Outer trapped surfaces are dense near MOTSs

We show that any vacuum initial data set containing a marginally outer trapped surface S and satisfying a"no KIDs"condition can be perturbed near S so that S becomes strictly outer trapped in the new vacuum initial data set. This, together with the results in [9], gives a precise sense in which generic initial data containing marginally outer trapped surfaces lead to geodesically incomplete spacetimes.


Introduction
In [8] results were obtained concerning the topology of three-dimensional asymptotically flat initial data sets (S , g, K). It is proven there that if the initial data 3-manifold S is not diffeomorphic to R 3 then it contains an immersed marginally outer trapped surface (MOTS), cf [8, theorem 4.1]. Recall that a MOTS in an initial data set (S , g, K) is a closed embedded two-sided hypersurface S in S such that the null expansion θ + with respect to one of the two null normal fields + to S vanishes, θ + = 0. An immersed MOTS is the image under a finite covering map p :Ŝ → S of a MOTSŜ in the initial data set (Ŝ ,ĝ,K), whereĝ and K are the pullbacks via p of g and K, respectively. By considering trivial covers, we see, in particular, that MOTSs are immersed MOTSs. Simple examples of immersed MOTSs that are not MOTSs are described in [8].
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A basic point of view espoused in [8] is that conditions on an initial data set that imply the existence of an immersed MOTS (such as theorem 4.1 in [8]) should be viewed as an 'initial data singularity result'. This view is justified by showing that a Penrose type singularity theorem holds for immersed MOTS cf, theorem 3.2 (for MOTSs) and corollary 3.5 (for immersed MOTSs) in [8]. By covering arguments the proofs can be reduced to the case of a MOTS that separates a Cauchy surface. In this situation there are several conditions on a MOTS that lead to incompleteness, as indicated in the following proposition (see [6] for related results). Proposition 1.1. Let (M , g) be a spacetime of dimension 3 such that the following hold.
(i) M satisfies the null energy condition, Ric(X, X ) = R i j X i X j 0 for all null vectors X, and admits a noncompact Cauchy surface S . (ii) There exists a closed, connected hypersurface S ⊂ S that is separating: For the sake of completeness, we indicate the proof of proposition 1.1, together with relevant definitions, in section 3.
The generic condition in point (iii)(a) above requires that there be a non-zero tidal acceleration somewhere along a null geodesic η as in the statement of the theorem. More precisely, it requires that there be a point p on η and a vector X at p orthogonal to η such that g(R(X, η )η , X ) = 0, cf [3,10]. The condition appears in the classical Hawking-Penrose singularity theorem [10, section 8.2, theorem 2], where it is required to hold along all inextendible causal geodesics. While this condition may seem physically reasonable, mathematically it is rather unsatisfactory, especially from an initial data point of view. For example, it is not known whether the maximal globally hyperbolic development of generic vacuum initial data satisfies the generic condition, even in the weak version of point (iii) (a) above.
Similarly, it is not known whether the remaining alternative conditions of point (iii) are satisfied for generic initial data sets with MOTS.
The purpose of this paper is to resolve the issue by showing that, given a vacuum initial data set (S , g, K) that contains a MOTS S and has no local Killing Initial Data (KIDs) near S, there exists an arbitrarily small perturbation of the data in a neighborhood of S which gives rise to a new vacuum initial data set (S , g , K ) in which S becomes strictly outer trapped.
We proceed to a more detailed statement. Given an open set , let K denote the kernel of the map The equations P * (Y, N) = 0 are called the vacuum KID equations on , and their solutions are called KIDs. We will say that there are no local KIDs near a set S if K = {0} for every open neighborhood of S. We note that the condition of non-existence of local KIDs is a generic property of several families of vacuum initial data sets, including the asymptotically flat ones [4, theorem 1.3]. In particular, every vacuum asymptotically flat smooth initial data set is the limit of a sequence of smooth such initial data sets, each of which has no local KIDs 3 . We have.  ) can be chosen to be vacuum on , respectively to satisfy J i = 0 on .
The proof of theorem 1.2 can be found in section 6. It should be pointed out that in theorem 1.2 we allow any value of the cosmological constant .
For simplicity we have assumed smoothness of all fields at hand, but an identical result holds for metrics with finite, sufficiently high degree of differentiability.
It is clear that an analogue of the vacuum result above holds for specific matter models, e.g. Einstein-Maxwell, or Einstein-scalar field equations, but definite claims seem to require a case-by-case analysis which we have not attempted to carry out.
As should be clear from the proof of proposition 1.1, theorem 1.2 (together with a covering argument from the proof of [8, theorem 3.2] when the MOTS is non-separating) implies the following: let (S , g, K) be a vacuum initial data set, with S noncompact, and suppose S is a MOTS in (S , g, K), without local KIDs near S. Then there exists an arbitrarily small smooth local perturbation of the initial data to a new vacuum initial data set (S , g , K ) whose maximal globally hyperbolic development is null geodesically incomplete. Or, to put in more colloquial terms, the maximal globally hyperbolic development of the vacuum initial data set (S , g, K) is on the verge of being null geodesically incomplete, if not already so.
An outline of the proof of theorem 1.2 might be in order: let S be a MOTS within an initial data (S , g, K) as in the theorem. We wish to show that there exists an arbitrarily small deformation of the initial data so that S will be outer-trapped. To accomplish this, as a first step we use the inverse function theorem and the conformal method for solving the constraints to produce initial data in a small neighborhood of S with unchanged energy-momentum content by imposing suitable Robin boundary conditions for the conformal factor. Supposing there are no KIDs on this neighborhood, when S is part of the boundary of S , a gluing smoothes-out the deformation near the other boundary of the neighborhood, preserving the outer-trapping condition on S. When S is an interior submanifold, a subsequent deformation-and-smoothing argument in the other direction provides the desired initial data.

