − + c i H i ≥ H i Φ − nn − + (cid:16) c i N i + 1 (cid:17) that is c i (cid:16) N i + 1 (cid:17) ≥ (cid:16) c i N i + 1 (cid:17) . However, since c i >
1, it is easy to observe that this inequality holds only for i ∈ { j + 1 , ..., k } . It remains to show that (3.1) is also true for i ∈ { , ..., j } .For this, consider the smooth metric defined on M n by g − = Φ n − − g with Φ − = 1 − N N is harmonic and asymptotically isotropic. Sim-ilarly to the metric g + , the scalar curvature R − of g − is nonnegative. More-over since g is asymptotically isotropic, it can be shown ([C2, C3, J]) thatone can insert a point p ∞ into ( M n , g − ) to obtain a compact Riemannianspin manifold ( M n ∞ := M n ∪ { p ∞ } , g −∞ ) whose metric is smooth on M n , C , at p ∞ and with boundary ∂M n ∞ = ∂M n . Then the mean curvature of ∂M n in ( M n ∞ , g −∞ ) computed with respect to the unit normal ν − = − Φ − n − − ν is H − i = − Φ − nn − − (cid:16) n − n − ∂ Φ − ∂ν + H i Φ − (cid:17) . For i = 1 , ..., i , we have that H − i = 2 nn − n − n − κ i > is a positive constant, once again because of (2.1). On the other hand, if i ∈ { i + 1 , ..., k } , we deduce using (2.3) that H − i = 12 H i Φ − nn − − (cid:16) c i N i − (cid:17) (4.8)is also a constant but it could, a priori, be negative for i ∈ { j + 1 , ..., k } .This is the main reason why we cannot apply directly the inequality (3.3).Instead, we use the following result whose proof is postponed to the nextsection. Lemma 2.
For i ∈ { , ..., j } , the first eigenvalue λ ( D/ − i ) of the Diracoperator D/ − i on (Σ i , g/ − i ) satisfies λ ( D/ − i ) ≥ H − i . Moreover, if equality holds, the boundary ∂M n is connected. As H − i = H + i for all i ∈ { , ..., i } because of (4.2) and (4.7), Lemma 2allows to deduce directly that (3.1) holds for nondegenerate static horizonssince g/ − i = g/ + i implies λ ( D/ − i ) = λ ( D/ + i ). If i ∈ { i + 1 , ..., j } , we remarkthat since Φ + and Φ − are constant on Σ i and g/ + i = (cid:16) Φ + Φ − (cid:17) n − g/ − i , the corresponding Dirac operators are related by D/ + i = (cid:16) Φ − Φ + (cid:17) n − D/ − i so that the corresponding first eigenvalues satisfy λ ( D/ + i ) = (cid:16) Φ − Φ + (cid:17) n − λ ( D/ − i ) . (4.9)Therefore, the inequality of Lemma 2 reads λ ( D/ + i ) ≥ (cid:16) Φ − Φ + (cid:17) n − H − i . Now, from (4.3) and (4.8), we remark that (cid:16) Φ − Φ + (cid:17) n − H − i ≥ H + i if and only if (cid:16) cN i − (cid:17) Φ + ≥ (cid:16) cN i + 1 (cid:17) Φ − which is true precisely if i ∈ { i + 1 , ..., j } . In conclusion, the assump-tion (3.1) of the Positive Mass Theorem 3 holds for every component of theboundary, so that the asymptotically flat manifold ( M n , g + ) with nonnega-tive scalar curvature and zero ADM mass has to be flat. Moreover, each Σ i is totally umbilical with constant mean curvature. IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 13
