Characterization of large isoperimetric regions in asymptotically hyperbolic initial data

Let $(M,g)$ be a complete Riemannian $3$-manifold asymptotic to Schwarzschild-anti-deSitter and with scalar curvature $R \geq - 6$. Building on work of A.~Neves and G.~Tian and of the first-named author, we show that the leaves of the canonical foliation of $(M, g)$ are the unique solutions of the isoperimetric problem for their area. The assumption $R \geq -6$ is necessary. This is the first characterization result for large isoperimetric regions in the asymptotically hyperbolic setting that does not assume exact rotational symmetry at infinity.


Introduction
The systematic study of stable constant mean curvature spheres in initial data sets for the Einstein equations has been pioneered in the work of D. Christodoulou and S.-T. Yau [11] and of G. Huisken and S.-T. Yau [18]. The existence of the canonical foliation of the end of initial data asymptotic to Schwarzschild-anti-deSitter has been established by R. Rigger [25]. A. Neves and G. Tian [22,23] have shown that the leaves of this foliation are the unique stable constant mean curvature spheres that enclose the center of the manifold and which satisfy a pinching condition that relates their inner and their outer radius. We refer the readers to Appendix A for notation and to Appendix C for a more detailed discussion of these results.
In Theorem 1.1, we observe that the pinching condition used in [22], stated here as (C.2), may be replaced by an integral condition in the form of an a priori bound on their Hawking mass.
Theorem 1.1. Let (M, g) be asymptotic to Schwarzschild-anti-deSitter with mass m > 0. Let Λ > 0. There is a constant r 0 > 1 with the following property. Every stable constant mean curvature sphere Σ in (M, g) that encloses B r 0 and with Hawking mass m H (Σ) ≤ Λ is a leaf of the canonical foliation.
Assume now that R ≥ −6 where R is the scalar curvature of (M, g). The existence of isoperimetric surfaces in (M, g) for every sufficiently large area has been proven by the firstnamed author [7], together with a bound of their Hawking mass. In conjunction with ideas from [19,10], we obtain from Theorem 1.1 our second main result in this paper: A fully global characterization of the leaves of the canonical foliation as the unique large solutions of the isoperimetric problem.
Theorem 1.2. Let (M, g) be a complete Riemannian 3-manifold asymptotic to Schwarzschildanti-deSitter with mass m > 0. We also assume that R ≥ −6 and that ∂M is connected and the only closed H = 2 surface in (M, g). There is V 0 > 1 with the following property. Let Ω ⊂ M be an isoperimetric region of volume V where V ≥ V 0 . Then Ω is bounded by ∂M and a leaf of the canonical foliation. In particular, the solutions of the isoperimetric problem in (M, g) for sufficiently large volumes are unique.
When (M, g) is exactly Schwarzschild-anti-deSitter, this result was proven by J. Corvino, A. Gerek, M. Greenberg, and B. Krummel in [12] building on the pioneering work of H. Bray [1]. When (M, g) is isometric to Schwarzschild-anti-deSitter outside of a compact set, Theorem 1.2 was proven by the first-named author in [7]. It is shown in Section 10 of [7] that Theorem 1.2 fails when the condition R ≥ −6 is dropped: There exist rotationally symmetric (M, g) with outermost H = 2 boundary that are equal to Schwarzschild-anti-deSitter outside of a compact set and in which no large centered coordinate sphere is isoperimetric. Finally, we note that S. Brendle has shown that in exact Schwarzschild-anti-deSitter (or Schwarzschild), the centered coordinate spheres are the unique embedded closed constant mean curvature surfaces [2].
We conclude with a brief account of the available results in the asymptotically flat setting.
The optimal, global uniqueness result for stable constant mean curvature spheres in initial data asymptotic to Schwarzschild has recently been established by the first-and the secondnamed authors in [8,9], building on earlier work of G. Huisken and S.-T. Yau [18], of J. Qing and G. Tian [24], of J. Metzger and the second-named author [14], of S. Brendle and the secondnamed author [4], as well as that of A. Carlotto and the first-and second-named authors [5]. We refer to the introduction of [8] for a comprehensive account and more detailed description of these and other important contributions in this context.
