The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass
In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes from a previous result of P. Chruściel and W. Simon, which states that the Penrose inequality we prove implies a static uniqueness theorem for negative mass Kottler metrics.
The Penrose inequality for asymptotically flat -manifolds with mass and nonnegative scalar curvature states that if is an outermost minimal surface (i.e., there are no compact minimal surfaces separating from infinity), then
where is the area of .
G. Huisken and T. Ilmanen  first proved this inequality for equal to the largest area of a connected component of . H. Bray  later proved the more general inequality described above using a different method, and this result was later extended to dimensions less than by Bray and the first author .
We are interested in an analog of this theorem for a class of asymptotically locally hyperbolic manifolds which we now define.
We say that a Riemannian metric on a smooth manifold is asymptotically locally hyperbolic if there exists a compact set and a constant curvature surface , called the conformal infinity of , such that is diffeomorphic to with the metric satisfying
is the constant curvature of ;
is the coordinate on ;
is a symmetric two-tensor on so that
where is the hyperbolic metric and are derivatives taken with respect to . We will use the notation as a convenient abbreviation;
is a symmetric two-tensor on depending on in such a way that there exists a function on , called the mass aspect function, such that , where the convergence is in .
For the sake of convenience, we assume that is , , or , and in the case , we further assume that . (These assumptions simply serve the purpose of normalization.)
Finally, we define the mass to be
and we also define
An important class of asymptotically locally hyperbolic manifolds is given by the Kottler metrics (see  for instance), which are static metrics with cosmological constant .
Let be a surface with constant curvature equal to , , or , with area equal to in the case. Let be large enough so that the function
has a nonnegative zero. Let be the largest zero of , and define the metric
on . Define to be the metric completion of this Riemannian manifold. We say that is a Kottler space with conformal infinity and mass .
The Kottler metrics have scalar curvature .
The most familiar situation is when is a sphere , in which case the metric is also called an anti-de Sitter–Schwarzschild metric in the literature.
As long as has a positive largest zero , we obtain with as an outermost minimal surface boundary. One can see that these metrics are asymptotically locally hyperbolic by performing a substitution, in which case one has .
In order for to have a positive zero, we must have when or . However, when , the parameter need not be positive but only greater than a critical mass .
When the largest zero of is exactly , we say that is a critical Kottler space. When and , the metric on is a two-ended complete Riemannian manifold, with one end asymptotically locally hyperbolic and the other end asymptotic to the cylindrical metric on . When and , the metric can be written as on after a coordinate change, and of course, when and , the completed metric is just hyperbolic space.
We can now state our main theorem.
Theorem 1.1 (Penrose Inequality for nonpositive mass).
Let be a asymptotically locally hyperbolic manifold with and conformal infinity , whose genus is .
Assume that , is an outermost minimal surface, and there is a boundary component of genus . Then
where is the area of , and is a topological constant.
Furthermore, equality occurs if and only if is isometric to the Kottler space with infinity and mass .
If one replaces by and removes the condition in our theorem,111Although we are unable to replace by in general, we do obtain a slightly stronger inequality than (1). See Lemma 3.13. one obtains the natural analog of Huisken and Ilmanen’s Penrose inequality  in the asymptotically locally hyperbolic setting. Versions of this statement have been conjectured by P. Chruściel and W. Simon [8, Section VI], and in the case, by X. Wang [19, Section 1]. The graph case of the conjecture has recently been established by L. de Lima and F. Girão when . See also a related volume comparison result by S. Brendle and O. Chodosh .
Let be a asymptotically locally hyperbolic manifold with conformal infinity of genus .
Assume that , is an outermost minimal surface, and there is a boundary component of genus . Then
If or and we obtain at once from Theorem 1.1 that , which is a contradiction.
