Initial data rigidity results
aa r X i v : . [ g r- q c ] S e p INITIAL DATA RIGIDITY RESULTS
MICHAEL EICHMAIR, GREGORY J. GALLOWAY, AND ABRA ˜AO MENDES
Abstract.
We present several rigidity results for initial data sets motivatedby the positive mass theorem. An important step in our proofs here is to estab-lish conditions that ensure that a marginally outer trapped surface is weaklyoutermost . A rigidity result for Riemannian manifolds with a lower bound ontheir scalar curvature is included as a special case. Introduction
In this paper, we study several rigidity questions for initial data sets that aremotivated by the spacetime positive mass theorem and its proofs.An initial data set is a triple (
M, g, K ) where (
M, g ) is a connected Riemannianmanifold and K is a symmetric (0 , M is oriented.Let ( M, g, K ) be an initial data set.The local energy density µ and the local current density J of ( M, g, K ) aregiven by µ = 12 (cid:16) R − | K | +(tr K ) (cid:17) and J = div ( K − (tr K ) g )where R is the scalar curvature of ( M, g ).We say that (
M, g, K ) satisfies the dominant energy condition (DEC), if theinequality µ ≥ | J | holds on M .Let Σ ⊂ M be a two-sided hypersurface with unit normal ν and associatedmean curvature scalar H Σ = div Σ ν . The future outgoing null expansion scalar θ +Σ and past outgoing null expansion scalar θ − Σ of Σ are given by θ +Σ = H Σ + tr Σ ( K ) and θ − Σ = H Σ − tr Σ ( K ) . We say that Σ is outer trapped if θ +Σ < weakly outer trapped if θ +Σ ≤ marginally outer trapped if θ +Σ = 0. In the latter case, we refer to Σ as amarginally outer trapped surface (MOTS). Unless otherwise stated, we assumethat MOTS are closed, i.e. compact and without boundary.We also define χ +Σ = K | Σ + A Σ and χ − Σ = − K | Σ + A Σ where A Σ is the second fundamental form of Σ. The sign convention here is suchthat H Σ = tr Σ A Σ so that, in particular, θ ± Σ = tr χ ± Σ .Initial data sets arise naturally in general relativity. Let M be a spacelikehypersurface in a spacetime, i.e. a time-oriented Lorentzian manifold, ( ¯ M , ¯ g ).Then associated to M ⊂ ¯ M is the initial data set ( M, g, K ) where g is the inducedRiemannian metric and K is the second fundamental form with respect to thefuture-pointing unit normal u of M . In this setting, χ ± Σ are called null secondfundamental forms with respect to the null normal fields ℓ ± = ν ± u along Σ. Theycompletely determine the second fundamental form of Σ as a co-dimension twosubmanifold of ¯ M . Note that the null expansion scalars are given by θ ± Σ = div Σ ℓ ± .An initial data set ( M, g, K ) is said to be time-symmetric or Riemannian if K = 0. In this case, the DEC asks that the scalar curvature of ( M, g ) be non-negative. Moreover, Σ is a MOTS if and only if it is a minimal surface in (
M, g ).Quite generally, MOTS share many properties with minimal surfaces, whichthey generalize; cf. e.g. [3].The following version of the spacetime positive mass theorem has been obtainedby L.-H. Huang, D. A. Lee, R. Schoen, and the first-named author in [12].
Theorem 1.1 ([12]) . Let ( M, g, K ) be an n -dimensional asymptotically flat initialdata set with ADM energy-momentum vector ( E, P ) . Assume that ≤ n ≤ . Ifthe dominant energy condition µ ≥ | J | is satisfied, then E ≥ | P | . In [17], J. Lohkamp has presented a different proof of this result (in all dimen-sions). His method is by reduction to, and proof of, the following result: Let(
M, g, K ) be an initial data set that is isometric to Euclidean space, with K = 0,outside some bounded set U . Then one cannot have µ > | J | on U ; cf. [17, Theo-rem 2]. In particular, in the case of general interest, in which ( M, g, K ) satisfiesthe DEC, there must be a point in U at which µ = | J | . The goal of our first resultis to show that a much strong conclusion holds in dimensions 3 ≤ n ≤ NITIAL DATA RIGIDITY RESULTS 3
Under the assumptions of Lohkamp’s result stated above, one may obtain, afteran obvious compactification, a compact manifold M with boundary ∂M = Σ ∪ Σ ,such that Σ and Σ are flat ( n − M, g ) and totally geodesic in thespacetime sense, χ ± Σ = 0 and χ ± Σ = 0. In particular, both are MOTS (withrespect to any choice of ν ). With this compactified picture in mind, we state ourfirst main rigidity result. Theorem 1.2.
