Asymptotically hyperbolic extensions

Asymptotically hyperbolic extensions and
an analogue of the Bartnik mass

Armando J. Cabrera Pacheco Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA. Department of Mathematics, Universität Tübingen, 72076 Tübingen, Germany. cabrera@math.uni-tuebingen.de Carla Cederbaum Department of Mathematics, Universität Tübingen, 72076 Tübingen, Germany. cederbaum@math.uni-tuebingen.de  and  Stephen McCormick Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden. Matematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden. stephen.mccormick@math.uu.se
Abstract.

The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant.

Following the ideas of Mantoulidis and Schoen [16], of Miao and Xie [20], and of joint work of Miao and the authors [6], we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.

Key words and phrases:
Quasi-local mass; asymptotically hyperbolic manifolds; bounded scalar curvature

1. Introduction

In a recent paper of Miao and the authors [6], we constructed extensions of constant mean curvature (CMC) Bartnik data with controlled ADM mass [2] in order to estimate Bartnik’s quasi-local mass [3]. By Bartnik data, we mean a triple consisting of a metric on a surface and a non-negative function (constant and positive in the CMC case) on . An admissible extension in the context of the Bartnik mass is then an asymptotically flat Riemannian manifold with non-negative scalar curvature whose boundary is isometric to with induced mean curvature , and with no closed minimal surfaces enclosing . The estimates in [6] are obtained by constructing examples of admissible extensions with controlled mass; a construction based on the ideas of Mantoulidis and Schoen [16], who recently proved that the Bartnik mass of stable minimal surface is exactly its Hawking mass, and on a collar construction by Miao and Xie [20] in which the growth of the Hawking mass along its level sets is well-controlled. In this article, we give an analogue of both of these results in the asymptotically hyperbolic case.

There is a natural analogue of the Bartnik mass for asymptotically hyperbolic manifolds, which to the best of our knowledge, has received little attention in the literature to date. We would like to remark that such a quantity motivates the work of Bonini and Qing [5] and ongoing work of Martin [18], but as we are unaware of a precise definition in the literature, we give such a definition in Section 2.

Here we give estimates for this asymptotically hyperbolic analogue of the Bartnik mass. The idea is to construct a “collar” manifold with two boundary components; one realising the Bartnik data, while the other is a round sphere with controlled hyperbolic Hawking mass (see (2.1) below). We then smoothly glue this collar to an AdS-Schwarzschild manifold with mass close to the Hawking mass of the end of the collar.

We will give estimates in two cases: First, we discuss the case (when the Bartnik data corresponds to a minimal surface). Then we provide estimates for Bartnik data when is a positive constant, which we refer to as CMC Bartnik data. The main results are indicated in Theorem 1.1 (minimal surfaces) and Theorem 1.3 (CMC Bartnik data). The precise statements are given as Theorem 4.1 and Theorems 5.2 and 5.4, respectively. All necessary definitions will be given in Section 2.

Theorem 1.1.

Let be Bartnik data satisfying or satisfying , where denotes the first eigenvalue of the operator on and denotes the Gaussian curvature of . Then its hyperbolic Bartnik mass satisfies

Remark 1.2.

The quantity on the right-hand side of this inequality is precisely the bound in the conjectured Riemannian Penrose inequality in the asymptotically hyperbolic case. This conjecture states that for an asymptotically hyperbolic manifold with scalar curvature , with outermost minimal surface , we have

where is the hyperbolic mass of the manifold (see Section 2.1). The interested reader is referred to the review article by Mars [17]. This inequality has since been proven in some cases (see for example [11], [12] and [1]), however the general case remains open. If the general case is established, Theorem 1.1 then implies the equality

in analogy with the asymptotically flat case established in [16].

Theorem 1.3.

Let be Bartnik data with Gaussian curvature and a positive constant. Assume that its hyperbolic Hawking mass satisfies

Then its hyperbolic Bartnik mass satisfies

where approaches zero as goes to zero, and approaches zero as tends to a round metric. In particular, the upper bound for the hyperbolic Bartnik mass approaches the hyperbolic Hawking mass as goes to zero or as tends to a round metric.

