On the geometry and topology of initial data sets

On the geometry and topology of initial data sets with horizons

Abstract.

We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set such that the boundary of is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions , a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary induced by . This result may be viewed as a generalization of Galloway and Schoen’s higher dimensional black hole topology theorem [17] to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in [10] in dimension , to higher dimensions.

1. Introduction

One of the interesting features of general relativity is that it does not a priori impose any restrictions on the topology of space. In fact, as was shown in [22], given a compact manifold of arbitrary topology and a point , there always exists an asymptotically flat solution to the vacuum Einstein constraint equations on . However, according to the principle of topological censorship, the topology of the domain of outer communications (DOC), that is the region outside of all black holes (and white holes), should, in a certain sense, be simple. The rationale for this is roughly as follows. Known results [18, 23] suggest that nontrivial topology tends to induce gravitational collapse. In the standard collapse scenario, based on the weak cosmic censorship conjecture, the process of gravitational collapse leads to the formation of an event horizon which shields the singularities from view. As a result, according to the viewpoint of topological censorship, the nontrivial topology gets hidden behind the event horizon, and hence the DOC should have simple topology. There have been a number of results supporting this point of view, the most basic of which establishes the simple connectivity of the DOC in asymptotically flat spacetimes obeying suitable energy and causality conditions [14, 15]. However, all the results alluded to here are spacetime results, that is, they involve conditions that are essentially global in time.

In [10] a result on topological censorship was obtained at the pure initial data level for asymptotically flat initial data sets, thereby circumventing difficult questions of global evolution; see [10, Theorem 5.1]. This result, which establishes, under appropriate conditions, the topological simplicity of -dimensional asymptotically flat initial data sets with horizons, relies heavily on deep results in low dimensional topology, in particular the resolution of the Poincaré and the geometrization conjectures. The aim of the present paper is to obtain some results of a similar spirit, but in higher dimensions. While similar in spirit, the methods we employ here are entirely different. Given an asymptotically flat initial data set which satisfies the dominant energy condition and contains an inner horizon (marginally outer trapped surface), we use Jang’s equation ([34, 2]), and other techniques, to deform the metric to one of positive scalar curvature on the manifold obtained by compactifying the end, such that the metric has a special structure near the horizon. As we shall discuss, one can then use known obstructions to the existence of such positive scalar curvature metrics to obtain restrictions on the topology of the original initial data manifold. Related approaches to the topology of asymptotically flat initial data sets without horizons have been considered in [33, 29]. Here we must overcome a number of difficulties due to the presence of a horizon.

An initial data set for Einstein’s equations consists of an -dimensional manifold , a Riemannian metric on , and a symmetric -tensor on . The energy density and the momentum density of are computed through

The initial data set satisfies the dominant energy condition if

Let be a compact -sided hypersurface in an initial data set . Then admits a smooth unit normal field in . By convention, refer to such a choice as outward pointing. Then the outgoing and ingoing null expansion scalars are defined in terms of the initial data as , where is the partial trace of along and is the mean curvature of , which, by our conventions, is the divergence of along . We call outer trapped if on , while if , is inner trapped. We call a marginally outer trapped surface (MOTS) if , while if , we call a marginally inner trapped surface (MITS). The distinction between MOTS and MITS is only meaningful when a choice, natural or otherwise, has been made between the notions of “outside” and “inside”.

Galloway and Schoen have proved the following extension of Hawking’s black hole topology theorem to higher dimensions.

Theorem 1.1 ([17]).

Let be an -dimensional, initial data set satisfying the dominant energy condition. If is a stable MOTS, in particular if is outermost, then, apart from certain exceptional circumstances, is of positive Yamabe type.

In the situation of this theorem, let denote the induced metric on . The conclusion is then that is conformal to a metric of positive scalar curvature. The “exceptional circumstances” can be ruled out in various ways [17, 16], in particular if is assumed to be strictly stable. See [2] and references therein for the notion of MOTS stability.

The main result of the present paper is an extension of the above theorem for asymptotically flat initial data sets, stating that the positive scalar curvature metric on can be extended to the one-point compactification of the asymptotically flat manifold consisting of the exterior of .

Definition 1.1.

A Cauchy Domain of Outer Communications (CDOC) is an asymptotically flat initial set such that the boundary of , is a collection of MOTSs and/or MITSs and such that contains no MOTSs or MITSs.

The precise form of asymptotic flatness we require is given in Definition 2.1.