Let ( M, g, K ) be an n -dimensional, ≤ n ≤ , compact-with-boundary initial data set. Suppose that ( M, g, K ) satisfies the DEC and that theboundary ∂M can be expressed as a disjoint union of hypersurfaces, ∂M = Σ ∪ S ,such that the following conditions are satisfied:(1) θ + K ≤ along Σ with respect to the unit normal pointing into M ,(2) θ + K ≥ along S with respect to the unit normal pointing out of M ,(3) M satisfies the homotopy condition with respect to Σ , and(4) Σ satisfies the cohomology condition.Then, the following hold:(i) M ∼ = [0 , ℓ ] × Σ for some ℓ > .(ii) The leaf Σ t ∼ = { t } × Σ is a MOTS for every t ∈ [0 , ℓ ] . In fact, the outwardnull second fundamental form of Σ t vanishes.(iii) Σ t is a flat torus with respect to the induced metric for every t ∈ [0 , ℓ ] .(iv) We have that µ K = | J K | and − J K ( ν t ) = | J K | on Σ t where ν t is the unitnormal in direction of the foliation. Here we have introduced subscript notation on µ , J and θ + , to show theirdependence on a particular initial data set, as different initial data sets, evenwithin a given proof, will be used.Precise definitions of the cohomology condition and the homotopy condition are given in Section 3. The cohomology condition ensures that Σ does not admita metric of positive scalar curvature. The homotopy condition will hold, forexample, if M has almost product topology , i.e. if M ∼ = ([0 , × Σ ) N where N is a compact manifold. The homotopy condition implies that Σ is connected. Apriori, we allow S to have multiple components.The assumptions of Theorem 1.2 are satisfied in the compactified picture de-scribed above. Note that Theorem 1.2 provides a relatively simple proof of [17, NITIAL DATA RIGIDITY RESULTS 4
Theoren 2] in dimensions 3 ≤ n ≤
7. In conjunction with [17, Corollary 2.11],this leads to an alternative proof of Theorem 1.1.Theorem 1.2 may be viewed as a global version of the local rigidity result forMOTS obtained in [13] in dimensions 3 ≤ n ≤
7; see Section 2 for a precisestatement. This local rigidity result is used in the proof of Theorem 1.2, inconjunction with Lemma 3.2, which establishes conditions under which the weaklyoutermost condition (defined in Section 2) is satisfied.By imposing a convexity condition on the spacetime second fundamental form K , one can obtain a stronger rigidity result such as the following. (Note also thedifference in the boundary conditions.) Theorem 1.3.
Let ( M, g, K ) be an n -dimensional, ≤ n ≤ , compact-with-boundary initial data set. Assume that ( M, g, K ) satisfies the DEC and that theboundary ∂M is the union of disjoint surfaces Σ and S such that the followinghold:(1) θ + K ≤ along Σ with respect to the unit normal pointing into M .(2) θ − K ≥ n − ǫ along S with respect to the unit normal pointing out of M , where ǫ = 0 or ǫ = 1 .(3) M satisfies the homotopy condition with respect to Σ .(4) Σ satisfies the cohomology condition.(5) K + ǫ g is ( n − -convex.Then, the following hold: (i) (Σ , g ) is a flat torus, where g is the induced metric on Σ . (ii) ( M, g ) is isometric to ([0 , ℓ ] × Σ , dt + e ǫ t g ) for some ℓ > . (iii) K = (1 − ǫ ) a dt − ǫ g on M , where a depends only on t ∈ [0 , ℓ ] . (iv) µ K = 0 and J K = 0 on M . We recall the definition of ( n − K + ǫ g is ( n − ǫ = 1 is relevant in the asymptotically hyperbolic(or asymptotically hyperboloidal) setting. By choosing a suitable initial data set,Theorem 1.3 has as a simple corollary the following purely Riemannian result. Corollary 1.4.
Let ( M, g ) be an n -dimensional, ≤ n ≤ , compact Riemannianmanifold with boundary. Suppose that the scalar curvature of ( M, g ) satisfies R ≥ NITIAL DATA RIGIDITY RESULTS 5 − n ( n − ǫ , where ǫ = 0 or ǫ = 1 . We also assume that ∂M is a disjoint unionof hypersurfaces ∂M = Σ ∪ S such that the following assumptions hold:(1) The mean curvature of Σ in ( M, g ) with respect to the unit normal thatpoints into M satisfies H Σ ≤ ( n − ǫ .(2) The mean curvature of S in ( M, g ) with respect to the unit normal thatpoints out of M satisfies H S ≥ ( n − ǫ .(3) M satisfies the homotopy condition with respect to Σ .(4) Σ satisfies the cohomology condition.Then (Σ , g ) is a flat torus, where g is the induced metric on Σ . Moreover, ( M, g ) is isometric to ([0 , ℓ ] × Σ , dt + e ǫ t g ) for some ℓ > . Corollary 1.4 is closely related to results in [14]. Note that no weakly outermostcondition is required here. It can roughly be regarded as a scalar curvature versionof Theorem 1 in the paper [9] of C. B. Croke and B. Kleiner in the Ricci curvaturesetting. The case of ǫ = 1 in Corollary 1.4 can be used to give an alternative proofof the rigidity result for hyperbolic space, Theorem 1.1 in [2].Some background material on MOTS is presented in Section 2. In Section 3we establish conditions to verify the weakly outermost condition. In Section 4we present a proof of Theorem 1.2. In Section 5 we present proofs of Theorem1.3 and Corollary 1.4 and consider some further results. Finally, in Section 6, weshow how the initial data sets in Theorem 1.3 arise in Minkowski space. Acknowledgments.
Michael Eichmair is supported by the START-ProjectY963-N35 of the Austrian Science Fund. Gregory J. Galloway acknowledges thesupport of NSF Grant DMS-1710808. Abra˜ao Mendes would like to express hisgratitude to the University of Miami where much of his work on this projectwas carried out. Abra˜ao Mendes was supported in part by the Coordena¸c˜ao deAperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code001. 2.
Preliminaries
Here we recall properties of and results for MOTS that are needed in this paper.As in the minimal surface case, a useful tool for the study of a MOTS is the firstvariation formula for θ + K . Let Σ be a MOTS in the initial data set ( M, g, K ) withoutward unit normal ν . Consider a normal variation of Σ in M , i.e. a variation NITIAL DATA RIGIDITY RESULTS 6 t Σ t of Σ = Σ with variation vector field V = φ ν , φ ∈ C ∞ (Σ). Let θ + K ( t )be the null expansion of Σ t in ( M, g, K ) with respect to ν t where ν = ν . Ascomputed in e.g. [5], ∂θ + K ∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 = − ∆ φ + 2 h X K , ∇ φ i + (cid:16) Q K − | X K | + div X K (cid:17) φ (2.1)where h · , · i is the induced metric on Σ, ∆, ∇ , and div are the Laplacian, thegradient, and the divergence on (Σ , h · , · i ), respectively. Moreover, X K is thevector field on (Σ , h · , · i ) dual to the 1-form K ( ν, · ) | Σ and Q K is the functiongiven by Q K = 12 R Σ − ( µ K + J K ( ν )) − | χ + K | , where R Σ is the scalar curvature of (Σ , h · , · i ). A MOTS Σ is said to be stable provided there exists a normal variation of Σ of the sort just described such that0 ≤ ∂θ + K ∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 with φ >
0; cf. [4, 5]. This notion leads to a stability inequality analogous to thatfor minimal surfaces; cf. [15]. A notion of stability for free boundary (or, moregenerally, capillary ) MOTS has been introduced in [1].Assume now that Σ separates M . We denote by M + the region in M thatis bounded by Σ and towards which ν is pointing. A MOTS Σ is said to be outermost in ( M, g, K ) if there are no weakly outer trapped surfaces ( θ + K ≤
0) in M + that are homologous to Σ. Also, Σ is said to be weakly outermost if there areno outer trapped surfaces ( θ + K <
0) in M + homologous to Σ. Finally, Σ is locallyoutermost if there exists a neighborhood V of Σ in M such that Σ is outermost in M + ∩ V . Similarly we can define the notion of a locally weakly outermost MOTS.We now state the local rigidity result for MOTS from [13] mentioned in theintroduction.
Theorem 2.1 ([13], Theorem 3.1) . Let ( M, g, K ) be an n -dimensional, n ≥ ,initial data set that satisfies the DEC, µ K ≥ | J K | . Suppose that Σ is a connectedlocally weakly outermost MOTS in ( M, g, K ) which does not support a metric ofpositive scalar curvature. There is a neighborhood U ∼ = [0 , δ ) × Σ of Σ in M suchthat the following hold: (ii) Σ t = { t }× Σ is a MOTS for every t ∈ [0 , δ ) . In fact, each Σ t has vanishingoutward null second fundamental form. NITIAL DATA RIGIDITY RESULTS 7 (iii) Σ t is Ricci flat with respect to the metric induced from g for every t ∈ [0 , δ ) . (iv) µ K + J K ( ν ) = 0 for every t ∈ [0 , δ ) , where ν is the unit normal of Σ t pointing towards increasing values of t . Theorem 2.1 implies the following result related to the topology of apparenthorizons.
Corollary 2.2.
Let ( M, g, K ) be an n -dimensional, n ≥ , initial data set satisfy-ing the DEC, µ K ≥ | J K | . If Σ is a possibly disconnected locally outermost MOTSin ( M, g, K ) , then Σ admits a metric of positive scalar curvature. We will need to make use of the following fundamental existence result forMOTS. It was obtained by L. Andersson and J. Metzger [6] in dimension n = 3 andthen, using different techniques, by the first-named author [10, 11] in dimensions3 ≤ n ≤
7. Both approaches are based on an idea of R. Schoen [20] to constructMOTS between suitably trapped hypersurfaces by forcing a blow up of the Jangequation. See also [3] for a survey of these existence results.
Theorem 2.3 ([6, 10, 11]) . Let ( M, g, K ) be an n -dimensional, ≤ n ≤ ,compact-with-boundary initial data set. Suppose that the boundary ∂M can beexpressed as a disjoint union of hypersurfaces, ∂M = Σ in ∪ Σ out , such that θ + K ≤ along Σ in with respect to the unit normal pointing into M , and θ + K > along Σ out with respect to the unit normal pointing out of M . Then there is an outermostMOTS in ( M, g, K ) homologous to Σ out . Finally, as noted in the introduction, some of our results require a convexitycondition on the second fundamental form. We say that a symmetric (0 , P is ( n − -convex if, at every point, the sum of the smallest ( n −
1) eigenvaluesof P with respect to g is non-negative. In particular, if P is ( n − Σ P ≥ ⊂ M . Such a convexity condition in thecontext of initial data sets has previously been considered by the third-namedauthor in [18]. 3. The Weakly Outermost Condition
In this section, we assume that (
M, g, K ) is a compact initial data set whoseboundary ∂M can be expressed as a disjoint union of hypersurfaces, ∂M = Σ ∪ S .While we do not assume a priori that M is topologically a product, we need to NITIAL DATA RIGIDITY RESULTS 8 impose a more general condition of a similar flavor. We say that M satisfies the homotopy condition with respect to Σ provided that there exists a continuousmap ρ : M → Σ such that ρ ◦ i : Σ → Σ is homotopic to id Σ , where i : Σ ֒ → M is the inclusion map. This condition implies that Σ is connected. (Recall that M is connected by assumption.) Note that this homotopy condition is satisfied ifthere exists a retraction of M onto Σ .Also, as suggested by the discussion of the positive mass theorem in the intro-duction, it is natural to assume that Σ is in a special class of manifolds that donot support metrics of positive scalar curvature.Let N be a closed orientable manifold of dimension m , 2 ≤ m ≤
7. Assumethat there are classes ω , . . . , ω m ∈ H ( N, Z ) whose cup product ω ⌣ · · · ⌣ ω m ∈ H m ( N, Z ) is nonzero . (3.2)R. Schoen and S.-T. Yau proved in [21] that such a manifold N does not admita metric of positive scalar curvature. They have recently extended their result toall dimensions m ≥ m -dimensional manifold N satisfies the coho-mology condition if there are classes ω , . . . , ω m ∈ H ( N, Z ) such that (3.2) holds.Note that every manifold diffeomorphic to T m or, more generally, to T m Q where Q is closed and orientable, satisfies this condition. Lemma 3.1.