If , then it is an easily verifiable fact that
This inequality and Theorem 1.1 imply at once that . If equality holds then we are in the equality case of Theorem 1.1 and so must be isometric to a critical Kottler space of mass . But this is impossible because a critical Kottler space has no compact minimal surfaces. (They are foliated by strictly mean convex surfaces.) ∎
Of course, the case of Corollary 1.2 is just the positive mass theorem for asymptotically hyperbolic manifolds, proved by Chruściel and M. Herzlich  and by Wang , for the case of a manifold with minimal boundary, proved by V. Bonini and J. Qing . While the assumption on the boundary is not desirable, we note that there exist examples (the AdS solitons due to G. Horowitz and R. Myers ) of asymptotically locally hyperbolic manifolds with , no boundary, and negative mass.
We note that the inverse mean curvature flow technique allows us to a give a new proof of a weakened version of the positive mass theorem for asymptotically hyperbolic manifolds that was mentioned above.
Theorem 1.3 (Positive theorem for asymptotically hyperbolic manifolds).
Let be a complete asymptotically hyperbolic222Here we define this to mean asymptotically locally hyperbolic with infinity equal to the round sphere. manifold (with or without a minimal boundary) with scalar curvature . Then . Moreover, if , then must be hyperbolic space.
We prove Theorem 1.1 following the inverse mean curvature flow theory developed by Huisken and Ilmanen in . The general idea is to flow outward with speed inversely proportional to the mean curvature and obtain a (weak) flow of surfaces (where ). To each compact surface one considers its Hawking mass to be
where denotes the area of , and is its mean curvature. Observe that our notation leaves out the dependence on . The key property of inverse mean curvature flow is that is non-decreasing in time. Therefore, since coincides with the right-hand side of the inequality in Theorem 1.1, the desired result follows if
In the asymptotically locally hyperbolic setting this inequality is subtle, and in fact the second author constructed well-behaved examples (with ) where the above inequality does not hold . The central observation of this paper is that this inequality holds if . The difference between our result here and the one in  can be traced to the fact that if the mass aspect is positive, then a desired inequality goes in the wrong direction, whereas for nonpositive mass aspect, it goes in the right direction. See the end of the proof of Lemma 3.13 to see the exact place where the condition is used.
Acknowledgements: The authors would like to thank Piotr Chruściel and Walter Simon for bringing this problem to our attention and for their interest in this work. We also thank Richard Schoen for some helpful conversations.
2 An application to static uniqueness
We say that is a complete vacuum static data set with cosmological constant if and only if is a smooth manifold (possibly with boundary) equipped with a complete Riemannian metric and a nonnegative function such that and
If is a vacuum static data set as above, then the Lorentzian metric on is a solution to Einstein’s equations with cosmological constant . In the case a 1987 result of G. Bunting and A. Masood-ul-Alam  shows that Schwarzschild spaces are the only asymptotically flat vacuum static data sets (with the case of connected boundary originally proved by H. Müller zum Hagen, D. Robinson, and H. Seifert in 1973 ).
We are interested in a similar characterization of the Kottler metrics defined in the previous section, which are known to be vacuum static data sets with cosmological constant . A static uniqueness theorem for the hyperbolic space was proved in work of Boucher-Gibbons-Horowitz , Qing , and Wang .
The following definition is equivalent to the one given in [8, Section III.A].
Let . We say that a complete vacuum static data set with cosmological constant is conformally compactifiable if and are and there exists a smooth compact manifold with boundary and a embedding of into such that ,
the function extends to a function on with at , and
the formula near defines a Riemannian metric.
If is a complete vacuum static data set, the fact that on implies that is constant on each component of , and we call this constant the surface gravity of that component.
The surface gravity is strictly positive for the following reason: Given , let be the unit speed geodesic that starts at perpendicular to and set . From the static equation we see the existence of and so that for all Standard o.d.e. comparison shows that if then for all which is impossible because is strictly positive on the interior of .
The (non-critical) Kottler metrics have exactly one component of and, for fixed , there is a bijection between possible surface gravities in and possible masses in when , or in when is or . Therefore, for fixed we can define a bijection according to the fixed relationship between mass and surface gravity for Kottler metrics whose infinities have curvature equal to . For this bijection, when one has and (see [8, Section II] for details).