Definition 1.1 is meant to capture, strictly on the level of initial data sets, the well known notion of the DOC as the region of spacetime outside of all the black holes (and white holes). More precisely, it is meant to model an asymptotically flat (partial) Cauchy surface within the DOC, with boundary on the event horizon (in the equilibrium case) or perhaps somewhat inside the event horizon (in the dynamic case). Our main theorem, as noted above, establishes the existence of a particular type of positive scalar curvature metric on a CDOC.

Theorem 1.2.

Let be an -dimensional, , CDOC whose boundary is connected and is a strictly stable MOTS. Suppose further that the initial data extends to a slightly larger manifold (which contains and a collar neighborhood of ) such that the dominant energy condition (DEC) holds on , .

Let denote with the asymptotically flat end compactified by a point. Let denote the metric on induced from . Then admits a positive scalar curvature metric

  1. whose induced metric on the boundary is conformal to a small perturbation of , and

  2. is a Riemannian product metric in a collar neighborhood of .

Remark 1.2.

Theorem 1.2 is stated for simplicity for a CDOC with a single outermost MOTS . The analogous statement for the case where is a collection of outermost MOTSs and MITSs can easily be proved along the same lines.

Remark 1.3.

The proof in the case of Theorem 1.2 requires a modification from the general case when . This is addressed in Remarks 2.2 and 3.1. The restriction that in Theorem 1.2 is the result of our use of existence results for smooth solutions of Jang’s equation (see in particular Theorems 2.1 and 2.2 below). This restriction is closely related to the partial regularity imposed in higher dimensions by the existence of the Simons cone, a singular area minimizing hypersurface in .

The existence of a positive scalar curvature metric on the compactification of (and its double, which follows immediately from the product structure near the boundary) gives restrictions on the topology of . In Section 4 we discuss such restrictions in more detail.

The black ring spacetime of Emparan and Reall [13], which is an asymptotically flat, stationary solution to the vacuum Einstein equations, illustrates certain features of our results. Let be the closure of a Cauchy surface for the domain of outer communications of the black ring. The boundary of coincides with the bifurcate horizon, which has topology . Further, as shown in [5] (see also [1]), the compactification of has topology . This is consistent with Theorem 1.2, as well as standard results on topological censorship, which require the domain of outer communications to be simply connected. Note that, while is trivial, .

An essential part of the argument is to show that we can specialize to the case in which dominant energy condition holds strictly, . This involves a perturbation of the initial data, as discussed in Section 2. It is here that we need the assumption that is strictly stable.

This paper is a contribution to the long history of results tying the existence of metrics of positive scalar curvature to the analysis of initial data sets in general relativity. One of the earliest and most important examples of this is the transition from Schoen and Yau’s work on topological obstructions to positive scalar curvature metrics [30, 32] to their proof, using minimal hypersurfaces, of the positive mass theorem [31, 34]. The results here, like Theorem 1.1, make strong ties between the dominant energy condition, the presence of marginally trapped surfaces and metrics of positive scalar curvature.

Acknowledgements

The authors wish to thank Michael Eichmair, Lan-Hsuan Huang and Anna Sakovich for many helpful comments on the work in this paper.

2. Deforming to strict dominant energy condition

We start by introducing an appropriate notion of asymptotically flat initial data. We shall use the conventions of [11] for function spaces, which agree with the conventions of [4]. All sections of bundles are assumed to be smooth unless otherwise stated. The following definition is an adaption of [11, Definition 3] to our situation.

Definition 2.1.

Let be an initial data set of dimension . Let be an integer, . Further, let , , , . We say that is asymptotically flat (of type ) if there is a compact set and a diffeomorphism identifying with for some closed ball , for which

(2.1)

where is the standard flat metric on , and

(2.2)

In order to avoid certain technical problems our definition of asymptotic flatness differs from that of [11] by assuming higher regularity. Note that our assumptions imply pointwise estimates for two derivatives of which is not valid under the assumptions of [11]. Note also that the condition (2.2) which is adapted from [11, Definition 3] implies additional fall-off for over that implied by (2.1).

We shall make use of the weighted Sobolev and Hölder spaces in the setting of manifolds with boundary. Definition 2.1 extends immediately to this situation.

The aim of this section is to establish the following perturbation result.

Theorem 2.1.

Let be an initial data set of dimension , , which is asymptotically flat in the sense of Definition 2.1 and such that is a manifold with boundary, whose boundary is a connected strictly stable MOTS. Suppose further:

  1. The initial data extends to a slightly larger manifold (which contains and a collar neighborhood of ) such that the dominant energy condition (DEC) holds on , .

  2. There are no MOTSs or MITSs in .

Then for there is an asymptotically flat initial data such that

and a manifold diffeomorphic to , with MOTS boundary , which is a small (in ) perturbation of , such that the following statements hold.