Let M be a compact orientable n -dimensional manifold with bound-ary, n ≥ . Suppose that the boundary ∂M can be expressed as a disjoint unionof hypersurfaces, ∂M = Σ ∪ S . If M satisfies the homotopy condition with re-spect to Σ and Σ satisfies the cohomology condition, then every hypersurface Σ homologous to Σ in M satisfies the cohomology condition. In particular, by the preceding discussion, Σ does not support a metric ofpositive scalar curvature.
Proof.
Let ρ : M → Σ be a continuous map such that ρ ◦ i ≃ id Σ , where i : Σ ֒ → M is the inclusion map. Let ω , . . . , ω n − ∈ H (Σ , Z ) be cohomologyclasses such that ω ⌣ · · · ⌣ ω n − = 0. Let j : Σ ֒ → M be the inclusion map.Define σ = ρ ◦ j : Σ → Σ and observe that σ induces a map from H n − (Σ , Z ) to NITIAL DATA RIGIDITY RESULTS 9 H n − (Σ , Z ) that satisfies σ ∗ [Σ] = ρ ∗ ( j ∗ [Σ]) = ρ ∗ ( i ∗ [Σ ]) = (id Σ ) ∗ [Σ ] = [Σ ] . Then, by standard properties of cup and cap products, σ ∗ ([Σ] ⌢ ( σ ∗ ω ⌣ · · · ⌣ σ ∗ ω n − )) = [Σ ] ⌢ ( ω ⌣ · · · ⌣ ω n − ) = 0 . In particular, σ ∗ ω ⌣ · · · ⌣ σ ∗ ω n − = 0. This concludes the proof. (cid:3) We use the preceding lemma in conjunction with Corollary 2.2 and Theorem2.3 to verify the weakly outermost condition from seemingly weaker assumptions.
Lemma 3.2.
Let ( M, g, K ) be an n -dimensional compact-with-boundary initialdata set, ≤ n ≤ . Suppose that ( M, g, K ) satisfies the DEC, µ K ≥ | J K | ,and that the boundary ∂M can be expressed as a disjoint union of hypersurfaces, ∂M = Σ ∪ S , such that the following conditions hold:(1) θ + K ≤ along Σ with respect to the unit normal that points into M ,(2) θ + K ≥ along S with respect to the unit normal that points out of M ,(3) M satisfies the homotopy condition with respect to Σ , and(4) Σ satisfies the cohomology condition.Then Σ is a weakly outermost MOTS in ( M, g, K ) .Proof. First, we show that Σ is a MOTS. Suppose, by contradiction, that θ + K does not vanish identically on Σ . It follows from [6, Lemma 5.2] that there is ahypersurface Σ obtained as a small perturbation of Σ into M such that θ + K < .Now, let W be the compact connected region bounded by Σ and S in M .Observe that θ + − K ≤ S with respect to the unit normal pointing into W and θ + − K > W .Applying Theorem 2.3 to the initial data set ( W, g, − K ), we obtain an outermostMOTS e Σ in (
W, g, − K ) homologous to Σ. In particular, e Σ is homologous to Σ in M . Then, by Lemma 3.1, ˜Σ does not support a metric of positive scalar curvature.This contradicts Corollary 2.2 applied to the initial data set ( W, g, − K ).We now show that the MOTS Σ is weakly outermost. Suppose, by contra-diction, that Σ is not weakly outermost. Then there is a hypersurface Σ in M homologous to Σ with θ + K < and Σ. Without loss of generality, we NITIAL DATA RIGIDITY RESULTS 10 may assume that each connected component of Σ is homologically nontrivial in M . Then, denoting by W the compact region bounded by Σ and S and applyingTheorem 2.3 for each component of W separately, we obtain an outermost MOTS˜Σ in ( W, g, − K ) that is homologous to Σ. As before, this contradicts Corollary2.2. (cid:3) Proof of Theorem 1.2
As mentioned in the introduction, Theorem 1.2 may be viewed as a globalversion of Theorem 2.1. We emphasize that Theorem 1.2 does not require the weakly outermost assumption.We start with the following observation.
Lemma 4.1.
Under the assumptions of Theorem 1.2, there is a neighborhood U of Σ in M diffeomorphic to [0 , δ ) × Σ for some δ > , such that the leaves Σ t ∼ = { t } × Σ satisfy properties (ii)-(iv) of the conclusion of Theorem 1.2.Proof. By Lemma 3.2, Σ is a weakly outermost MOTS. By Theorem 2.1, eachΣ t satisfies property (ii) of Theorem 1.2 and is Ricci flat with respect to theinduced metric. Using the cohomology condition, together with Poincar´e dualityand the fact that H n ( M, Z ) is torsion free, one sees that Σ t has first Betti number b (Σ t ) ≥ n . Conversely, by a classical result of Bochner, see e.g. [19, p. 208], itholds that b (Σ t ) ≤ n with equality if and only if Σ t is isometric to a flat torus.Hence, property (iii) of Theorem 1.2 holds. Finally, the DEC in conjunctionwith part (iv) of Theorem 2.1 easily implies that property (iv) of Theorem 1.2holds. (cid:3) Proof of Theorem 1.2.