If we chose a nondegenerate static horizon Σ l with l ∈ { , ..., i } , sinceequality holds in (3.1), we also have equality in Lemma 2 and so ∂M n = Σ l isconnected. The Gauss formula relative to the immersion of ∂M n in ( M n , g + )implies that ∂M n has positive constant sectional curvature and it has to beisometric to the quotient of a round sphere. Arguing as in [He2] we concludethat ( M n , g + ) is (up to an isometry) the exterior of a round ball with radius s l = n − nn − κ l . The metric g + being the Euclidean one, it turns out that the function Ψ :=Φ − satisfies the boundary value problem ∆ δ Ψ = 0 on R n \ B s l (0)Ψ( s ) = 1 + O ( s − n ) as s → ∞ Ψ | ∂B sl (0) = 2where ∆ δ denotes the Euclidean Laplacian. The maximum principle impliesthe uniqueness of solutions to this elliptic equation so that it is straightfor-ward to check that Ψ( s ) = 1 + (cid:16) s l s (cid:17) n − is the only one. The Riemannian manifold ( M n , g ) has to be isometric tothe exterior of the Schwarzschild metric with mass m = 2 s n − l = 12 (cid:16) Aω n − (cid:17) n − n − > N = N m . Here A denotes the area of the inner boundary ( ∂M n , g/ ).Now assume that we chose a quasilocal photon surface Σ l with l ∈ { i +1 , ..., j } . As before, since (3.1) is an equality, a straightforward computa-tion implies that equality also holds in Lemma 2 and the boundary ∂M n has to be connected, that is ∂M n = Σ l . Moreover, since the boundary istotally umbilical with constant mean curvature, one can argue as in the pre-vious paragraph to conclude that ( M n , g + ) is isometric to the exterior of anEuclidean ball with radius e s l = ( n − − n − c − n n − l (cid:0) c / l + 1 (cid:1) n − H l since N l = c − / l . Once again, writing g = Ψ n − δ with Ψ := Φ − and δ theEuclidean metric, we deduce that Ψ is the unique solution of the problem ∆ δ Ψ = 0 on R n \ B e s l (0)Ψ( s ) = 1 + O ( s − n ) as s → ∞ Ψ | ∂B e sl (0) = 2(1 + c − / l ) − . It is then straightforward to check thatΨ( s ) = 1 + (cid:16) b s l s (cid:17) n −
24 SIMON RAULOT with b s l = ( n − − n − c − n n − l (cid:0) c l − (cid:1) n − H l . We conclude that ( M n , g ) is isometric to the exterior { s ≥ e s l } of theSchwarzschild metric with mass m = 2 b s n − l = 12 (cid:16) − c l (cid:17)(cid:16) Aω n − (cid:17) n − n − > N = N m . Here we used the Stokes’ theorem and the harmonicity of N to compute that Z ∂M n ν ( N ) = ( n − ω n − m (4.10)for any asymptotically isotropic pseudo-static systems of mass m with con-nected inner boundary. Then, using (2.3) and the fact that N l = 1 / √ c l ,yields H l = 2( n − ω n − A √ c l c l − m and the last expression of the mass follows.Finally, if we chose a quasilocal photon surface Σ l with l ∈ { j + 1 , ..., k } then, since (3.1) is an equality, we directly get that the equality is alsoachieved in (4.5). This implies that N i = c − / i for all i ∈ { j + 1 , ..., k } andthus from (4.8) that H − i is a positive constant. Now since all the componentsof the boundary have positive mean curvature, the Hijazi-Montiel-Zhanginequality (3.3) applies (see Section 5) and it is in fact an equality. Theboundary has therefore to be connected and totally umbilical with constantmean curvature. One can conclude exactly as in the previous situation andthis finish the proof of Theorem 2.5. Proof of Lemma 2