  1. The dominant energy condition holds strictly on , that is

  2. There exists a smooth solution to Jang’s equation , such that

    (2.3)

    and on approach to .

Remark 2.2.

In the statement of Theorem 2.1 we have excluded the case . The reason for this is that in the proof we are making use of the density theorem [11, Theorem 22], which yields deformed data ) satisfying the strict dominant energy condition, and with the same asymptotic behavior as , in particular , which in case is in general incompatible with having a bounded solution (near infinity) to Jang’s equation. This problem does not arise for in which case for some . In Theorem 2.2 below, which does not rely on the just mentioned density theorem, we have avoided this technical point by including the additional assumption (2.4). In Remark 3.1 below we describe the modifications necessary to prove Theorem 1.2 in the case .

The proof involves several elements. We begin with some comments about Jang’s equation. Schoen and Yau [34] studied in detail the existence and regularity of solutions to Jang’s equation in their proof of the positive mass theorem in the general (not time-symmetric) case. They interpreted Jang’s equation geometrically as a prescribed mean curvature equation, and discovered that the only possible obstruction to global existence are MOTSs in the initial data, where, in fact, the solution may have cylindrical blow-ups.

Given an initial data set , consider graphs of functions in the initial data set of one dimension higher, where , , and is the pullback of to by the projection to . Jang’s equation may then be written as

where is the mean curvature of , with respect to the downward pointing normal, in and is the partial trace of over the tangent spaces of .

The fundamental existence result of Schoen and Yau [34, Proposition 4] for Jang’s equation may now be applied. We also rely on the work of Metzger [25] to allow for an interior barrier, and the regularity theory (up to dimension 7) of Eichmair [7, 8] (see also [12, 9]). This together yields the following existence result for Jang’s equation in our setting.

Theorem 2.2.

Let be an initial data set of dimension , which is asymptotically flat in the sense of definition 2.1. In case , we require that satisfies the additional decay condition

(2.4)

for some . Further, we assume that is a manifold with boundary, whose boundary is a compact connected outer trapped surface.

Then there exist open pairwise disjoint sets , and , with containing a neighborhood of infinity, and an extended-real valued function whose domain includes the union (and is realized as a limit of solutions to the capillarity regularized Jang equation (2.5)) such that

  1. .

  2. on , where contains a neighborhood of , and on .

  3. Each boundary component of is a MOTS (except for ), and each boundary component of is a MITS. (Here “outside” is determined by the outward normal to these open sets.)

  4. is a smooth solution to Jang’s equation such that as , as , and as . The boundary components of are smooth and form a subcollection of the MOTSs and MITSs in point (3).

Remark 2.3.

To prove Theorem 2.2 one considers the capillarity regularized Jang equation

(2.5)

and studies the limit as . This regularized equation satisfies an a priori height estimate that allows one to construct a smooth global solution on such that on the asymptotically flat end (uniformly in ), and as on a fixed neighborhood of ; see [34, 3, 7, 25, 12]. To get smooth convergence up to dimension 7, one applies the method of regularity introduced in the study of MOTSs by Eichmair [7], based on the -minimizing property. By a calibration argument, the graphs obey the -minimizing property (this is true in general for graphs of bounded mean curvature). By the compactness and regularity theory of -minimizers as described in [7, Appendix A], a subsequence of these graphs converges to a smooth hypersurface , consisting of cylindrical components (which occur at the intersection of and , where is not defined) and graphical components which are also -minimizing. It is this hypersurface that determines the open sets , and . Considering translations of the graphical components of gives rise to further cylinders obeying the -minimizing property. The collections of MOTSs and MITSs in part (3) of Theorem 2.2 arise from the intersection of all these cylinders with in . Of further importance to us, as observed in [7], the -minimizing property of these cylinders descends to the collection of MOTSs and MITSs .

2.1. Proof of Theorem 2.1

Since is strictly stable in , assumption (i) of the theorem enables us to construct an enlarged manifold , where is an exterior collar attached to , such that is outer trapped for all and , see Figure 2.1. For more details, see the discussion following Definition 3.1 in [2]. Then is an asymptotically flat initial data set with boundary . On the asymptotically flat end we let denote the radial sphere .

{lpic}

[]FIG1(0.20) \lbl[]230,160; \lbl[]170,85; \lbl[]195,115; \lbl[]150,105;

Figure 2.1. .
Lemma 2.3.

With the assumptions of Theorem 2.1, there exists a sequence of initial data sets such that converges to in and such that the following holds.