We use ν to denote the unit normal field of the foliation { Σ t } t ∈ [0 ,δ ) of U from Lemma 4.1. Note that the divergence of ν evaluated alongΣ t is equal the mean curvature of Σ t . Since every leaf Σ t is a MOTS, we see thatthe divergence of ν is bounded. By the divergence theorem,vol(Σ t ) = vol(Σ ) + Z U t div( ν )where U t ∼ = [0 , t ] × Σ is the collar between Σ and Σ t . This argument shows thatvol(Σ t ) is bounded independently of t ∈ [0 , δ ).Note that the second fundamental form of each Σ t is bounded independentlyas well, since the null second fundamental form of each Σ t vanishes. NITIAL DATA RIGIDITY RESULTS 11
To proceed, we briefly recall a standard fact. Given
C >
0, there is a smallconstant r >
M, g ) with the following property.Let Σ ⊂ M be a closed and two-sided surface whose second fundamental form isbounded by C . Let p ∈ M be such that Σ ∩ B r ( p ) = ∅ . Then Σ contains thegraph of a function f : { y ∈ R n − : | y | < r } → R with | f (0) | < r , | Df | ≤
1, and | D f | ≤ C where an appropriately rotated geodesic coordinate system centeredat p is used to identify B r ( p ) with the Euclidean ball { x ∈ R n : | x | < r } . Infact, Σ ∩ B r ( p ) is covered by such graphs, all with respect to the same geodesiccoordinate system.It follows from these facts that the leaves { Σ t } t ∈ [0 ,δ ) have a smooth immersed limit Σ δ as t ր δ . Using the weakly outermost condition (ensured by Lemma 3.2)and an idea of L. Andersson and J. Metzer [6], we now argue that Σ δ is embedded.For if not, we can find for every η > t and p ∈ M such that Σ t ∩ B r ( p )contains the graphs of two functions f , f with the properties stated above andsuch that | f (0) − f (0) | < η . In fact, we can arrange for the layer between thesetwo graphs to lie to the outside of Σ t . Arguing exactly as in Section 6 of [6], if η > t across thislayer so as to obtain a surface with non-positive expansion and negative expansionaround the neck. By flowing this surface outward at the speed of its expansionas in Lemma 5.2 of [6], one obtains a surface homologous to Σ t with everywherenegative expansion. This contradicts the fact that Σ is weakly outermost.It is easy to see from the proof of Theorem 2.1 that { Σ t } t ∈ [0 ,δ ] is a foliation.Recall that M and Σ δ ∼ = Σ are connected. By the strong maximum principleas in e.g. [7, Proposition 3.1] or [6, Proposition 2.4], we have that Σ δ = S ifΣ δ ∩ S = ∅ . Note that the assumptions of the theorem continue to hold if wereplace Σ by Σ δ and M by the complement of U in M . The result now followsby a continuity argument. (cid:3) Example 4.2.
We now describe a class of examples that illustrate certain aspectsof Theorem 1.2. For ease of notation, we restrict to 3-dimensional initial data sets.Let L be Minkowski space with standard coordinates t, x, y, z . In the slice t = 0,consider the box B = { ( x, y, z ) : 0 ≤ x ≤ , ≤ y ≤ , ≤ z ≤ } . Let f : B → R be a smooth function which vanishes near the boundary of B and whose graphis spacelike. Let M be the manifold obtained from the graph of f by identifyingopposite sides of B . Note that M ∼ = T . NITIAL DATA RIGIDITY RESULTS 12
Consider the initial data set (
M, g, K ), where g and K are, respectively, the in-duced metric and second form fundamental of graph f in L . Note that ( M, g, K )satisfies the DEC. In fact, µ K = 0 and J K = 0 since L is a vacuum spacetime.Moreover, all the other conditions of Theorem 1.2 are satisfied, where, say, Σ isthe torus obtained by setting t = z = 0. Hence, the conclusions of Theorem 1.2must hold, as well. In fact, by intersecting graph f with the null hypersurfaces H c : t = z + c , we obtain a foliation of M by flat tori with vanishing null secondfundamental forms. This may be roughly understood as follows (cf. e.g. [8, Appen-dix A]). The hypersurfaces H c are totally geodesic null hypersurfaces, i.e. each hasvanishing null second fundamental form with respect to any null vector field K c tangent to H c . Since K c is orthogonal to every spacelike cross section, it followsthat all these cross sections have vanishing null second fundamental form, and,in particular, are MOTS. Moreover, again because H c is totally geodesic, the in-duced metric on every spacelike cross section is invariant under the flow generatedby K c . It follows that any two such cross sections are isometric.This shows that there is still a fair amount of flexibility in the initial datasets covered by Theorem 1.2. As described in Theorem 1.3, by imposing a con-vexity condition on the second fundamental form and slightly different boundaryconditions, a substantially stronger rigidity result can be obtained.5. Proof of Theorem 1.3 and Further Consequences
Theorem 1.3 follows from the following local rigidity result.
Lemma 5.1.