Definitions
Let S be a spacelike hypersurface in (M , g), and consider a submanifold S ⊂ S of codimension 1. Assume that S is two-sided in S , which means that there exists a globally defined field m of unit normals to S within S . There are actually two such fields, m and −m, we arbitrarily choose one and call it outer pointing. We let H denote the mean extrinsic curvature of S within S , where D is the Levi-Civita connection of the Riemannian metric g induced by g on S . Now, in the discussion above we have assumed that S is embedded into a Lorentzian spacetime (M , g), so that the objects defined in this section have a clear spacetime meaning. However, no such embedding is actually necessary for our purposes, and a triple (S , g, K) by itself suffices to define all the quantities relevant for our argument. This will be our framework for theorem 1.2.
We say that S is outer-future trapped if where K is the extrinsic curvature tensor of S in M . The submanifold S is said to be a marginally future-outer trapped surface, or simply MOTS, or a MOTS, if the inequality in (2.2) is an equality. Finally, S is said to be weakly outer trapped if strict inequality '<' in (2.2) is replaced by ' '. We emphasize that marginally trapped surfaces are assumed to be compact throughout this paper.

Proof of proposition 1.1 (iii)(a):
We start by noting that the conclusion of proposition 1.1 follows immediately from the proof of [8, theorem 3.2] when (iii)(a) is replaced by the requirement that the genericity condition holds along all null geodesics in the direction of + . Now, in the absence of the generic condition one has a rigidity statement: if all null geodesics in the direction of + are future complete then they form a totally geodesic null hypersurface emanating from S see [8, theorem 7.1]. The claim easily follows from this.
(iii)(b): Assume that S is a strictly stable MOTS in S , by which we mean that the principal eigenvalue of the MOTS stability operator L of [1, equation (1)] is strictly positive. Then S can be perturbed within S to a strictly outer trapped (θ + < 0) surface, and the future null geodesic incompleteness of (M , g) follows (cf [6]).
(iii)(c): We define the null second fundamental form of S, say χ + , as where X, Y are tangent to S. Assume that χ + does not vanish identically. Then S can be perturbed to an outer trapped surface in a slight deformation S of S : indeed, as follows from Raychaudhuri's equation, by pushing S an arbitrarily small amount along the future directed null normal geodesics of S we obtain a surface S ⊂ S which is weakly outer trapped, θ + 0, with strict inequality at some points. Then S can be perturbed outward in S to a strictly outer trapped surface (cf [2, lemma 5.2]), and the result follows. We note that when (iii)(a) holds, then at least one null geodesic in the direction of + is future incomplete. To see that this statement does not hold in general, consider a Cauchy surface S in the extended Schwarzschild spacetime. Then S meets the event horizon with respect to one of the asymptotically flat ends in a MOTS S. All outward future directed null normal geodesics to S, which correspond to the generators of the horizon, are future complete.