  1. The dominant energy condition holds strictly, .

  2. For each data set , is outer trapped for all .

  3. There exists such that for all , is inner trapped () and outer untrapped () with respect to each .

Proof.

The proof of Lemma 2.3 is based on [11, Theorem 22]. By considering the double of , one sees that can be compactly “filled-in” beyond its boundary to obtain a complete manifold without boundary. Extend the data arbitrarily, but smoothly to . Then is an asymptotically flat manifold such that dominant energy condition holds on . Thus, by [11, Theorem 22]1 and the remark following the statement of [11, Theorem 18] concerning higher regularity, there exists a sequence of asymptotically flat initial data sets satisfying

such that part (1) of Lemma 2.3 holds. Moreover, as follows from [11, Equation (40)], for each , can be made sufficiently -close to on the collar so that part (2) holds.

Now consider the null mean curvatures (resp., ) of the coordinate spheres in the initial data set (resp., ). Since the null mean curvatures are polynomials in and its first derivatives, and (and similarly for with respect to and ) the weighted Sobolev embedding provided , implies that

for a constant independent of . This implies that

Since the mean curvature of large spheres falls off linearly with the radius this implies that part (3) of Lemma 2.3 holds. This concludes the proof of the lemma. ∎

We now apply Theorem 2.2 to each initial data set guaranteed by Lemma 2.3. Thus, for each there exist open sets , and and an extended-real valued function as in the theorem. In particular, is a smooth solution to Jang’s equation. Let be the component of containing the asymptotically flat end. We are primarily interested in the smooth solutions . The boundary consists of MOTSs and MITSs . (Consistent with Theorem 2.2, the “outside” is determined by the normal pointing into .) Here we use indices to enumerate the MOTS and MITS components of . We have on approach to the MOTSs and on approach to the MITSs .

Let be the component of containing . From Theorem 2.2, for all . In fact, as in [25], the maximum principle implies that . This implies, in particular, that for all . Moreover, by part (3) of Lemma 2.3 and the maximum principle, , where is the compact region bounded by . By the convergence of the data to on , the sequence obeys a uniform -minimizing property, see Remark 2.3. Hence, by the compactness theory presented in [7, 8] (which provides area bounds, curvature bounds and injectivity bounds), by passing to a subsequence if necessary, the sequence converges in to which is a combination of MOTSs and MITSs in . Note that no component of enters the collar region exterior to in since for all .

We shall now prove that . By the above there is a unique smallest collection of components of surrounding in the sense that separates from infinity, and hence a unique smallest collection of components of surrounding . Since consists of MOTSs and MITSs, and since our assumptions exclude any MOTSs or MITSs in the exterior of , it follows that and that each component of meets at some point.

{lpic}

[]FIG2(0.20) \lbl[]105,120; \lbl[]205,105; \lbl[]170,160;

Figure 2.2. meets at .

Let now be one of the components of , which by the above must meet at some point , see Figure 2.2. Suppose that is a MOTS. Since the outward pointing normal of must agree with that of , the maximum principle implies that in this case and we are done. It remains to consider the case when is a MITS. By construction, is the limit of a sequence of components of the boundary of , each of which is separated from by a part of . Pulling back slightly from the limit, i.e. for very large and for values of sufficiently small, in a small neighborhood of , the capillarity regularized Jang graph, , and its vertical translates come uniformly close to the vertical cylinders over the MITS and a MITS component of inside , which converges to (see Figure 2.3). For very large and values of sufficiently small, this can be seen to lead to a violation of the uniform (in both and ) -minimizing property of these capillarity regularized Jang graphs [7, 8] (for example by gluing in a tube.) Thus, the case that is a MITS meeting is precluded by this -minimizing property. Hence we find that . (We remark that instead of considering the limits and separately, a simultaneous limit in can be taken. Arguing along the same lines as above, and making use of the uniformity of the C-minimizing property with respect to , one concludes that the graphs have a smooth subsequential limit which satisfies the C-minimizing property. This again precludes the MITS meeting .)

Figure 2.3. The capillarity regularized Jang graph for .

It follows from the above that for all sufficiently large , must have only one component which is a MOTS surrounding . In fact, by the conclusions of the compactness results for MOTS in [8] we know that, for all large, must be a graph over . To prove Theorem 2.1 we set and for sufficiently large. It remains to prove the regularity claimed in point (2). This follows by writing Jang’s equation in the form

making use of the fact that the barrier argument used in constructing the solution to Jang’s equation yields for some and using elliptic estimates. This completes the proof of Theorem 2.1.

3. Proof of Theorem 1.2

In this section we prove our main theorem. Assume that is an initial data set of dimension as in Theorem 1.2. For technical reasons we first restrict to the case . The extension of the proof to the case is discussed in Remark 3.1 below.