The precise form of the error term appearing in Theorem 1.3 can be found in Theorems 5.2 and 5.4.

The outline of this paper is as follows. In Section 3, we provide a tool to smoothly glue together two spherically symmetric Riemannian manifolds while keeping the scalar curvature bounded from below. In Section 4, we consider the case where the Bartnik data corresponds to a stable minimal surface (apparent horizon) and prove Theorem 4.1.

In Section 5, we give a collar construction motivated by that of Miao and Xie [20] and study its behaviour with respect to the variation of several parameters. Then we smoothly glue it to a spatial AdS-Schwarzschild manifold to prove Theorems 5.2 and 5.4.

It should be remarked that, while we consider only -dimensional Bartnik data (-dimensional asymptotically hyperbolic manifolds) here, we do not rely on the dimension in any critical way. Miao and the first-named author have shown that the construction method of asymptotically flat extensions in the work of Mantoulidis and Schoen [16] can be extended to higher dimensions in [7]. Provided one can obtain the corresponding smooth path of metrics needed for the collar extension, following this work would naturally lead to a higher dimensional analogue of the work considered here. However, for the sake of exposition, we do not pursue any higher dimensional result here, except in Section 3 where the proofs are nearly identical in higher dimensions and no additional definitions nor concepts are needed.


Acknowledgements All three authors thank the Erwin Schrödinger Institute for hospitality and support during our visits in 2017 in the context of the program Geometry and Relativity. AJCP and CC thank the Carl Zeiss foundation for generous support. The work of CC and SM was partially supported by the DAAD and Universities Australia. The work of AJCP was partially supported by the NSF grant DMS 1452477. The work of CC is supported by the Institutional Strategy of the University of Tübingen (Deutsche Forschungsgemeinschaft, ZUK 63). SM is grateful for support from the Knut and Alice Wallenberg Foundation.

2. Definitions and Facts

From the perspective of general relativity, an asymptotically hyperbolic Riemannian -manifold represents time-symmetric initial data for a gravitating system in the presence of a (negative) cosmological constant. Generally, the cosmological constant is set to , in which case time-symmetric initial data satisfying the dominant energy condition corresponds to asymptotically hyperbolic Riemannian -manifolds with scalar curvature bounded below by . One may view the cosmological constant as a (negative) vacuum energy density. Therefore, if one seeks to measure the quasi-local mass of a bounded domain in such a manifold, the vacuum energy density should somehow be compensated for. As an important example of such a compensation, one may compare the Hawking mass [14]

with the following hyperbolic version that is known in the literature

(2.1)

2.1. Hyperbolic Bartnik mass

The natural analogue of the usual Bartnik mass of Bartnik data is given by the infimum, over a space of “admissible asymptotically hyperbolic extensions” of , of the well-known total hyperbolic mass – see below. We briefly describe the notion of asymptotically hyperbolic Riemannian -manifolds and the mass of such a manifold.

A Riemannian -manifold is called asymptotically hyperbolic if there exists a compact set , a closed ball , and a diffeomorphism from to such that the push-forward of the metric can be written as the hyperbolic metric plus a well-controlled error term. Then, the (total) hyperbolic mass is given by a quantity computed from this error term which turns out to be an invariant under change of diffeomorphism. For precise definitions the reader is referred to X. Wang [23]; see also Chruściel and Herzlich [10]. Since all the extensions considered here will be exactly 3-dimensional spatial AdS-Schwarzschild manifolds (see below) outside a compact set (which are well-known to be asymptotically hyperbolic of mass ), we do not go into any detail here.

Now let be the set of admissible extensions ; namely, of smooth asymptotically hyperbolic -manifolds with scalar curvature , containing no minimal surfaces enclosing the boundary (except possibly the boundary itself), and whose boundary is isometric to with induced mean curvature . We then define the hyperbolic Bartnik mass

where denotes the total hyperbolic mass of .