Conformal deformations
We consider the usual conformal approach to constructing solutions of the constraint equations. Given initial data (g, K), let L denote the trace-free part of K, and let τ be the trace of K: where n = dim S . For any function φ and traceless symmetric tensorL i j we set 3) and in applications that we have in mindL i j will take the form We define the momentum vector J of ( g, K) by the formula The energy density μ of a pair ( g, K) is defined as where is the cosmological constant. If μ and the matter-current J i have been prescribed (e.g. in vacuum μ = 0 = J i ), we obtain a system of equations forL i j and φ: where (4.10) We will need the linearizations of the partial differential operators above, including the boundary operators to be described in section 4.1. Given (g, K) with energy and momentum (μ, J i ) (possibly, but not necessarily, vanishing), consider a differentiable one-parameter family of solutions (φ(λ),L i j (λ)) of (4.8) and (4.9) with λ-independent ( μ, Differentiating (4.8) and (4.9) at λ = 0 we obtain (4.13) Equation (4.5) leads to fields δL i j of the form which turns (4.12) and (4.13) into an elliptic system for (δφ, δY ).
By assumption φ = 1 satisfies (4.9) with μ = μ, which gives It follows that (4.13) can be rewritten as Taking into account (4.14), equations (4.12) and (4.16) form a homogeneous linear system of PDEs for (δφ, δY ). We will seek solutions (φ(λ),L i j (λ)) so that the MOTS S becomes future outer trapped for λ < 0. This will be done by applying the implicit function theorem to a family of nonlinear boundary value problems, where the solution is driven by appropriate boundary conditions at S, see section 6. To apply the implicit function theorem it suffices to show that the map defined by (4.12) and (4.16) is an isomorphism. Our boundary conditions will be elliptic, which implies that the problem is Fredholm. They will also be self-adjoint, which reduces the problem to checking that the system (4.12), (4.16) has no kernel. The boundary conditions will be introduced in section 4.1, while uniqueness will be established in section 5.

Conformal deformations of θ +
Let θ + be the future outer-expansion of S in an n-dimensional initial data set (M, g, K): (4.17) Recall that K is decomposed as K i j = L i j + τ n g i j , where L i j is traceless, thus tr g K = τ . Under the rescaling g i j = φ − 4 (n−2) g i j as in (4.2), we have As already pointed out, we will be using the implicit function theorem to solve our problem, and so we need to find the linearized boundary operators. For this, given initial data (g, K) consider again a one-parameter differentiable family (φ(λ),L i j (λ)) satisfying As before, let δ of a quantity denote a partial derivative with respect to λ at λ = 0. We find and, of course, all the quantities should be evaluated at S. Equivalently, on S.

Integral identities and uniqueness
We wish to prove uniqueness of the solutions of the linearized boundary problem for the equations for (δφ, δY ) above, with δθ + prescribed at the boundary, when the domain of interest is a sufficiently small collar neighborhood of S. This will be done via integration by parts, using standard functional inequalities. We will be working in an exterior collar neighborhood of S, with the normal m of the previous section pointing toward on S, and therefore it is convenient to choose m to be the inwards-pointing normal to ∂ throughout ∂ . Multiply (4.16) by δφ and integrate by parts over a set with smooth boundary : (Here, and elsewhere, the Riemannian measure associated to the metric g is used unless explicitly stated otherwise.) Similarly multiply (4.12) by δY j and integrate by parts to obtain where Z j = 2 (n−2) (8π (n + 1)J j + (n − 1)D j τ ). Let S ⊂ ∂ be a compact hypersurface in a Riemannian manifold (S , g). Let (x, y A ) be a Gauss coordinate system near S, 3) with S given as {x = 0}, and y A being local coordinates on S propagated to a tubular neighborhood of S 0 along geodesics normal to S. Let = [0, ] × S, for some > 0 that will be chosen shortly. For reasons that will become apparent in section 6, we will seek solutions of (4.12) and (4.16) with the boundary conditions, in the adapted coordinates (x, 4) where α, β and η are smooth functions and where m = ∂ x , consistently with our previous notation. On the remaining part of ∂ we will assume δY | { }×S = 0, δφ| { }×S = 0. (5.5) The integral identities (5.1) and (5.2) and the boundary conditions (5.4) and (5.5) with η = 0 lead to Adding, we obtain Let h be the x-independent Riemannian metric, Recall the Poincaré inequality, with some constant C, for all smooth functions δφ and vector fields δY satisfying (5.5) with = 1 there. To make things clear we wrote |δY | h and |∂ x δY | h for the norm of the vector fields δY = δY i ∂ i and ∂ x δY := (∂ x δY k )∂ k in the metric h, but in what follows we will simply write |δY | instead of |δY | h , etc. In (5.9) it is convenient to use the Riemannian measure associated with h, as then by scaling we obtain for a constant C which can be chosen independently of . To see this, note that Using our boundary conditions on ∂ , we find Inserted into (5.12) this gives The boundary term above may be handled as follows; here it is convenient to use the product h-measure. With the current boundary conditions we have This can be estimated in modulus by with some other constant C . As before, replacing C by a larger constant if necessary, (5.14) remains true when the measure of h is replaced by that of g.
Using the above estimate and (5.10) in (5.13), (5.11) easily follows for small enough. We conclude that for some constant C.
To close the inequalities it remains to estimate the integrals over S in (5.8). In fact, one of those integrals has just been estimated above, and the remaining ones may be handled in an essentially identical manner. Using (5.10), (5.15) and these boundary estimates it is straightforward to show now that, choosing small enough, the right-hand side of (5.8) is dominated by one-half of the left-hand side. One concludes that δφ = δY = 0, whence uniqueness.