In order to prove Lemma 2, we first need to show that the boundaryvalue problem (3.4) for the Dirac operator D − on ( M n ∞ , g −∞ ) under theAtiyah-Patodi-Singer condition P − > admits an unique strong solution. Here P − > denotes the L -orthogonal projection on the positive eigenspaces of theboundary Dirac operator D/ − . Since the metric g −∞ is smooth on M n and W ,qloc for q > n/ p ∞ (since C , at this point), the re-sults of Bartnik and Chru´sciel recalled in Section 3.2.2 apply on ( M n ∞ , g −∞ ).So proving the existence of such a solution is reduced to show that (3.5)holds under the assumptions of Lemma 2.For this, we recall the integral version of the famous Schr¨odinger-Lichne-rowicz formula (see [BC1, HMZ] for a proof) which states that Z M n ∞ (cid:16) |∇ − ϕ | + R − | ϕ | − | D − ϕ | (cid:17) = k X i =1 Z Σ i (cid:16) h D/ − i ϕ i , ϕ i i − H − i | ϕ i | (cid:17) (5.1) IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 15 for all ϕ ∈ Γ( S − ) and where ϕ i := ϕ | Σ i ∈ Γ( S / − i ). The notations in thisformula are directly derived from those of Section 3.Consider now ψ ∈ Ker ( D − , P −≥ ) that is ψ ∈ H and satisfies (cid:26) D − ψ = 0 on M n ∞ P −≥ ψ i = 0 along Σ i (5.2)for all i ∈ { , ..., k } . Since the scalar curvature of the metric g −∞ is nonneg-ative, the formula (5.1) applied to ψ yields k X i =1 Z Σ i (cid:16) h D/ − i ψ i , ψ i i − H − i | ψ i | (cid:17) ≥ . (5.3)On the other hand, it is straightforward to check that for all ϕ ∈ Γ( S − ), wehave Z Σ i h D/ − i ϕ i , ϕ i i ≤ Z Σ i h D/ − i (cid:0) P − > ϕ i (cid:1) , P − > ϕ i i (5.4)with equality if and only if the L -projection of ϕ i on the negative eigenspacesof D/ − i is zero. So we can rewrite (5.3) as k X i = j +1 Z Σ i (cid:16) h D/ − i ψ i , ψ i i − H − i | ψ i | (cid:17) ≥ H − i > i ∈{ , ..., j } . Note that we didn’t use (5.4) directly for i ∈ { j + 1 , ..., k } , sincethe mean curvature H − i may be negative and the inequality would then beuseless. Instead, we remark that since the L -projection of ψ i on the kernelof D/ − i is zero, the upper bound (5.4) can be refined to Z Σ i h D/ − i ψ i , ψ i i ≤ − λ ( D/ − i ) Z Σ i | ψ i | (5.6)for i ∈ { j + 1 , ..., k } with equality if and only if ψ i is an eigenspinor for D/ − i associated to the eigenvalue − λ ( D/ − i ) or is zero on Σ i . This allows torewrite (5.3) as k X i = j +1 (cid:16) λ ( D/ − i ) + H − i (cid:17) Z Σ i | ψ i | ≤ . (5.7)Assume for a moment that λ ( D/ − i ) + H − i > i ∈ { j + 1 , ..., k } . We thus have equality in (5.7) and so ψ i = 0for all i ∈ { j + 1 , ..., k } . Moreover, equality also holds in (5.5) and sothe L -projection of ψ i on the negative eigenspaces of D/ − i vanishes for all i ∈ { , ..., j } . But ψ i satisfies the boundary condition in (5.2) and then wededuce that ψ i = 0 for all i ∈ { , ..., k } . Finally, equality occurs in (5.1) and so we get a parallel spinor field ψ which vanishes along ∂M n . Since ψ is smooth on M n , its norm is constant on M n . But since it is zero onthe boundary, it has to be zero on the whole of M n and so (3.5) is fulfilled.It remains to show that (5.8) is satisfied. For this, we put together (3.1)(which holds for i = j + 1 , ..., k without using Lemma 2 by the Friedrichinequality) and the formula (4.9) which relates the first eigenvalue of Diracoperators for homothetic metrics to get λ ( D/ − i ) ≥ (cid:16) Φ + Φ − (cid:17) n − H + i . This lower bound and the expressions (4.3) and (4.8) of the mean curvaturesof Σ i with respect to the metrics g + and g − in terms of the metric g yield λ ( D/ − i ) + H − i ≥ H i N i N i Φ − nn − − ( c i − . This implies (5.8) since the right-hand side of the previous inequality isclearly positive because H i as well as N i are positive constants and c i > i ∈ { j + 1 , ..., k } .Now the previous discussion ensures that for a fixed l ∈ { , ..., j } , theboundary value problem (cid:26) D − ξ = 0 on M n ∞ P − > ξ | ∂M n = η l along ∂M n admits a solution ξ ∈ Γ( S − ), where η l ∈ Γ( S / − ) is defined by η l = (cid:26) ζ l on Σ l i for i = l with ζ l ∈ Γ( S / − l ) is a smooth eigenspinor for D/ − l associated to λ ( D/ − l ). Notethat as recalled in Section 3.2.2, ξ is smooth on M n . Applying the integralversion of the Schr¨odinger-Lichnerowicz formula (5.1), the upper bounds(5.4) and (5.6) respectively for i ∈ { , ..., j } and for i ∈ { j + 1 , ..., k } to thespinor field ξ give (cid:16) λ ( D/ − l ) − H − l (cid:17) Z Σ l | ξ l | ≥ k X i = j +1 Z Σ i (cid:16) H − i | ξ i | − h D/ − ξ i , ξ i i (cid:17) ≥ k X i = j +1 (cid:16) λ ( D/ − i ) + H − i (cid:17) Z Σ i | ξ i | . From (5.8), we immediately obtain the inequality of Lemma 2. We shouldpay attention to the fact that the upper bound (5.6) holds for spinor fields ψ i ∈ Γ( S / − i ) such that P −≥ ψ i = 0 and so we have to check that ξ satisfiedthis property. Since P − > ξ i = 0 for all i = l , it is enough to show thatthe projection of ξ i on the kernel of the Dirac operator D/ − i is reduced tozero for i ∈ { j + 1 , ..., k } . In fact, for such i , the Friedrich inequality (3.2)combined with the positivity (2.4) of the scalar curvature of (Σ i , g/ i ) ensures IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 17 that the kernel of D/ i is reduced to zero. But the dimension of this spacebeing invariant under conformal changes of the metric on Σ i (see [Hi]), wededuce that the kernel of D/ − i is also reduced to zero and so P −≥ ξ i = P − > ξ i = 0 . Now if equality occurs in Lemma 2, it it immediate to see that equality alsoholds in (5.1) and the spinor field ξ has to be parallel. On the other hand,since H − i > i ∈ { , ..., j } , the kernel of D − i is also reduced to zeroand so Ker ( D/ − ) = { } , where D/ − is the full boundary Dirac operator of( ∂M n , g/ − ). This allows to conclude that ξ | ∂M n = P − > η l = η l since equality has to hold in (5.4) and (5.6). As the squared norm of theparallel spinor ξ is smooth on M n , it has to be a positive constant because itsrestriction to Σ l is a nonzero eigenspinor for D − l . However, this is impossibleif ∂M n is disconnected since otherwise it is also zero on the other connectedcomponents. We conclude that the boundary ∂M n is connected and theproof of Lemma 2 is now complete. Appendix A. Simon’s proof of the static black hole uniquenesstheorem of [MHRS]In this section, we give the proof of the (3 + 1)-dimensional static blackhole uniqueness theorem in the connected boundary case by Walter Simon[Si1] which, in our conventions and notations, can be stated as follows:
Theorem 4.
