Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions
The question of isoperimetry What is the largest amount of volume that can be enclosed by a given amount of area? can be traced back to antiquity.111We refer the reader to  for a beautiful collection of materials on the history of the isoperimetric problem. The first mathematically rigorous results are as recent as the nineteenth century. The question of isoperimetry and its close relative, the analysis of minimal surfaces, are two of the model problems of the geometric calculus of variations.
The list of geometries where an explicit answer to the question of isoperimetry is available is short. We provide an overview of available results in Appendix H. In  and this paper, we extend this list by a class of Riemannian manifolds for which we describe all large isoperimetric regions completely.
We refer the reader to Section 2 for the precise definitions of all terms in the statement of our first main theorem:
Let be an -dimensional initial data set with that is -asymptotic to Schwarzschild of mass . There exists such that for every the infimum in
is achieved by a smooth isoperimetric region . The boundary of is close to a centered coordinate sphere where is such that . If is -asymptotic to Schwarzschild of mass , then can be chosen such that for every , is the unique minimizer of (1), and such that the hypersurfaces foliate smoothly. If is also asymptotically even, then the centers of mass of the boundaries converge to the center of mass of as .
Theorem 1.1 follows from Theorems 5.12 and 6.1. The class of Riemannian manifolds to which Theorem 1.1 applies appears naturally in mathematical relativity as initial data for the Einstein equations. It also appears naturally in conformal geometry: If is a closed Riemannian manifold of positive Yamabe type and if either or if is locally conformally flat, then lies in this class. Here, and is the Green’s function with pole at of the conformal Laplace operator of , cf. Theorem V.3.6 in  and Propositions 3.3 and 4.4 in .
The critical points for the isoperimetric problem are exactly the constant mean curvature surfaces. Stable critical points are volume preserving stable constant mean curvature surfaces. The study of such surfaces also has a rich (if relatively recent) history. We mention in particular Alexandrov’s theorem which shows that closed constant mean curvature surfaces in Euclidean space are round spheres. In their seminal paper , G. Huisken and S.-T. Yau proved that the complement of a bounded set of a three dimensional initial data set that is -asymptotic to Schwarzschild of mass is foliated by strictly volume preserving stable constant mean curvature spheres. Moreover, the leaves of this foliation are the unique volume preserving stable constant mean curvature spheres of their mean curvature within a large class of surfaces, including all nearby ones. This uniqueness has been extended to a larger class of surfaces in important work by J. Qing and G. Tian . G. Huisken and S.-T. Yau have also shown in  that the centers of mass of their surfaces have a limit, the “Huisken-Yau geometric center of mass”. H. Bray conjectured in  that the surfaces found in  are in fact isoperimetric. We have confirmed this conjecture in  by establishing an effective volume comparison result for initial data sets that are -asymptotic to Schwarzschild of mass , building on H. Bray’s characterization  of the isoperimetric regions in the exact spatial Schwarzschild geometry. We refer the reader to the introductions of [21, 22] for a more extensive discussion including the physical significance of these results and further references.
Theorem 1.1 here extends these results in several ways:
Our result holds in all dimensions. The existence of a foliation by volume preserving stable constant mean curvature surfaces when is -asymptotic to Schwarzschild of mass has been shown by R. Ye in  in all dimensions. The proofs of the uniqueness results for large volume preserving stable constant mean curvature surfaces in [32, 49] depend delicately on some tools that are special to three dimensional initial data sets.
The uniqueness of the leaves in the class of isoperimetric surfaces in our result is global. The uniqueness results of [32, 49] apply only to surfaces that lie far in the asymptotic regime of the initial data set where the geometry is close to that of the exact spatial Schwarzschild geometry. An important ingredient in our proof is the recent characterization of closed constant mean curvature surfaces in the exact spatial Schwarzschild geometry by S. Brendle .
Unlike , we only require that is -asymptotic to Schwarzschild of mass . To accomplish this, we rely on strong a priori position estimates for large isoperimetric regions in initial data sets that are -asymptotic to Schwarzschild of mass . These estimates come from our effective volume comparison result, see Theorems 3.5 and 4.1. In the case , we also rely on an idea from  which in turn depends on the effective version of Schur’s lemma of C. De Lellis and S. Müller in .
When and is -asymptotic to Schwarzschild of mass , the convergence of the centers of mass of the leaves of the foliation was established in . L.-H. Huang showed in [28, 29] that the Huisken-Yau geometric center of mass coincides with the usual center of mass [50, 4]. Our proof here, which works in all dimensions, uses ideas from [28, 29], but it is both shorter and more elementary. In particular, we do not rely on the delicate density theorem of .
One key ingredient in our proof of Theorem 1.1 is an all-dimensional analogue of the effective volume comparison theorem for -dimensional initial data sets that are -asymptotic to Schwarzschild of mass that was obtained by the authors in . This result is established in Section 3.
In , M. Ritoré has shown that in a complete Riemannian surface with non-negative curvature, isoperimetric regions exist in for every volume . Our second main result is the existence of large isoperimetric regions in arbitrary -dimensional initial data sets with non-negative scalar curvature:
Assume that is a three dimensional initial data set that has non-negative scalar curvature. There exists a sequence of isoperimetric regions with .
The proof of Theorem 1.2, which is given in Section 7, is indirect and uses recent deep insights of G. Huisken’s on the isoperimetric mass of initial data sets. Note that our theorem implies in particular that contains large volume preserving stable constant mean curvature surfaces. Using arguments as for example in  it follows that appropriate homothetic rescalings of these large isoperimetric regions to a fixed volume are close to coordinate balls. The existence of such surfaces in this generality seems to lie deep and out of reach of e.g. implicit-function type arguments.