Theorem 2.1 (Static uniqueness with nonpositive mass).
Let be a complete vacuum static data set with cosmological constant , and assume that it is conformally compactifiable with conformal infinity of constant curvature .
Assume that there is a component of such that is homeomorphic to and has the largest surface gravity of any component.
If (or equivalently ), then must be isometric to the Kottler metric with infinity and mass , while is equal to the usual static potential of the Kottler metric, up to a constant multiple.
Before we present the proof some comments are in order.
It would be interesting to remove the condition on .
2.1 Proof of Theorem 2.1
The following proposition is a consequence of Theorem I.1 and Proposition III.7 of , together with a coordinate change.
Let . Let be a conformally compactifiable complete vacuum static data set with cosmological constant . Further assume that the induced metric on (as defined in the previous section) has locally constant Gauss curvature equal to , , or .
Then is connected, is asymptotically locally hyperbolic (as defined in the Introduction) with conformal infinity , and
where is the coordinate used in the definition of asymptotically locally hyperbolic, and is the mass aspect.
In particular, static data sets with have a well-defined mass and .
The key theorem of  for the purposes of this article is Theorem I.5:
Theorem 2.3 (Chruściel-Simon).
Let be a conformally compactifiable complete vacuum static data set with cosmological constant , conformal infinity of constant curvature , and .
Let denote the boundary component with the largest surface gravity and suppose (i.e., ).
If denotes the Kottler space with infinity and mass (i.e., the one with surface gravity ), then
where is the area with respect to .
In the case where has the same genus as , this theorem provides a simple comparison between the masses and boundary areas of a vacuum static data set and its so-called reference solution . For the sake of completeness we provide the proof of this theorem in Section 4.
Let be a asymptotically locally hyperbolic, complete vacuum static data set with cosmological constant . If , then is an outermost minimal surface. In fact, there are no compact minimal surfaces in the interior of .
First note that the static equations imply that is totally geodesic, so we need only show that there are no other compact minimal surfaces. Consider the trapped region of , which is the union of all compact minimal surfaces in , together with all regions of that are bounded by these minimal surfaces. The boundary of the trapped region, , must itself be a smooth compact minimal surface. (See the proof of Lemma 4.1(i) of .) Following , we define the exterior region of to be the metric completion of . Thus is a vacuum static data set, except for the requirement that . The exterior region has a strictly outward minimizing minimal boundary and no interior compact minimal surfaces (see ), where strictly outward minimizing means that is strictly less than the area of any other surface that encloses it.
Suppose that has a compact minimal surface other than . Since away from , it follows from the definition of that does not vanish identically on . We consider the outward normal flow of surfaces with initial condition that flows with speed . Since does not vanish on , this flow is nontrivial. According to the formula for the variation of mean curvature, we see that the mean curvature of evolves according to
where is the outward unit normal, and is the second fundamental form. Since , it follows that must have for all small . By the first variation of area formula, must have area less than or equal to that of . But this contradicts the strictly outward minimizing property of . ∎
Let be as in the statement of the theorem. In particular, all of the hypotheses of Chruściel-Simon’s Theorem (Theorem 2.3) are satisfied and so . Hence Proposition 2.2 and Lemma 2.4 imply that all of the hypotheses of our Penrose inequality (Theorem 1.1) are also satisfied.
Since we are assuming that is homeomorphic to , Theorem 2.3 tells us that
where is the area of and is the area of in the reference solution. Moreover, since the curvature of is equal to our Penrose inequality (Theorem 1.1) tells us that
It is convenient to define constants
so that we have
Inequality (5) then becomes
Meanwhile, on the reference space we know that is the largest root of
and so using elementary reasoning. Thus
Therefore all of the inequalities must be equalities and so it follows from the rigidity part of Theorem 1.1 that is isometric to the Kottler space with infinity and mass .
It is simple to check that any two static potentials on the Kottler space must be proportional and so is the usual static potential, up to a constant multiple.