An asymptotically flat static vacuum triple ( M , g, N ) with anondegenerate connected and compact inner boundary ∂M := N − ( { } ) isisometric to the exterior { s ≥ ( m/ n − } of the Schwarzschild manifold ofpositive mass m . Here the 3-dimensional static vacuum triple ( M , g, N ) is asymptoticallyflat in the sense that the manifold M is diffeomorphic to the union of acompact set and an open end E which is diffeomorphic to R \ B where B is an open ball in R . Furthermore, we require that, with respect tothe coordinates induced by this diffeomorphism, the metric g and the lapsefunction N satisfy g ij − δ ij ∈ W k,q − τ ( E ) N − ∈ W k +1 ,q − τ ( E ) (A.1)for some τ > / τ / ∈ Z , k ≥ q >
4. Furthermore, nondegenerate meanhere that the surface gravity κ = ν ( N ) of the inner boundary is nonzero.Then, since ( M , g, N ) is a static vacuum triple, it is a well known fact thatthis implies that the inner boundary is a nondegenerate static horizon in thesense of Section 2.One of the idea of Simon was to apply the positive mass theorem withboundary of Herzlich to the metric g + defined by (4.1). As done before, it isenough to show that (3.1) holds in this situation and one way to obtain such a lower bound is to use the Friedrich inequality (3.2). Then we are broughtto compare the scalar curvature R/ + of ( ∂M , g/ + ) with the mean curvature ∂M in ( M , g/ + ) whose value is H + = 16 κ (A.2)as computed in (4.2). This can be achieved by using a divergence identityregarding the static vacuum triple in the compactified metric g −∞ defined inSection 4. It turns out from the works of Beig and Simon [BS1, BS2] andKennefick and ´O Murchadha [K ´OM] that since ( M , g, N ) is an asymptot-ically flat static vacuum triple with non-zero mass, its one-point compact-ification ( M ∞ , g −∞ ) admits a smooth (even analytic) extension to p ∞ . Thefact that the mass is non-zero in our situation follows directly by formula(4.10). Then from the divergence identity (see [Si2] and Remark 4 for ageneralization of this formula)div g − (cid:16) N − (1 + N ) ∇ − W (cid:17) = 18 V | Ric − | g − (A.3)where W and V are the respectively smooth and continuous nonnegativefunctions on M ∞ given by W = |∇ N | (1 − N ) and V = N (1 − N ) (1 + N ) , one can deduce the following lemma which plays a crucial role in whatfollows. Here ∇ − , div g − and Ric − are respectively the covariant derivative,the divergence and the Ricci curvature of the manifold M ∞ with respect tothe metric g −∞ . Lemma 3.
The scalar curvature R/ of the nondegenerate static horizon ∂M with respect to the metric g satisfies R/ ≥ κ .Proof. Applying the weak maximum principle to (A.3) on the smooth com-pact manifold M ∞ , we find that W is constant on M ∞ or W must takeits maximum on the nondegenerate static horizon where it is constant. Ineither case, the derivative of W along the inward normal ν is nonpositivenear ∂M ∞ and the same applies for the following limitlim N → κ − N − g ( ∇ W, ∇ N ) ≤ . (A.4)On the other hand, a direct computation using the static equations (1.1)yields g ( ∇ W, ∇ N ) = (1 − N ) − (cid:16) ∇ N (cid:0) |∇ N | (cid:1) + 8 N |∇ N | (1 − N ) − (cid:17) = 2(1 − N ) − N (cid:16) Ric ( ∇ N, ∇ N ) + 4 |∇ N | (1 − N ) − (cid:17) which allows to rewrite (A.4) as0 ≥ κ (cid:16) Ric ( ν, ν ) + 4 κ (cid:17) = κ (cid:16) − R/ κ (cid:17) . IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 19
Here we used the fact that ν = κ − ∇ N and, in the last equality, the Gaussformula (recall that ∂M is totally geodesic in ( M , g ) which is scalar flat).This conclude the proof of this lemma. q.e.d.Now, since the manifold M is 3-dimensional, it is automatically endowedwith a spin structure so that the first eigenvalue of the Dirac operator on ∂M with the metric g/ + satisfies the Friedrich inequality λ ( D/ + ) ≥
12 inf ∂M R/ + . (A.5)However, it is now immediate from the formula (4.4) that the Lemma 3implies that R/ + = 16 R/ ≥ κ . Combining this estimate with the inequality (A.5) yields λ ( D/ + ) ≥ κ which, because of (A.2), is exactly (3.1). We conclude the proof of Theorem4 as usual. Remark 4.