The global uniqueness of large isoperimetric surfaces in asymptotically flat manifolds with non-negative scalar curvature has been established recently in joint work [10] of H. Yu and the first-, second-, and third-named authors. Building on the work of H. Bray [1] for metrics which are exactly Schwarzschild outside of a compact set, global uniqueness of large solutions of the isoperimetric problem in (M, g) asymptotic to Schwarzschild with mass m > 0 has been shown by J. Metzger and the second-named author in any dimension and with no assumption on the scalar curvature [15,16]. These results in [15,16,10] resolve a long-standing conjecture of G. Huisken.
Finally, we note that there are very few geometries where we have complete understanding of the isoperimetric problem in the large. To our knowledge, the results in [15,16,10]  Throughout this section, we let (M, g) be a Riemannian 3-manifold that is asymptotic to Schwarzschild-anti-deSitter with mass m > 0.
We assume that Σ is a stable constant mean curvature sphere in (M, g). The mean curvature of Σ with respect to its outward pointing unit normal ν is denoted by H. We also assume that Σ and ∂M together bound a compact region Ω in M , and that B r 0 ⊂ Ω where r 0 > 1 is a large numerical constant that depends only on (M, g).
is the Hawking mass of Σ. Note that (2.1) is equivalent to either one of the bounds Lemma 2.1. We haveˆΣ Proof. This follows from (B.2) and R + 6 = O(e −5 r ).
Proof. Integrating (B.5) and using the Gauss-Bonnet formula, we obtain The estimate follows in conjunction with (2.3), using that m > 0.  Proof. We use (2.4) to sharpen the steps leading to Proposition 4.2 (iii) in [22]: We denote by Σ δ ⊂ B 1 (0) the surface with Ψ(Σ δ ) = Σ and by Ω δ the compact region enclosed by it. (We recall this notation in Appendix A.) From (2.4), we obtain where we have also used (A.1) and The claim follows from the Euclidean isoperimetric inequality . Corollary 2.5. We haveˆΣ Proof. We use our stronger estimates to sharpen the steps leading to Proposition 4.2 in [22].
Corollary 2.6. We have where we use the notation explained in Appendix A.
Proof. The Euclidean area estimate follows from (2.7) together with (2.11) and (2.4). The Euclidean volume estimate is a restatement of (2.8).
We now rescale Σ homothetically,ĝ wherer > is the hyperbolic area radius The same rescaling is studied by A. Neves and G. Tian in Section 5 of [22]. Instead of their pinching estimate (C.2), we use our bound (2.1) on the Hawking mass to estimate the Gaussian curvature ofĝ Σ and to produce a good conformal parametrization.
The following is now an immediate consequence of Theorem A.1 in [21].
We also consider the conformal rescaling Note that Σ with the Riemannian inner productg Σ is isometric to the Euclidean surface Σ δ ⊂ B 1 (0). Here we use the notation explained in Appendix A.
The conformal rescaling (2.15) is also considered by A. Neves and G. Tian in Section 6 of [23]. They use it in conjunction with a result of C. De Lellis and S. Müller [13] to show that Σ is close to a coordinate sphere in the chart at infinity. In Proposition 2.9 below, we apply results from [13] rather differently to obtain a suitable conformal parametrization ofg Σ .
Proof. We have thatḡ Here we use the bound sup from (B.4) to obtain the second estimate. Using also (2.3) and (2.4), we find The result now follows from Proposition 3.2 in [13].
We recall from [22, p. 929], cf. [6, Lemma 1], the definition of the functional and its conformal invariance: A straightforward computation gives that Note that the conclusion of Theorem 1.1 is equivalent to the bound We establish this bound in three strides.
Proof. In view of (2.17), we have Using conformal invariance and (2.10), we get Using conformal invariance of energy as well as estimates (2.14) and (2.16), we find .14) and (2.16). Putting these estimates together, we obtain (2.18).
Combining (B.3) and (2.5), we deduce From the conformal invariance of the Laplace-Beltrami operator on surfaces, we obtain From this, we verify the asserted bound term by term. First, note that by (2.14) and Sobolev embedding. Using (2.4) and Hölder's inequality, we obtain This bounds the contribution to ∆ S 2 w L p (S 2 ) from the first term in (2.19). To estimate the contribution from the third term, we apply (2.9) to obtain Here we also use that ν, ∇r > 0 by Lemma B.5. The bounds for the other terms follow from these.
Proposition 2.12. Let p be such that 1 < p < 3/2. Then In particular, w is bounded.