Remark 2.1.

The condition that an admissible extension must not contain minimal surfaces enclosing the boundary is introduced to rule out extensions where the Bartnik data is hidden behind a horizon (minimal surface). As Bartnik pointed out in his original definition in the asymptotically flat case, extensions like this could have arbitrarily small (positive) mass, rendering the definition somewhat meaningless.

It is worth remarking that if the boundary is outer minimising then, in the asymptotically flat case, it is known that the Hawking mass of the boundary provides a lower bound for the ADM mass via Huisken and Ilmanen’s proof of the Riemannian Penrose inequality [15]. For this reason, sometimes the class of admissible asymptotically flat extensions is further restricted to manifolds where the boundary is outer-minimising. It is conjectured that this restriction does not change the infimum.

Remark 2.2.

Bartnik’s original definition of quasi-local mass assigned a mass to a domain with compact boundary contained in an asymptotically flat manifold – as opposed to assigning the mass to Bartnik data, as described above. The extensions that he considered were asymptotically flat manifolds in which this bounded domain could be isometrically embedded. However, if a minimising extension exists then it will generically fail to be smooth at , which leads one to consider extensions that are not smooth at provided that an appropriate version of the positive mass theorem holds for such a non-smooth manifold. In this case the appropriate positive mass theorem for such a manifold with corner along a hypersurface was established in the asymptotically flat case independently by Miao [19], and by Shi and Tam [21]. More recently, Bonini and Qing [5] have proven a positive mass theorem for asymptotically hyperbolic manifolds with corners.

Therefore, one often considers “extensions” to be asymptotically flat manifolds with boundary satisfying certain geometric boundary conditions [4]. The Bartnik mass is then considered to be a quantity associated with Bartnik data, as described in Section 1, both in the asymptotically flat and the asymptotically hyperbolic setting.

2.2. Spatial AdS-Schwarzschild manifolds

Arguably the most important example of an asymptotically hyperbolic manifold is the spatial AdS-Schwarzschild manifold. This is the asymptotically hyperbolic analogue of the standard spatial Schwarzschild manifold, and represents canonical initial data for a static, spherically symmetric black hole sitting in the presence of a negative cosmological constant, and an otherwise vacuum spacetime. For later convenience, we will introduce a larger class of Riemannian -manifolds with two parameters, and . The AdS-Schwarzschild metric will arise as a special case.

Now consider the family of metrics on given by

(2.2)

where is the largest root of if and otherwise . Here denotes the standard round metric on with area .

The metric is a static solution to the vacuum Einstein constraint equations in the presence of a cosmological constant or “radius” when considered as asymptotic to a hyperboloid. In particular, it has scalar curvature and by special choices of the parameters and , we recover several well-known manifolds: When and , we recover a hyperbolic space of radius ; when and we recover the spatial Schwarzschild metric of mass ; when and we recover the AdS-Schwarzschild metric of mass . We remark that the hyperbolic Hawking mass (2.1) is constant along the centred spheres in the spatial AdS-Schwarzschild manifold just as the usual Hawking mass is constant along the centred spheres in the spatial Schwarzschild manifold.

We will now discuss further properties of the metrics , which will be used later in the collar extensions constructed in Section 5. For any , define

on . Then we can write the metric as

(2.3)

where , defined on , is the inverse of on .

One can directly check that the function satisfies the following properties:

  1. ,

  2. , and

  3. .

In addition, combining (a), (b), and (c) above, one can directly check that satisfies

(2.4)

which is equivalent to the condition , as we will see below.

2.3. Collar extensions

The asymptotically flat extensions constructed in [16] and [6] provide a way to construct admissible extensions (in the sense of the Bartnik mass) with control on the ADM mass for minimal and CMC Bartnik data, respectively. A key step in the construction in the minimal case is to obtain a collar extension of a given in such a way that its initial boundary is isometric to and minimal, while controlling the growth of the area along the collar. For Bartnik data with constant , one is more generally interested in controlling the growth of the Hawking mass along the collar. For our construction, we will apply the general idea first put forward in [16], using appropriate collar extensions as in [6] (see also [20]) in the CMC case.