Assumptions as in Theorem 1.3. Then (Σ , g ) is a flat torus, where g is the metric on Σ induced from g . Moreover, there is a neighborhood U of Σ in M such that the following hold:(i) ( U, g ) is isometric to ([0 , δ ) × Σ , dt + e ǫ t g ) , for some δ > .(ii) K = (1 − ǫ ) a dt − ǫ g on U , where a depends only on t ∈ [0 , δ ) .(iii) µ K = 0 and J K = 0 on U .Proof. By assumption, θ − K = H − tr S K ≥ n − ǫ along S , where H is the mean curvature of S with respect to the unit normalpointing out of M . Using also the assumption that K + ǫ g is ( n − NITIAL DATA RIGIDITY RESULTS 13 obtain H ≥ tr S K + 2 ( n − ǫ = tr S K + ( n − ǫ + ( n − ǫ ≥ ( n − ǫ. Therefore, θ + K = H + tr S K ≥ ( n − ǫ + tr S K ≥ S . It follows from Lemma 3.2 that Σ is a weakly outermost MOTS in( M, g, K ). By Theorem 2.1, Σ has a neighborhood U ∼ = [0 , δ ) × Σ in M suchthat the following hold:– We have that g = φ ds + g ( s )on U , where g ( s ) is the metric on Σ ( s ) ∼ = { s } × Σ induced by g .– Every leaf Σ ( s ) is a MOTS. In fact,0 = χ + K ( s ) = K | Σ ( s ) + A ( s ) , where A ( s ) is the second fundamental form of Σ ( s ) in M computed withrespect to the unit normal ν ( s ) in direction of the foliation.– We have that µ K + J K ( ν ( s )) = 0 . Consider now the initial data set (
M, g, P ), where P = − K − ǫ g. Note that (
M, g, P ) satisfies the DEC. In fact, µ P − | J P | = µ K − | J K | +2 ( n −
1) (tr K + n ǫ ) ǫ ≥ K + ǫ g is ( n − θ + K = H + tr Σ K ≤ , where H is the meancurvature of Σ with respect to the unit normal that points into M , we have H ≤ − tr Σ K ≤ ( n − ǫ. Then, θ + P = H + tr Σ P = H − tr Σ K − n − ǫ ≤ . Also, θ + P = θ − K − ǫ ( n − ≥ NITIAL DATA RIGIDITY RESULTS 14 along S . We see from Lemma 3.2 that Σ is a weakly outermost MOTS in( M, g, P ). It follows from Theorem 2.1 that there is a neighborhood U of Σ in M diffeomorphic to [0 , δ ) × Σ for some δ >
0, such that the following hold:– We have that g = φ dt + g ( t )on U , where g ( t ) is the metric on Σ ( t ) ∼ = { t } × Σ induced by g .– Every leaf Σ ( t ) is a MOTS. In fact,0 = χ + P ( t ) = P | Σ ( t ) + A ( t ) , where A ( t ) is the second fundamental form of Σ ( t ) in M computed withrespect to the unit normal ν ( t ) in direction of the foliation.– (Σ ( t ) , g ( t )) is Ricci flat.– We have that µ P + J P ( ν ( t )) = 0 . Decreasing δ >
0, if necessary, we may assume that U ⊂ U . Fix t ∈ (0 , δ )and note that Σ ( s ) ∩ Σ ( t ) = ∅ for some s ∈ (0 , δ ), since Σ ( t ) ⊂ U ⊂ U andΣ (0) ∩ Σ ( t ) = Σ ∩ Σ ( t ) = ∅ . Let s = s ( t ) = inf { s ∈ (0 , δ ) : Σ ( s ) ∩ Σ ( t ) = ∅} and note that Σ ( s ) ∩ Σ ( t ) = ∅ . In particular, s >
0. Also, Σ ( s ) ∩ Σ ( t ) = ∅ for all s ∈ [0 , s ). This means that Σ ( t ) is contained in the region outside ofΣ ( s ).The mean curvature of Σ ( s ) is given by H ( s ) = tr χ + K ( s ) − tr Σ ( s ) K = − tr Σ ( s ) K ≤ ( n − ǫ. For the mean curvature of Σ ( t ), we have the estimate H ( t ) = tr χ + P ( t ) − tr Σ ( t ) P = tr Σ ( t ) K + 2 ( n − ǫ ≥ ( n − ǫ. In particular, H ( s ) ≤ H ( t )so that Σ ( s ) = Σ ( t )by the maximum principle. NITIAL DATA RIGIDITY RESULTS 15
We see that the foliations { Σ ( s ) } s ∈ [0 ,δ ) and { Σ ( t ) } t ∈ [0 ,δ ) are the same afterreparametrization. Below, we will denote this foliation of a neighborhood U ofΣ in M by { Σ( t ) } t ∈ [0 ,δ ) . Note that χ + K = 0 and χ + P = 0 along each leaf Σ( t ). Let ν ( t ) be the unit normal field of Σ( t ) in direction of the foliation, g ( t ) the inducedmetric, A ( t ) the second fundamental form with respect to ν ( t ), and φ the lapsefunction of the foliation. By (2.1), we have that0 = ∂θ + K ∂t = − ∆ φ + 2 h X K , ∇ φ i + (cid:16) div X K − | X K | (cid:17) φ where Q K = 12 R Σ( t ) − ( µ K + J K ( ν ( t ))) − | χ + K | vanishes. Arranging terms as in [15, (2.9)], we obtain thatdiv( X K − ∇ ln φ ) − | X K − ∇ ln φ | = 0 . Integrating both sides of this equation over Σ( t ) and applying the divergencetheorem, we obtain that X K = ∇ ln φ along Σ( t ). By the same argument, we find X P = ∇ ln φ. From the definition of P , X P = − X K µ P = µ K + 2 ( n −