Proof. By Lemma 2.10 and Lemma 2.11, we have that Testing the equation with v and using Cauchy-Schwarz and the Poincaré inequality, we obtain It follows that In conjunction with S(w) = O(1), we find Putting these estimates together, we obtain from the Poincaré inequality. Standard elliptic theory now gives (2.20).
Since w is bounded, the pinching condition (C.2) is satisfied, and Theorem 1.1 follows from the uniqueness results in [22].

Proof of Theorem 1.2
In this section, we assume that (M, g) is a complete Riemannian 3-manifold asymptotic to Schwarzschild-anti-deSitter with mass m > 0, that R ≥ −6, and that ∂M is connected and the only closed H = 2 surface in (M, g).
We need a large amount of area to bound a large amount of volume in (M, g). This follows by comparison with hyperbolic space in the chart at infinity together with cut-and-paste arguments. Comparison with hyperbolic space also gives that We may thus use either area or volume to specify large solutions to the isoperimetric problem.
We recall from Lemma E.3 that every large enough isoperimetric region has a unique large component. The residue is a small collar about ∂M by Lemma E.4.
The crucial ingredient for the proof of Theorem 1.2 is the characterization of isoperimetric spheres as leaves of the canonical foliation given in the following proposition. Proposition 3.1. There is A 0 > 1 with the following property. Let Ω be the unique large component of an isoperimetric region for area A ≥ A 0 . Its outer boundary Σ = (∂Ω) \ (∂M ) is connected. If Σ is a sphere, then it is a leaf of the canonical foliation and Ω coincides with the original isoperimetric region.
Proof. The connectedness of Σ follows from the discussion in Appendix D. By Lemma F.1 and Lemma F.3, there is r 0 > 1 such that Ω ∩ B r 0 = ∅ and m H (Σ) ≤ 4 m provided that V > 0 is sufficiently large. Taking r 0 > 1 larger, if necessary, Theorem 1.1 shows that Σ is a leaf of the canonical foliation if B r 0 ⊂ Ω. Theorem 4.3 from [19] rules out the scenario Σ ∩ B r 0 = ∅. Finally, Lemma E.4 shows that Σ and ∂M bound the original isoperimetric region.
We can now complete the proof of Theorem 1.2 following the strategy of Section 9 in [7], which in turn develops an idea of H. Bray [1]. We sketch the full argument from [7], where (M, g) is assumed to be exactly Schwarzschild-anti-deSitter outside of a compact set, below, including the minor but necessary adaptations to our more general setting. Since the strategy is technical, we begin with an outline.
Let {Σ A } A>A 0 be the canonical foliation of (M, g). We use Ω A to denote the compact region bounded by Σ A together with ∂M .
The derivative of is the inverse of the mean curvature H A of Σ A . We obtain an explicit estimate for H A from the expansion of the Hawking mass along the canonical foliation discussed in Lemma C.1. Assume now that I ⊂ (A 0 , ∞) is an open interval such that, for every A ∈ I, there is an isoperimetric region the boundary of whose unique large component has non-zero genus. The derivative of the isoperimetric profile on such an interval I is appreciably smaller than anticipated by (3.1). Integrating up and comparing with the volume enclosed by the centered coordinate sphere S r(A) of area A, cf. Lemma F.4, it follows that the interval I is bounded. From this, we conclude that a maximally extended such interval I far out has the form (A 1 , A 2 ) where Ω A 1 , Ω A 2 are isoperimetric. We can rule out the existence of such intervals by studying the derivative of the isoperimetric profile at the endpoins A 1 , A 2 .
We now proceed to make this argument precise.
It follows from standard compactness properties of isoperimetric regions that is open. Let A ∈ I. By Proposition 3.1, the unique large component of an isoperimetric region for area A has boundary of non-zero genus. Using this input, we estimate the derivative of the isoperimetric profile on connected components of I. Lemma 3.2 (Cf. Lemma 9.1 and Proposition 9.3 in [7]). There is A 0 > 1 with the following property. Let (A 1 , A 2 ) ⊂ [A 0 , ∞) be such that, for every A ∈ (A 1 , A 2 ), there exists an isoperimetric region for area A and the boundary of whose unique large component has non-zero genus. Then Proof. Assume first that the isoperimetric profile is smooth on [A 1 , A 2 ]. Fix A ∈ (A 1 , A 2 ) and let Σ be the boundary of the unique large component of an isoperimetric region whose boundary has area A. Following the proof of Lemma 9.1 in [7], using Lemma F.3 instead of Proposition 8.3 in [7], we obtain Recall from the discussion in Appendix D that the isoperimetric profile is increasing. Dropping the non-negative second term on the right-hand side and absorbing the third term into the first, we arrive at . This gives (3.3) in the special case when the isoperimetric profile is smooth. In the general case, we can argue using weak derivatives exactly as in the proof of Proposition 6.3 in [7] to arrive at the same conclusion.