One main ingredient of the construction of an extension à la Mantoulidis and Schoen [16] is the existence of a path of -metrics connecting a given -metric to a round metric of the same area on and using it to construct a collar extension realising the given Bartnik data as the inner boundary of the collar. More concretely, for a -metric (obeying certain curvature conditions), they prove existence of a path connecting to a round metric while preserving the curvature conditions such that

  1. on for some , and

  2. on ,

where the latter condition implies that the area form is preserved along the path. Existence of such a path under the curvature conditions we will impose is obtained in the minimal surface case using the Uniformisation Theorem as in [16] (see Section 4) and in the CMC case using Ricci flow (see Section 5).

A collar extension is then given by the manifold with metric

(2.5)

where and are smooth positive functions, and satisfies and . Throughout this work, we will use the notation to denote the -level set of . Here, induces the metric on and induces a round metric on for all . For simplicity we avoid writing explicitly the dependence on of various quantities.

A direct computation shows that the scalar curvature of the collar extension is given by

(2.6)

(see [16, 20]). The mean curvature of the -level set along the collar extension is given by

(2.7)

We will be particularly interested in the hyperbolic Hawking mass of the -level sets in the case that is constant on each , which in that case is given by

(2.8)

Expressions (2.8) and (2.6) suggest that by choosing carefully the functions and , one can control the growth of the hyperbolic Hawking mass of the -level sets of the collar extension as well as its scalar curvature. We also note that, as in [6], condition (1) is not necessary as one can approximate by a family of paths with . However, since the the proof of our main result is constructive, we keep imposing this condition. In the following section we provide tools to smoothly glue a spatial AdS-Schwarzschild manifold to a certain type of collar extension, resulting in admissible extensions of the given data . We remark that the hyperbolic mass of such an extension provides an upper bound for .

3. Gluing via an bridge in spherical symmetry

The main results of this paper are obtained by constructing a collar manifold with one boundary component realising the given Bartnik data and then gluing the collar to an exterior AdS-Schwarzschild manifold via an bridge. In this section, we provide the gluing tools to be used in Sections 4 and 5. The following lemma is a natural generalisation of Lemma 2.2 in [16] (see also Lemma 2.1 of [6]), and the proof closely follows the original one. We state and prove it in dimensions as it may be of interest for other applications, although we will only apply it for .

Lemma 3.1 (Smooth gluing lemma).

Let , be two smooth positive functions, let be the standard metric on (), and let be some constant. Suppose that

  1. the metric has scalar curvature ,

  2. ,

  3. and

  4. .

Then, after translating appropriately, there is a smooth positive function on such that

  1. on and on and

  2. the metric has scalar curvature on .

In addition, if on then is foliated by mean convex CMC spheres.

Proof.

First note that it is possible to translate to ensure that as well as the existence of a function satisfying (see [6, Lemma 2.1])

  • ,

  • ,

  • on , and

  • .

Using , we then define the function

which clearly satisfies

  • and ,

  • and ,

  • on , and

  • on .

From the condition and , we have on . On , define

(3.1)

Then clearly , is away from and . We now smooth out using an appropriate mollification (as in [7, 6]):

Let satisfy

Now let be a smooth cut-off function that equals on , vanishes on the set , and satisfies elsewhere. Let be a standard smooth mollifier with compact support in and .