Proof of theorem 1.2
We start with a preliminary result. Furthermore, for every k ∈ N the fields ( g s , K s ) converge to (g, K) in C k ( ) as s approaches zero.
Proof. Given a function φ and a vector field Y let ( g, K) be given by (4.2)-(4.5) with, as elsewhere, K i j = L i j + τ n g i j . Let β ∈ (0, 1), k ∈ N. Consider a Banach manifold E 1 ⊕ E 2 defined as follows: as E 1 we take the space of all functions in C k+2,β (¯ ) equal to one on { } × S. As E 2 we take the space of C k+2,β (¯ ) sections of the bundle of vectors tangent to S over satisfying the boundary conditions in adapted coordinates as in (5.3). Let be the differentiable map which to (φ, Y ) ∈ E 1 × E 2 assigns the energy and momentum ( μ, J i ) of the pair ( g, K), as in (4.6)-(4.7), and the outer-future expansionθ + of S in ( g, K).
The derivative of C with respect to (φ, Y ) at φ = 1 and Y = 0 is the operator which has been studied in section 5, and has been shown to be injective there when is chosen small enough. This linear operator is formally self-adjoint. It is Fredholm by standard theory (see, e.g., the a priori estimates of [9, theorem 6.30], and the comments at the end of section 6.7 there) hence a linear isomorphism. We can apply the inverse mapping theorem [11, theorem 5.9] to conclude that C is a diffeomorphism near (g, K). The pair ( g s , K s ) with the required properties is obtained as the image by C −1 of (μ, J i , s).
We are ready to pass to the proof of our main result.
Proof of theorem 1.2 Let > 0. Suppose, first, that S is a component of the boundary of S . Let s i 0 and, replacing by a smaller number if necessary, consider the corresponding sequence of data ( g s i , K s i ) → (g, K) on a small tubular neighborhood of S, as given by proposition 6.1. Using the results in [5] (compare [7]), the hypothesis that there are no local KIDs near S allows us to glue, for i large enough, ( g s i , K s i ) with (g, K), in a way such that the resulting initial data set coincides with (g, K) outside of , and with ( g s i , K s i ) near S, and has the same energy-momentum content (μ, J i ). In particular we haveθ + = s i < 0, and S is outer-trapped in the new initial data set. The energy conditions μ = 0 = J i , or μ 0 and J i = 0, or μ > |J| will be satisfied (at least for i large enough in the last case) by construction, which establishes our claim when S ⊂ ∂S .
Suppose, next, that S is a submanifold of S . The construction just done on the outer side of S makes S outer-trapped in each of the resulting data sets, labeled by s i . Increasing i if necessary, the construction described in [5, section 8.6] allows us to find smooth pairs (g i , K i ) which coincide with the data sets already constructed on the outer side of S, and which are equal to (g, K) outside of a small inner tubular neighborhood of S, with μ and J i unchanged.