Lemma 3.3 (Cf. Proposition 3.3 in [7]
). For every A 0 > 1 there is A ≥ A 0 with the following property. There does not exist an isoperimetric region for area A such that the boundary of its unique large component has non-zero genus.
Proof. Assume that the conclusion fails. It follows from Lemma 3.2 that there is A 0 > 1 large such that for all A ≥ A 0 . Indeed, we have (3.5) for all A 1 , A 2 with A 0 < A 1 < A 2 . We now take A 2 → ∞ in (3.3) and use (D.1) and (E.2) to conclude (3.5). Using that A → V g (A) is strictly increasing and absolutely continuous, we conclude that for all A ≥ A 1 . This estimate contradicts (3.2).
Lemma 3.4 (Cf. [7, p. 433]). There can be no A 1 , A 2 as in Lemma 3.2 such that the leaves Σ A 1 and Σ A 2 of the canonical foliation both bound isoperimetric regions.
Proof of Theorem 1.2. Lemma 3.3 shows that I ⊂ (A 0 , ∞) is not connected at infinity. On the other hand, Lemma 3.4 gives that I has no bounded components provided that A 0 > 0 is sufficiently large. It follows that I is empty as long as A 0 > 1 is taken sufficiently large. Thus every leaf of the canonical foliation Σ A bounds an isoperimetric region. Thus for all A > A 0 . In particular, the isoperimetric profile is a smooth function on (A 0 , ∞). Using the estimates for the lapse function of the canonical foliation and the geometry of the leaves from Section C, we compute that as A → ∞. Assume that there exists another isoperimetric regionΩ A for area A > A 0 . We know from Proposition 3.1 that the boundary of its unique large component has non-zero genus. From (3.4), we obtain the estimate This contradiction shows that Ω A is the unique isoperimetric region for area A.
Appendix A. Asymptotically hyperbolic initial data Just as A. Neves and G. Tian do in [22], we work with two different standard models for three-dimensional hyperbolic space. We useḡ to denote the hyperbolic metric on R 3 given bȳ g = d r ⊗ d r + sinh 2 r g S 2 in polar coordinates. We will also use the disk model for hyperbolic space with metric tensor in polar coordinates on B 1 (0). The radial map s → r(s) = log 1 + s 1 − s induces an isometry Ψ : B 1 (0) → R 3 between these models. In particular, When Σ ⊂ R 3 is a surface, we use Σ δ ⊂ B 1 (0) to denote the Euclidean surface with Ψ(Σ δ ) = Σ.
We say that a Riemannian 3-manifold (M, g) is asymptotic to Schwarzschild-anti-deSitter of mass m > 0 if it is connected and if there are a bounded open set U ⊂ M and a diffeomorphism such that, in polar coordinates, where R is the scalar curvature of (M, g).
Our convention here differs from that used in [22] by a factor of 2 for the mass. We usually require in addition that (M, g) is complete and such that ∂M is connected and the only closed H = 2 surface in (M, g). It can be shown 1 that, in this case, M itself is diffeomorphic to R 3 \ B 1 (0).
Of course, Schwarzschild-anti-deSitter initial data itself satisfies these conditions. We recall that a closed form of the Schwarzschild-anti-deSitter metric (with boundary H = 2) is given by We recall in passing that (M, g) is said to be asymptotically hyperbolic if, in place of (A.2), we have that We use S r ⊂ M to denote the image of the centered coordinate sphere S r (0) under this diffeomorphism, and let B r be the bounded component of M \ S r .

Appendix B. Estimates for stable CMC spheres
In this section, we recall several estimates for stable constant mean curvature surfaces Σ in Riemannian 3-manifolds (M, g) that are used throughout the paper. Let H denote the mean curvature of Σ with respect to the (designated or natural) outward pointing unit normal ν.