Given any , we define by

(3.2)

Note that is smooth on , on and

(3.3)

As is everywhere and except possibly at and , by standard mollification arguments we have for

(3.4)

using , where . Moreover, for where is smooth, we have

(3.5)

We will now show that for sufficiently small , the metric has scalar curvature . Given a metric of the form with smooth positive , a direct computation shows that the scalar curvature condition for some given is equivalent to (see [7] Eq. (4.13))

(3.6)

Now we define the quantity

as a shorthand for the right-hand side of (3.6). In analogy to the asymptotically flat case, the key to this proof will be the fact that

holds for the function defined by (3.1) and some number . To see this, recall first that . Since on , attains its maximum at some and thus . Moreover on and on . If , since and (iii) holds we know that

and hence on . If , by the previous argument on and since is strictly decreasing on we have , which implies

Combining this with (iv) and the fact that , we also obtain on . Now, as on and on by construction of , and as has scalar curvature on for , we have

(3.7)

on . In fact, can be bounded from below by

(3.8)

as the inequality (3.7) holds separately on the compact intervals and and because is on these intervals, separately. It follows that

everywhere where is defined.

Now we consider in order to show that (3.7) also holds for so that . By (3.2) and (3.3), we have in as . Hence, in , which shows, for sufficiently small ,

Furthermore, by (3.4) and (3.5), and from in , we find

for sufficiently small . It then follows that, for all ,

noting that, in the second-to-last inequality, we also used the uniform continuity of on , and hence for any with provided is small enough.

We have therefore shown that (3.7) and thus (3.6) holds on with replaced by for small on . That is, the metric has scalar curvature and thus can act as the sought after profile function satisfying the conclusions of the lemma for sufficiently small .

In addition, if then , which directly implies as in , proving that the resulting warped product is foliated by mean convex CMC spheres. ∎

In order to apply Lemma 3.1 to smoothly glue an AdS-Schwarzschild manifold to the collars that we will construct in Section 4 ( case) and Section 5 ( case), we need to slightly bend a piece of the AdS-Schwarzschild metric to increase the scalar curvature in order to make room for smoothing out the gluing zone. This is the analogue of Lemma 2.3 in [16], and the proof is very similar.

Lemma 3.2 (Bending Lemma).

Let be a metric on a cylinder , where is a smooth positive function and is the standard round metric on . Assume that the scalar curvature of satisfies for a fixed . Then for any with , there is a and a metric , such that on and on . If in addition, for some positive constant and , then and .

Proof.

First, from (2.5) and (2.6) with and , we know that the scalar curvature of a metric of the form

is well-known (see, for example, [7]) to be given by

(3.9)

Notice that if already holds at , nothing needs to be done and we can set and choose such that in . Otherwise, we seek to bend the metric near the cross-section . We do this with a strictly increasing reparametrisation function such that on some interval , the metric

has scalar curvature , given via (3.9) by

Denoting -derivatives by and -derivatives by , expanding this gives

which can be rearranged to

Now note that the second group of terms is exactly the scalar curvature of the warped product metric we started with, multiplied by . As by assumption, we have

That is, in order to achieve on , we must choose so that

(3.10)

on . For this, it is sufficient to use the same function as is used for bending the Schwarzschild manifold in the asymptotically flat case [16, 6]. Define the translated bump function , such that . For such that and , define

where . This choice of constant ensures that can be smoothly extended to for . With this choice of , (3.10) becomes

(3.11)

for all .

We see that (3.11) is satisfied provided is taken sufficiently small, as the last term on the left-hand side is positive and dominates as goes to zero.

If in addition , then defined by clearly satisfies , taking smaller if necessary. For the last assertion in the lemma, note that

on , which can also be made positive by making smaller if . Then, in this case, is increasing on , which implies . ∎

We now combine the above lemma with the concrete profile of AdS-Schwarzschild manifolds. As we do not intend to discuss the hyperbolic Hawking mass in higher dimensions, we will from now on restrict our attention to .

Proposition 3.3.

Let be a smooth positive function on an interval with , be the standard metric on , and define the metric on . Suppose

  1. has scalar curvature ,

  2. has positive mean curvature, and

  3. .

Then for any , there exists a rotationally symmetric, asymptotically hyperbolic -manifold with inner boundary and scalar curvature , such that