The proof is broken up into a number of steps.

Step 1: Apply Theorem 2.1 to deform to strict DEC. By Theorem 2.1, may be deformed to an initial data set satisfying the dominant energy condition with strict inequality, , while preserving the MOTS boundary, and so that there is a solution to Jang’s equation which blows up at the MOTS boundary and has no further blow-up. We denote the deformed data set again by .

Let be the graph of , and let be the induced Riemannian metric on . Then is asymptotically flat and near it is asymptotic to the cylinder . From the Schoen-Yau identity [2, Section 3.6] it follows that

(3.1)

for every compactly supported smooth function on .

Step 2: Deform the metric to exactly cylindrical ends. Near the Jang graph is asymptotic to the cylinder , where .

We can write as where . Using the normal exponential map of in the asymptotically cylindrical end can be written as a graph of a function . In [34, Corollary 2] it is proven that for every there is a so that

for and . The result we refer to is stated in dimension , but its proof holds in all dimensions.

By deforming the function to be identically zero for large we can replace by a metric, which we still denote by , such that on , for . Under the deformation of the inequality (3.1) is almost preserved, so we get, for sufficiently large,

(3.2)

for some small and all smooth compactly supported functions .

In each of the remaining steps the metric is replaced by a modified metric , which is then renamed as .

Step 3: Deform the metric to be flat on the asymptotically flat end. Working in the asymptotically flat end of , let denote the difference between and the flat background metric . By construction, the metric on the Jang graph is , where is the solution of Jang’s equation which in our situation satisfies (2.3). It follows that if we write , then .

Let be a smooth cut-off function such that for , for . For asymptotically flat we define where is the Euclidean radial coordinate. This is defined for large and sufficiently far out in the asymptotically flat end, and then extended to all of by inside the end. We have that is supported in the annulus . Let denote the characteristic function of .

For large let

Then we have for and for , while in we have the estimates

In particular, we have that

and hence

where is the function such that . From (3.2) we have

(3.3)

where

is chosen so that the left hand side of (3.3) is conformally invariant. The integrand in the second term on the right hand side of (3.3) can be estimated in terms of . Using we get

(3.4)

for large enough, where

We note that may be negative for due to the contribution from the difference of the scalar curvatures. By construction, we have

for , and

(3.5)

for some constant .

Next we will estimate the right hand side of (3.4) using a weighted Hardy inequality. Let be the smooth cut-off function on defined by where is the function introduced above. The following lemma follows by the standard Hardy inequality (see for example Section 2.1.6. of [24]) applied to , followed by an application of the Cauchy-Schwartz inequality.

Lemma 3.1.

There is a constant depending only on , so that

for all .

If is asymptotically flat we have

for sufficiently large from the identification of the end with . Applying Lemma 3.1 gives the following corollary.

Corollary 3.2.

Let be an asymptotically flat Riemannian manifold of dimension , as in Definition 2.1 (with ). For sufficiently large and , there is a constant depending only on such that

for all .

Fix some large . Corollary 3.2 with gives us the estimate

for . With , where is the parameter in the definition of , the inequality

holds trivially on the region inside (since there) and by (3.5) it holds on the region if we choose sufficiently large. Similarly, keeping fixed and choosing sufficiently large, we have

in , since then on so is positive and independent of there.

Redefining as

we have that by the above choices. Redefining as we get from (3.4) that

(3.6)

for all compactly supported smooth functions . By construction we now have that is flat on the asymptotically flat end and

for large.

Step 4: Conformal compactification of the asymptotically flat end. The flat background metric compactifies to the standard round metric on by the conformal change

where and is the round metric on . Define

in coordinates on the asymptotically flat end and extend to a positive function on all of with on the cylindrical end. Set and let be with the asymptotically flat end compactified by adding a point at infinity. Then is isometric to the standard round metric on in a neighbourhood of the new point at infinity.

The conformal Laplacians and are related by

Further, we have

These equations, together with inequality (3.6) gives us

So with it holds that

for all smooth compactly supported functions on whose support does not contain .

In terms of the spherical radial coordinate at , we have that and . This means that which is compatible with the fact that we made use of the Hardy inequality in the construction of . We can now modify by decreasing its values in a neighborhood of the point at infinity, and thereby replace it by a bounded smooth function which is uniformly positive on . We finally get the inequality

(3.7)

for . A cut-off function argument shows that (3.7) is valid for .

We redefine and as and . This is then a metric with the asymptotically flat end compactified by a point, and an exact cylindrical end, such that