It may be interesting to note that the divergence identity (A.3)can be generalized to n -dimensional static vacuum triples ( M n , g, N ) for themetrics g ± . Namely it holds that div g ± (cid:16) N − (1 ∓ N ) ∇ ± W (cid:17) = 2 n − n − V ± | Ric ± | g ± (A.6) where W and V ± are the functions defined by W = |∇ N | (1 − N ) n − n − and V ± = N (1 ± N ) n − (1 ∓ N ) n − n − . Indeed, from the second Bianchi identity and the fact that g ± is scalar flat,one computes that div g ± (cid:16) Ric ± ( X ) (cid:17) = 12 h Ric ± , L X g ± i g ± (A.7) for all X ∈ Γ( T M n ) and where L X is the Lie derivative in the direction of X and Ric ± ( X ) is the vector field on M n defined by g ± ( Ric ± ( X ) , Y ) = Ric ± ( X, Y ) for all Y ∈ Γ( T M n ) . On the other hand, a straightforward (but lengthy)computation using the static equations (1.1) gives
12 Tf (cid:0) L X g ± (cid:1) = V ± Ric ± (A.8) where X = 2 n − (1 − N ) nn − ∇ N and Tf denotes the trace-free part of a symmetric tensor. Then puttingtogether (A.7) and (A.8), we deduce that div g ± (cid:16) Ric ± ( X ) (cid:17) = V | Ric ± | g ± , and (A.6) follows from the expression of the vector field X . It is importantto note that the proof of these divergence identities relies on the full set ofthe static equations (1.1) and does not hold for pseudo-static systems. Appendix B. Uniqueness without positive mass theorem
In this section, we present a simple proof of Theorem 4 which is clearlyin the spirit of the original works [I, MHRS, R] but which puts forward anew geometric point of view. As we shall see, this method also applies whenstudying uniqueness questions for connected quasilocal photon surface in3-dimensional static vacuum triple.B.1.
The nondegenerate static horizon case.
If we assume that theassumptions of Theorem 4 are fulfilled, one can integrate on ∂M the in-equality in Lemma 3 which, with the help of the Gauss-Bonnet formula,yields π χ ( ∂M ) ≥ κ A. Here χ ( ∂M ) denotes the Euler characteristic of the nondegenerate statichorizon. The formula (4.10) for n = 3 implies that the last inequality canbe rewritten as A π χ ( ∂M ) ≥ m . (B.1)On the other hand, a direct computation using the asymptotic of the lapsefunction N gives that W ( p ∞ ) = 116 m . Since the maximum of the smooth function W defined on M ∞ is reached on ∂M , we obtain κ ≥ m , which from (4.10), rewrites as the classical Penrose inequality m ≥ A π . Combining this inequality with (B.1) gives χ ( ∂M ) = 2 and m = A π , IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 21 and then W has to be constant on M ∞ . In particular, ∂M is homeomorphicto a 2-sphere. From (A.3), ( M ∞ , g −∞ ) is Ricci-flat and so flat since it is 3-dimensional. This allows to conclude that ( M , g ) is the desired exterior ofthe Schwarzschild manifold.B.2. The quasilocal photon surface case.
We are now in position togive a similar proof of the following result:
Theorem 5.