3 Proof of Theorem 1.1
Assume that be a asymptotically locally hyperbolic manifold with such that is an outermost minimal surface. In order to establish existence of the weak inverse mean curvature flow, we first need to find a weak subsolution.
Let be a asymptotically locally hyperbolic metric with radial coordinate as in the definition of asymptotically locally hyperbolic. There exists so that for all ,
are, respectively, subsolutions and supersolutions for inverse mean curvature flow with initial condition
We will prove that is a supersolution. (The proof for is similar.) Using the asymptotics of , one can see that the inverse mean curvature of the constant sphere in is
Since that is just the constant sphere with , we see that the speed of the flow is just
Clearly, for sufficiently large , this speed is less than , showing that is a supersolution. ∎
Given Lemma 3.1, we may now apply Huisken and Ilmanen’s Weak Existence Theorem 3.1 of  to find a weak solution for inverse mean curvature flow with initial condition . More precisely, is a proper, locally Lipschitz nonnegative function defined on with on that satisfies a certain variational property (defined on page 365 of ). The surfaces are and strictly outward minimizing,333Huisken and Ilmanen instead describe the region enclosed by as a strictly minimizing hull . as defined in the proof of Lemma 2.4. In particular, each is mean convex. There are only countably many “jump times,” that is, values of for which does not equal . In a nonrigorous sense, may be regarded as flowing by smooth inverse mean curvature flow, except when it ceases to be strictly outward minimizing, at which time it “jumps” to a strictly outward minimizing surface of equal area.
In case is not connected, Huisken and Ilmanen explained how one can single out a component of as the initial surface while treating the other components of as “obstacles.” See Section 6 of  for details. Essentially, we arbitrarily “fill in” all other components of to obtain a new space and then run the weak inverse mean curvature flow in with initial condition , except that whenever the surface is about to enter the filled-in region, we jump to a connected strictly outward minimizing surface enclosing both and one or more of the filled-in regions. We then restart the flow with initial condition .
There is another important alteration introduced by Huisken and Ilmanen [12, Section 4]. We consider the exterior region of , as defined in the proof of Lemma 2.4. Since was the outermost minimal surface of , it follows that is still part of the boundary of , but now there might be more minimal boundary components. The exterior region is an improvement over because it is completely free of compact minimal surfaces in its interior. We will actually run the weak inverse mean curvature flow in the exterior region of rather than in itself. So for our proof of Theorem 1.1, we may assume without loss of generality that is an exterior region.
3.1 Monotonicity of inverse mean curvature flow
Fix an integer and recall our definition of the Hawking mass of a surface in to be
The proof of the Geroch Monotonicity Formula 5.8 in  adapts straightforwardly to the locally hyperbolic setting to show the following:
Theorem 3.2 (Huisken-Ilmanen).
Let be a complete, one-ended, asymptotically locally hyperbolic manifold with outermost minimal boundary, and let be one of its boundary components. Let be a weak solution to inverse mean curvature flow (possibly with obstacles, as described above), with initial surface . Then for , if there are no obstacles between and , then
where , and is the trace-free part of the second fundamental form.
To make use of this theorem we need the following lemma.
Let be a complete, one-ended, asymptotically locally hyperbolic manifold, which is an exterior region. Let be a component of with genus , and let be a weak solution to inverse mean curvature flow (possibly with obstacles), with initial surface . For all , the surface is connected and has genus at least . In particular, .
Let be the manifold with the obstacles filled in. Let be the function defining the weak flow (possibly with obstacles). For each , let be the closure of , so that . We claim that is connected. If it were not connected, one of the components of would be disjoint from . By the variational property that characterizes , one can deduce that must be constant over . (See the proof of [12, Connectedness Lemma 4.2(i)].) Since is connected, must meet , and thus on , which is a contradiction to the definition of . Since is connected, it follows that is connected. Note that , where is some labeling of the other components of that touch .