The Christodoulou-Yau estimate for stable constant mean curvature spheres stated as (B.2) below is derived in [11]. The proof of the weaker estimate (B.1) for surfaces of non-zero genus follows the same lines, using in addition the Brill-Noether theorem exactly as in the proof of Theorem 6 in [26].
Lemma B.1 (Cf. [11,26]). We have The bound on the right-hand side can be sharpened to 16 π if Σ has genus zero, so in this case For the remaining results included in this section, we assume that (M, g) is asymptotic to Schwarzschild-anti-deSitter with mass m > 0, that Σ is a stable constant mean curvature sphere, and that Σ encloses the centered coordinate ball B 2 .
Lemma B.2 (Cf. Proposition 3.4 in [22]). We have It has been shown by R. Rigger [25] that there is a family of stable constant mean curvature spheres that foliate the complement of a compact subset of M . A. Neves and G. Tian [22] have shown that every stable constant mean curvature sphere Σ in (M, g) that encloses B r 0 and with is a leaf of this canonical foliation. Here, r 0 > 1 and C > 0 are constants depending only on (M, g). They also give an alternative proof of the existence of the canonical foliation.
We will use some of the estimates from [22] to estimate the Hawking mass along the foliation. For every r > 0 sufficiently large, consider the leaf Σ of the canonical foliation [22] with mean curvature From this, we obtain the estimate for the Hawking mass of Σ.
Let v ∈ C ∞ (M ) be the lapse function of the foliation with respect to the parametrization above. Thus where is the stability operator of Σ. Note that Moreover, Analyzing the spectrum of L Σ as in Lemma 3.13 of [18] or Section 8 of [22], we obtain Indeed, the distance of 0 from the spectrum of L Σ is at least 48 m e −3 r (1 + o (1)).
In our application below, it seems more natural to work with the lapse function u ∈ C ∞ (Σ) for the original parametrization by the area A of the canonical foliation. Note that is continuously differentiable and Proof. From a standard computation using the first and second variation of area along with the Gauss equation, we obtain

Estimate (C.3) for the lapse function giveŝ
and from which the assertion follows.
For convenient reference, we collect several properties of the isoperimetric profile of asymptotically hyperbolic Riemannian 3-manifolds (M, g). For simplicity, we assume that ∂M is connected and the only closed H = 2 surface in (M, g).
First, recall the definition of the isoperimetric profile A region Ω ∈ F A with V g (A) = area g (∂Ω).
is called an isoperimetric region for area A, and its boundary an isoperimetric surface. It is well known and discussed in e.g. [1, p. 24] or [7, p. 428] (see also [20]) that V g (A) is absolutely continuous and that for every A > 0, the left derivative V g (A) ′ − and the right derivative V g (A) ′ + exist, and that where H is the mean curvature of any isoperimetric surface of area A.
Using the assumption on ∂M and standard arguments, it follows that the isoperimetric profile is strictly increasing. From this, we see that the complement of an isoperimetric region has no bounded components. In particular, the boundary of a component of an isoperimetric region has either two or one component, depending on whether it includes the horizon or doesn't.
By Theorem 1.1 in [7], if we assume in addition that R ≥ −6, then isoperimetric regions for area A exist provided that A > area g (∂M ) is sufficiently large.

Appendix E. Some generalities about large isoperimetric regions
For convenient reference, we collect several generalities about large isoperimetric regions in asymptotically hyperbolic manifolds.
Lemma E.1 (Cf. Lemma 2.2 in [19]). Let (M, g) be a complete Riemannian 3-manifold that is asymptotically hyperbolic. There is a constant C > 0 with the following property. For every isoperimetric surface Σ ⊂ M in (M, g), we have In particular, for every p > 2,ˆΣ Lemma E.2. Let (M, g) be a complete Riemannian 3-manifold that is asymptotically hyperbolic. As A → ∞, where H is the mean curvature of an isoperimetric surface Σ with area A.
The following result has been obtained in [7]. We include an alternative, more elementary derivation below. Lemma E.3 (Cf. Proposition 6.4 of [7]). Let (M, g) be a complete Riemannian 3-manifold that is asymptotically hyperbolic. There is a constant A 0 > 1 with the following property. Every isoperimetric region for area A ≥ A 0 has a unique component Ω with area g (∂Ω) ≥ A 0 . Moreover, (∂Ω) \ (∂M ) is connected.