An asymptotically flat static vacuum triple ( M , g, N ) with aconnected quasilocal photon surface as inner boundary is isometric to theexterior of a suitable piece of a Schwarzschild manifold of positive mass. First recall that since ∂M is a quasilocal photon surface, it is a totallyumbilical surface with positive constant mean curvature H on which thelapse function is also a positive constant. Moreover, the identities (2.2) and(2.3) are fulfilled for a constant c >
1. Then since ν ( N ) >
0, the formula(4.10) implies that the mass of ( M , g ) is non-zero. Moreover, from theseproperties, the Gauss formula reads as2 Ric ( ν, ν ) = 1 − c H so that computations as in Lemma 3 yields N ≤ c − , where we let N := N | ∂M . Once again using (2.2), (2.3) and (4.10), it is straightforward tocheck that this inequality is equivalent to m ≤ (cid:16) − c (cid:17) A ω R/ which, once integrated over ∂M , gives m ≤ (cid:16) − c (cid:17) Aω χ ( ∂M ) . (B.2)On the other hand, since the function W reaches its maximum on the bound-ary (unless it is constant) we have W ( p ∞ ) = 116 m ≤ W | ∂M = ν ( N ) (1 − N ) ≤ (cid:16) − c (cid:17) − ω A m that is m ≥ (cid:16) − c (cid:17) Aω . Combining this inequality with (B.2) allows to conclude that ∂M is a topo-logical 2-sphere and that (B.2) is in fact an equality. The maximum princi-ple implies that W is constant on the whole of M and then the divergenceidentity (A.3) that ( M ∞ , g −∞ ) is flat. Finally, it is not difficult to show that( M ∞ , g −∞ ) is isometric to a flat ball with radius 2 /H − and we can conclude that ( M , g ) is isometric to the exterior { s ≥ s } (with s = e s l given inSection 4) of the Schwarzschild metric with mass m = 12 (cid:16) − c (cid:17)(cid:16) Aω (cid:17) . References [B] R. Bartnik,
The mass of an asymptotically flat manifold , Comm. Pure Appl.Math. (1986), no. 5, 661–693.[BC1] R. Bartnik, P. Chru´sciel, Boundary value problems for Dirac-type equations , J.Reine Angew. Math. (2005), 13-73.[BC2] R. Bartnik, P. Chru´sciel,
Boundary value problems for Dirac-type equations ,arXiv:math/0307278.[BS1] R. Beig, W. Simon,
Proof of a multipole conjecture due to Geroch , Comm. Math.Phys. (1980), no. 1, 75–82.[BS2] R. Beig, W. Simon, The multipole structure of stationary space-times , J. Math.Phys. (1983), no. 5, 1163–1171.[BHMM] J.P. Bourguignon, O. Hijazi, J.-L. Milhorat, A. Moroianu, A Spinorial Approachto Riemannian and Conformal Geometry , Monographs in Mathematics, EMS,Z¨urich, 2015.[BM] G. Bunting, A.K.M. Masood-ul-Alam,
Nonexistence of multiple black holes inasymptotically Euclidean static vacuum space-time , Gen. Relativity Gravitation (1987), no. 2, 147–154.[C1] C. Cederbaum, Uniqueness of photon spheres in static vacuum asymptoticallyflat spacetimes , Complex analysis and dynamical systems VI, Contemp. Math. (2015), AMS, 86-99.[C2] C. Cederbaum,
Rigidity properties of the Schwarzschild manifold in all dimen-sions , in preparation.[C3] C. Cederbaum,
A geometric boundary value problem related to the static equa-tions in General Relativity , Oberwolfach report (2017).[CG1] C. Cederbaum, G. J. Galloway,
Uniqueness of photon spheres via positive massrigidity , Comm. Anal. Geom. (2017), no. 2, 303-320.[CF] C. Cederbaum, A. Fehrenbach, Photon sphere and equipotential photon surfaceuniqueness in asymptotically flat static vacuum spacetimes `a la Robinson , per-sonal communication.[CG2] C. Cederbaum, G. J. Galloway,
Photon surfaces with equipotential time-slices ,arXiv:1910.04220.[Ch] P. Chru´sciel,