The rest of the proof does not use inverse mean curvature flow. It is essentially a topological argument that relies only on the following facts about : There exists a connected manifold whose boundary is , is mean convex, each is minimal, and the ’s are the only compact minimal surfaces in . This last part is where we use the assumption that is an exterior region.
Let be the connected components of . We minimize area in the isotopy class of in . Note that Theorem 1 of Meeks, Simon, and Yau applies because has mean convex boundary, as explained in Section 6 of . According to Theorem 1 and Remark 3.27 of , there exists some surface obtained from via isotopy and a series of -reductions such that each component of is a parallel surface of a connected minimal surface, except for one component that may be taken to have arbitrarily small area. Recall that a -reduction is a surgery procedure that deletes an annulus and replaces it with two disks in such a way that the annulus and two disks bound a ball in . (See [13, Section 3] for details.)
Note that -reduction preserves homology class. Since the only compact minimal surfaces in are the ’s, and because a surface of small enough area must be homologically trivial, it follows that
for some integers . Using the long exact sequence for the pair , we have exactness of
Since is connected, must be generated by
where and are oriented using the outward normal in as usual. Since equation (7) says that , it follows that must be connected and hence equal to . In particular,
Since each component of is either isotopic to one of the ’s (with some orientation) or is null homologous, and since there are no relations among in , the previous equation implies that at least one component of is isotopic to . Finally, since -reduction can only reduce the total genus of all components of a surface, we know that has genus at least as large as that of . ∎
Note that if we apply the reasoning in the proof of Lemma 3.3 above to the “conformal infinity” of , we see that the genus of is at least as large as the genus of .
Corollary 3.4 (Geroch monotonicity).
Let be a complete, one-ended, asymptotically locally hyperbolic manifold, which is an exterior region. Let be a component of with genus , and let be a weak solution to inverse mean curvature flow (possibly with obstacles), with initial surface . Then the Hawking mass of is nondecreasing in .
The result follows immediately from Theorem 3.2 and Lemma 3.3 in the absence of obstacles. Since there are only finitely many obstacles, all that is left to show is that the mass cannot drop when we jump over an obstacle. Let be the first time that we have to jump over an obstacle, and let denote the strictly outward minimizing surface that jumps to. Then we know from [12, Equation 6.1] that
where the second inequality essentially follows from the fact that the strictly minimizing hull of a surface should be minimal away from where it agrees with the original surface. In the case , these inequalities combine with nonnegativity of (from monotonicity in the absence of obstacles) to immediately show that , just as in the asymptotically flat case [12, Section 6].
To handle the case we proceed as follows.
Therefore it only remains to show that . Note that for any ,
Taking and using monotonicity in the absence of obstacles, we then have
On the other hand, observe that a stable compact minimal surface of genus in a 3-manifold with must have area at least . (This follows from a standard computation using the second variation of area, see [17, Section 2]). Thus
which completes the proof. ∎
3.2 The long-time limit of inverse mean curvature flow
First we compute the asymptotics of Ricci curvature for asymptotically locally hyperbolic manifolds. Proceeding as in Lemma 3.1 of , we deduce the following:
Let be a asymptotically locally hyperbolic, with radial coordinate . If is an orthonormal frame at a point in , then
In order to make certain computations easier, we consider a conformal compactification of the exterior region of . Then if we set , we have
on the space for small enough , where
There is a constant such that for sufficiently large , the coordinate on lies in and the coordinate on must lie in .
This follows immediately from the subsolutions and supersolutions of inverse mean curvature flow described in Lemma 3.1. ∎
We will use the following area bound repeatedly.
Let denote the surface endowed with the metric induced from . The area of is uniformly bounded in time.
This follows immediate from the fact that , the definition of , and the previous Lemma. ∎
is uniformly bounded above and below.
The upper bound follows easily from Corollary 3.4: Since
To prove the lower bound, we consider the Gauss-Codazzi equations in :
Performing the same computation in the compactified metric and using the fact that integral of the left-hand side is conformally invariant, we see that
By Lemma 3.5, we see that
for some constant independent of , where we used the fact that the curvature of is bounded. ∎