Proof. Suppose that there are two components Ω i with A i = area g (∂Ω i ) ≥ 1 where i = 1, 2. For definiteness, let us assume that A 1 ≤ A 2 . Let x : R 3 \ B 1 (0) → M be a chart at infinity of (M, g). We may choose regionsΩ i ⊂ R 3 such that where we have used the previous lemma for the area estimate. By the hyperbolic isoperimetric inequality, Using that Ω 1 , Ω 2 are components of an isoperimetric region and that (M, g) is asymptotically hyperbolic, we see that vol g (Ω 1 ) + vol g (Ω 2 ) is at least as large as the volume of a ball of area A 1 + A 2 − area g (∂M ) in hyperbolic space. It follows that From this and the previous estimate, we obtain Put another way, The connectedness of the outer boundary follows from the monotonicity of the isoperimetric profile at infinity; see Lemma 3.5 in [7].
Lemma E.4. Let (M, g) be a complete Riemannian 3-manifold that is asymptotically hyperbolic.
The distance between Σ i and ∂M tends to zero as i → ∞.

Proof.
We have H i = 2 + o(1) for the mean curvature of Σ i by (E.2). The diameter of Σ i is a priori bounded by the monotonicity formula. If the sequence has a subsequential limit in M , then this limit is a closed surface of constant mean curvature 2 and hence a component of ∂M . If the sequence is divergent, then we can follow a subsequence (in the sense of pointed geometric convergence) to a closed surface of constant mean curvature 2 in hyperbolic space. Such surfaces do not exist.
Appendix F. Extensions of results from [7] In this section, we collect several extensions of results in the work of the first-named author [7] to the case where (M, g) is asymptotic to Schwarzschild-anti-deSitter, rather than exactly Schwarzschild-anti-deSitter outside of a compact set.
Lemma F.1 (Cf. the proof of "Case 3" in Theorem 1.1 in [7] and Proposition 3.1 in [19]). Let (M, g) be a complete Riemannian 3-manifold with R ≥ −6 that is asymptotically hyperbolic, but not hyperbolic space, and such that ∂M is connected and the only closed H = 2 surface in (M, g). There are A 0 > 1 and r 0 > 1 with the following property. Let Ω be the unique large component of an isoperimetric regionΩ for areaÃ ≥ A 0 . Then Ω ∩ B r 0 = ∅.
The quantity ( * ) is positive by Proposition 5.3 in [7] (building on the earlier work [3] of the first-named author with S. Brendle). These estimates are not compatible.
Remark F.2. We expect that the assumption that ∂M be connected in Lemma F.1 can be removed by using the inverse mean curvature flow with forced jumps along with computations as in [7, p. 427].
Lemma F.3 (Cf. Proposition 8.3 in [7]). Let (M, g) be a complete Riemannian 3-manifold that is asymptotic to Schwarzschild-anti-deSitter with mass m > 0. We also assume that R ≥ −6 and that ∂M is connected and the only closed H = 2 surface in (M, g). There is A 0 > 1 with the following property. Assume that Ω is the unique large component of an isoperimetric region for area A ≥ A 0 . Let Σ = (∂Ω) \ (∂M ). Then Σ is connected and m H (Σ) ≤ 4 m.
Proof. We describe the minor modifications to the proof of Proposition 8.3 in [7], where the same result is shown under the additional assumption that (M, g) is equal to Schwarzschildanti-deSitter outside of a compact set. We use Lemma F.4 below instead of Lemma A.2 in [7]. We use Lemma F.1 instead of applying S. Brendle's characterization of closed constant mean curvature surfaces in exact Schwarzschild-anti-deSitter in [7, p. 427].
Lemma F.4 (Cf. Lemma A.2 in [7]). Let (M, g) be a complete Riemannian 3-manifold that is asymptotic to Schwarzschild-anti-deSitter with mass m > 0. For A > 0 large, let r(A) > 0 be such that the centered coordinate sphere S r(A) has area A. Denoting the renormalized volume of (M, g) by V (M, g), we have the expansion vol g (B r(A) ) = 1 2 A − π log A + π (1 + log π) + V (M, g) − 8 π Proof. The proof of Lemma A.2 in [7] for (M, g) equal to Schwarzschild-anti-deSitter outside of a compact set extends to the present generality.