1) (tr K + n ǫ ) ǫ J P = − J K . Thus, µ K = − J K ( ν ( t )) = J P ( ν ( t )) = − µ P = − µ K − n −
1) (tr K + n ǫ ) ǫ and ∇ ln φ = X K = − X P = −∇ ln φ. It follows that | J K |≤ µ K = − ( n −
1) (tr K + n ǫ ) ǫ ≤ ∇ ln φ = 0 . From this, we conclude that µ K = 0 J K = 0 (tr K + n ǫ ) ǫ = 0on U . Moreover, the lapse function φ is constant on Σ( t ) for every t ∈ [0 , δ ). NITIAL DATA RIGIDITY RESULTS 16
Using that0 = χ + K = K | Σ( t ) + A ( t ) and 0 = χ + P = − K | Σ( t ) − ǫ g ( t ) + A ( t ) , we obtain A ( t ) = ǫ g ( t ) = − K | Σ( t ) and thus g = dt + e ǫ t g . Using also that K ( ν ( t ) , · ) | Σ( t ) = 0 since X K = 0, we see that K = a dt − ǫ g ( t )on U . If ǫ = 1, we have tr K = − n . Thus a = − K = − g on U . If ǫ = 0, we use that d (tr K ) = div K (since J K = 0) to see that a is constant onevery leaf Σ( t ). Finally, the same argument as in the proof of Lemma 4.1 showthat (Σ , g ) is a flat torus. This completes the proof. (cid:3) Proof of Theorem 1.3.
Let ℓ = sup { δ : the conclusion of Lemma 5.1 holds with this value of δ > } . Note that ℓ < ∞ since M is compact. Reasoning the embeddedness of the finalsheet as in the proof of Theorem 1.2 (note that the warped product structuresimplifies the discussion of limit sheets), we see that ( M, g ) is isometric to thewarped product ([0 , ℓ ] × Σ , dt + e ǫ t g ). Moreover, we see that (iii) and (iv) ofTheorem 1.3 hold. (cid:3) Proof of Corollary 1.4.
Let K = − ǫ g and note that ( M, g, K ) satisfies the DEC.In fact, a straightforward calculation gives that µ K = 12 ( R + n ( n − ǫ ) ≥ J K = 0 . The expansion θ + K of Σ in ( M, g, K ) computed with respect to the unit normalthat points into M satisfies θ + K = H Σ + tr Σ K = H Σ − ( n − ǫ ≤ . The expansion θ − K of S in ( M, g, K ) computed with respect to the unit normalthat points out of M satisfies θ − K = H S − tr Σ K = H S + ( n − ǫ ≥ n − ǫ. NITIAL DATA RIGIDITY RESULTS 17
We see that (
M, g, K ) satisfies the assumptions of Theorem 1.3, which implies theassertion. (cid:3)
The next theorem establishes a rigidity result under the boundary conditionsof Theorem 1.2, assuming a volume minimizing condition on Σ . Theorem 5.2.
Let ( M, g, K ) be an n -dimensional, ≤ n ≤ , compact-with-boundary initial data. Suppose that ( M, g, K ) satisfies the DEC, µ K ≥ | J K | , andthat the boundary ∂M can be expressed as a disjoint union of hypersurfaces, ∂M =Σ ∪ S , such that the following hold:(1) θ + K ≤ along Σ with respect to the unit normal pointing into M ,(2) θ + K ≥ along S with respect to the unit normal pointing out of M ,(3) M satisfies the homotopy condition with respect to Σ ,(4) Σ satisfies the cohomology condition,(5) K is ( n − -convex, and(6) Σ is volume minimizing in ( M, g ) .Then, (i) (Σ , g ) is a flat torus, where g is the metric on Σ induced from g , (ii) ( M, g ) is isometric to ([0 , ℓ ] × Σ , dt + g ) , for some ℓ > , (iii) K = a dt on M , where a depends only on t ∈ [0 , ℓ ] , and (iv) µ K = 0 and J K = 0 on M . As shown in the following example, Theorem 5.2 fails to hold if one drops eitherthe ( n − Example 5.3.
Let (Σ , g ) be the square flat ( n − M, g ) be thecylinder ([0 , ℓ ] × Σ , dt + e ǫ t g ) and K = − ǫ g , where ǫ = − ǫ = 1. Thesecond fundamental form of Σ t = { t } × Σ in ( M, g ) with respect to the unitnormal in direction of increasing values of t is given by A ( t ) = ǫ e ǫ t g . Then( M, g, K ), 3 ≤ n ≤
7, satisfies all the assumptions of Theorem 5.2 except for thevolume minimizing assumption in the case where ǫ = − n − ǫ = 1. Proof of Theorem 5.2.