Boundary conditions at spatial infinity from a Hamiltonian pointof view , Topological properties and global structure of space-time (Erice, 1985),NATO Adv. Sci. Inst. Ser. B Phys., vol. 138, Plenum, New York, 1986, pp.49-59.[F] T. Friedrich,
Der erste Eigenwert des Dirac-Operators einer kompakten Rie-mannschen Mannigfaltigkeit nichtnegativer Skalarkr¨ummung , Math. Nach. ,117–146 (1980).[GIS] G. W. Gibbons, D. Ida, T. Shiromizu, Uniqueness and nonuniqueness of staticblack holes in higher dimensions , Phys. Rev. Lett. (2002), no. 4, 041101, 4pp.[G] N. Ginoux, The Dirac spectrum , Lecture Notes in Mathematics , Springer-Verlag, Berlin, 2009.[He1] M. Herzlich,
A Penrose-like inequality for the mass of Riemannian asymptoticallyflat manifolds , Commun. Math. Phys. (1997), no. 1, 121–133.[He2] M. Herzlich,
Minimal surfaces, the Dirac operator and the Penrose inequality ,S´eminaire de Th´eorie Spectrale et G´eom´etrie (2002), 9–16. IGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD 23 [HMZ] O. Hijazi, S. Montiel, X. Zhang,
Dirac operator on embedded hypersurfaces ,Math. Res. Lett. (2001), 195–208.[HM] S. Hirsch, P. Miao, A positive mass for manifolds with boundary , Pacific J. Math. (2020), no. 1, 185–201.[Hi] N. Hitchin,
Harmonic spinors , Adv. Math. (1974), 1–55.[Hw] S. Hwang, A rigidity theorem for Ricci flat metrics , Geom. Dedicata (1998),no. 1, 5–17.[I] W. Israel, Event horizons in static vacuum space-times , Phys. Rev. (1967),no. 5, 1775-1779.[J] S. Jahns,
Photon sphere uniqueness in higher dimensional electrovaccum space-times , Classical Quantum Gravity (2019), no. 23, 235019.[K ´OM] D. Kennefick, N. ´O Murchadha, Weakly decaying asymptotically flat static andstationary solutions to the Einstein equations , Class. Quantum Grav. (1995),no. 1, 149.[LM] H.B. Lawson, M.L. Michelsohn, Spin Geometry , Princeton Math. Series, vol. 38,Princeton University Press, 1989.[MS] D. McFeron, G. Sz´ekelyhidi,
On the positive mass theorem for manifolds withcorners , Comm. Math. Phys. (2012), no. 2, 425–443.[M] P. Miao,
Positive mass theorem on manifolds admitting corners along a hyper-surface , Adv. Theor. Math. Phys. (2002), no. 6, 1163–1182.[MHRS] H. M¨uller zum Hagen, D.C. Robinson, H.J. Seifert, Black holes in static vacuumspace-times , Gen. Relativity Gravitation (1973), 53–78.[R] D.C. Robinson, A simple proof of the generalization of Israel’s theorem , Gen.Relativ. Gravitation (1977), 695–698.[SY1] R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in GeneralRelativity , Commun. Math. Phys. (1979), no. 1, 45-76.[SY2] , Proof of the positive mass theorem II , Comm. Math. Phys. (1981),no. 2, 231-260.[SY3] R. Schoen, S.-T. Yau, Positive scalar curvature and minimal hypersurface sin-gularities , arXiv:1704.05490.[Sc] K. Schwarzschild, ¨Uber das Gravitationsfeld eines Massepunktes nach der Ein-steinschen Theorie , Sitzungsberichte der K¨oniglich-Preussischen Akademie derWissenschaften (1916), 189–196.[Si1] W. Simon, A spinorial proof of the static black hole uniqueness theorem , unpub-lished, University of Vienna, 1998.[Si2] W. Simon,
Criteria for (in)finite extent of static perfect fluids , The conformalstructure of space-time, Lecture Notes in Phys. , Springer, Berlin, 2002.[T] F.R. Tangherlini,
Schwarzschild field in n dimensions and the dimensionality ofspace problem , Nuovo Cim. (1963), 636–651.[W] E. Witten, A new proof of the positive energy theorem , Commun. Math. Phys. (1981), 381-402.(Simon Raulot) Laboratoire de Math´ematiques R. Salem UMR
CNRS-Universit´e de Rouen Avenue de l’Universit´e, BP. Technopˆole du Madrillet
Saint-´Etienne-du-Rouvray, France.
E-mail address ::