It follows from Lemma 3.2 that Σ is weakly outermost.Then, by Theorem 2.1, there exists a neighborhood U of Σ in M diffeomorphicto [0 , δ ) × Σ for some δ >
0, such that:
NITIAL DATA RIGIDITY RESULTS 18 – We have that g = φ dt + g ( t )on U , where g ( t ) is the metric on Σ( t ) ∼ = { t } × Σ induced from g .– Every leaf Σ( t ) is a MOTS. In fact, χ + K ( t ) = K | Σ( t ) + A ( t ) = 0 , where A ( t ) is the second fundamental form of Σ( t ) in ( M, g ).– (Σ( t ) , g ( t )) is Ricci flat.– We have that µ K + J K ( ν ( t )) = 0 , where ν ( t ) is the unit normal field on Σ( t ) in direction of increasing values of t .Since tr Σ( t ) K ≥
0, we have H ( t ) ≤ H ( t ) + tr Σ( t ) K = tr χ + K ( t ) = 0where H ( t ) is the mean curvature of Σ( t ) in ( M, g ). The first variation formulafor the volume of (Σ( t ) , g ( t )) gives ddt Vol(Σ( t ) , g ( t )) = Z Σ( t ) φ H ( t ) d vol g ( t ) ≤ . In particular, Vol(Σ( t ) , g ( t )) ≤ Vol(Σ , g )(5.3)for every t ∈ [0 , δ ). Since Σ is volume minimizing by assumption, we obtainVol(Σ( t ) , g ( t )) = Vol(Σ , g )for all t ∈ [0 , δ ). Then, by (5.3), we have H ( t ) = 0, which implies tr Σ( t ) K = 0,for each t ∈ [0 , δ ). Therefore θ + K = θ − K = 0 along Σ( t ), for each t ∈ [0 , δ ).As in the proof of Lemma 5.1, the first variation of θ + K gives that X K = ∇ ln φ onΣ( t ). On the other hand, the first variation of θ − K = θ + − K gives that X − K = ∇ ln φ on Σ t . Proceeding as in the proof of Lemma 5.1, we obtain the following localrigidity :– (Σ , g ) is a flat torus, where g is the metric on Σ induced from g .– ( U, g ) is isometric to ([0 , δ ) × Σ , dt + g ).– K = a dt on U , where a depends only on t ∈ [0 , δ ).– µ K = 0 and J K = 0 on U . NITIAL DATA RIGIDITY RESULTS 19
Observe that Σ( t ) is also volume minimizing in ( M, g ). The assertion follows fromthis local rigidity as in the proof of Theorem 1.3. (cid:3) Embedding of the initial data into Minkowski space
In this section we show how, under the assumptions of Theorem 1.3, (
M, g ) canbe isometrically embedded into a quotient of the Minkowski spacetime in a suchway that K is exactly its second fundamental form. This, together with Theorem1.3, characterizes the geometry – both intrinsic and extrinsic – of the initial dataset ( M, g, K ) under natural conditions. The same holds under the assumptions ofTheorem 5.2.
Theorem 6.1.
Assumptions as in Theorem 1.3 or Theorem 5.2. There is anisometric embedding of ( M, g ) into a quotient of Minkowski space in a such waythat K is its second fundamental form. Consider the Minkowski spacetime R n of dimension n + 1, i.e. R × R × R n − with the Lorentzian metric g M = − dt + dr + d x where d x is the standard Euclidean metric on R n − .Given a smooth function t : R → R , define r : R → R by r ( s ) = Z s q t ′ ( σ ) dσ. Consider the spacelike hypersurface N = { ( t ( s ) , r ( s ) , x ) : s ∈ R , x ∈ R n − } ⊂ R n . Note that ( N , h ) is isometric to ( R × R n − , ds + d x ) where h is the metricinduced by g M . Straightforward calculations show that the second fundamentalform of N in R n with respect to ∂∂s is given by P = b ds , where b : R → R is thefunction b = t ′′ √ t ′ . Now, consider the hyperbolic space H n of dimension n , that is, the n -manifold R + × R n − endowed with the metric g H = 1 x ( dx + d x ) . NITIAL DATA RIGIDITY RESULTS 20
Using the change of variables x to s = − ln x , we may write g H = ds + e s d x . Thus ( N , h ) = ( R × R n − , ds + e s d x ) is isometric to hyperbolic space.Consider the hypersurface H = { ( t, r, x ) ∈ R × R × R n − : − t + r + | x | = − , t < } , where | x | = x + · · · + x n − . Recall that H with the metric induced from g M isisometric to H n and that the second fundamental form of H in R n with respectto the future directed unit normal is given by − g M | H . Proof of Theorem 6.1.
From Theorem 1.3 and Theorem 5.2, we know that (
M, g )is isometric to ([0 , ℓ ] × Σ , ds + e ǫ t g ), where (Σ , g ) is a flat torus and K =(1 − ǫ ) a dt − ǫg for some function a : [0 , ℓ ] → R . Therefore, ( M, g ) is isometric toa quotient of ([0 , ℓ ] × R n − , ds + e ǫ t d x ). In the case where ǫ = 0, we can take t : [0 , ℓ ] → R to be the solution of t ′′ √ t ′ = a with initial condition t (0) = 0 and t ′ (0) = 0. Therefore, identifying ( N , h ) with( R × R n − , ds + d x ), it follows from the above remarks that we can embed ( M, g )into a quotient of R n in a such way that the second fundamental form of M isgiven by P = a ds = K . In the case where ǫ = 1, it suffices to identify ( N , h )with ( H, h ), where h is the metric on H induced from g M . (cid:3) References
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Faculty of Mathematics, University of Vienna, Vienna, Austria
E-mail address : [email protected] Department of Mathematics, University of Miami, Coral Gables, FL, USA
E-mail address : [email protected] NITIAL DATA RIGIDITY RESULTS 22
Instituto de Matem´atica, Universidade Federal de Alagoas, Macei´o, AL, Brazil
E-mail address ::