We are very grateful to Hubert Bray, Simon Brendle, Gerhard Huisken, Manuel Ritoré, Brian White, and Shing-Tung Yau for useful conversations, encouragement, and support. We also thank the referees for their careful reading and valuable comments. Michael Eichmair gratefully acknowledges the support of NSF grant DMS-0906038 and of SNF grant 2-77348-12. Also, Michael Eichmair wishes to express his sincere gratitude to Christina Buchmann, Katharina Halter, Madeleine Luethy, Alexandra Mandoki, Anna and Lisa Menet, Martine Verwey, Markus Weiss, and his wonderful colleagues in Group 6 at ETH for making him feel welcome and at home in Zürich right from the start.
2. Definitions and Notation
Let . An initial data set is a connected complete boundaryless -dimensional Riemannian manifold such that there exists a bounded open set so that , and such that in the coordinates induced by we have that
Given , , and an integer , we say that an initial data set is -asymptotic to Schwarzschild of mass at rate if
We say that an initial data set that is -asymptotic to Schwarzschild of mass at rate is asymptotically even, if
Here, is the scalar curvature of .
We extend as a smooth regular function to the entire initial data set such that . We use , to denote the surface and the region in respectively. We will not distinguish between the end of and its image under .
If is Borel and has locally finite perimeter, then its reduced boundary in is denoted by .
The isoperimetric area function is defined by
A Borel set with finite perimeter such that and is called an isoperimetric region of of volume .
3. A refinement of Bray’s isoperimetric comparison theorem for Schwarzschild in all dimensions
Throughout this section we will use the notation for the spatial Schwarzschild manifold of mass ,
set forth in Appendix D.
In his thesis , H. Bray has proven that the centered coordinate spheres in the three-dimensional Schwarzschild manifold of mass are isoperimetric. His argument also applies to compact perturbations of Schwarzschild provided the coordinate spheres are sufficiently large. The following proposition follows from straightforward modifications of H. Bray’s original, three dimensional arguments in , see also [21, Section 3]. The proof uses the analogue of the Hawking mass for rotationally symmetric Riemannian manifolds in higher dimensions, which we review in Appendix B. See also [6, 15, 1] for other extensions of the results in H. Bray’s thesis.
Proposition 3.1 (Bray’s volume preserving charts in higher dimension).
Let and be given. There exist , , , , and with the following properties:
The sphere in the metric cone has the same area and the same (positive) mean curvature as with respect to .
for . is a smooth non-decreasing function on and . The derivative of at exists and equals .
, for , the derivative of at exists and is , , and is smooth on .
Define the metric on . Then is isometric to under a rotationally invariant map that sends to .
As quadratic forms, we have that on .
The additional observations about , and in the following proposition, in particular (e), are the key to making H. Bray’s characterization of isoperimetric regions in Schwarzschild into an effective volume comparison theorem. We omit the proofs, which are simple adaptations of the arguments in .
Proposition 3.2 (Cf. [21, Section 3]).
We have that
The scalar curvature of the conical metric equals .
The Schwarzschild volume between and the horizon is greater than the volume of with respect to the metric . The difference is .
Fix and let . Then
provided that is sufficiently large (depending only on and ), and where is a constant depending only on .
With these preparations, it is a simple matter to carry over the derivation of the effective volume comparison result [21, Proposition 3.3] to arbitrary dimensions. The result is based on the concept introduced in the following definition:
Definition 3.3 (Cf. [21, Definition 3.2]).
Let be an initial data set that is -asymptotic to Schwarzschild of mass . Let be a bounded Borel set with finite perimeter in . Given parameters and we say that such a set is -off-center if
is so large that there exists a coordinate sphere with and , and if
There are several measures of asymmetry in the literature that lead to effective versions of the classical isoperimetric inequality in Euclidean space, cf. Appendix H.8. The following effective version of Bray’s characterization of the isoperimetric regions in Schwarzschild is not a consequence of an effective isoperimetric inequality in Euclidean space; it depends on the positivity of the mass in a crucial way.
Proposition 3.4 (Effective Volume Comparison in Schwarzschild, cf. [21, Proposition 3.3]).
Given and there exists so that the following holds: Let and let be such that and let be a bounded Borel set with finite perimeter such that and . If is -off-center, i.e. if , then
Here, is a constant that only depends on .
The proof of the main theorem in this section below is literally the same as in , except for adapting various exponents throughout the proof. Since the modifications are delicate, we include the full argument.
Theorem 3.5 (Cf. [21, Theorem 3.4]).
Let be an initial data set that is -asymptotic to Schwarzschild of mass . For every tuple and every constant there exists a constant such that the following holds: Given a bounded Borel set with finite perimeter in and with that is -off-center with and such that holds for all , one has
where is such that , and where is a constant that only depends on .
For ease of exposition we only consider smooth regions . The result for sets with finite perimeter follows from this by approximation. We will use here that as , which follows from the isoperimetric inequality in Lemma E.4. Note also that .
We break into several steps:
Let . Let be the corresponding region in Schwarzschild.
Note that and . Moreover, satisfies for all where depends only on and .
By Corollary A.2 with ,
By Lemma A.3 with , .
By Lemma A.3 with and choice of , .
The Schwarzschild region is -off-center provided that is sufficiently large. Hence by (2).
where the inequality follows by explicit computation from .
. This is obvious.
The conclusion follows from this since and since . ∎
4. Large isoperimetric regions center
Theorem 4.1 (Cf. [21, Theorem 5.1]).
Let be an initial data set that is -asymptotic to Schwarzschild of mass . There exists a constant so that if is an isoperimetric region with , then is smooth and is a connected smooth hypersurface that is close to the coordinate sphere , where is such that . The scale invariant norms of functions that describe such as normal graphs above the corresponding coordinate spheres tend to zero as .
It follows exactly as in the proof of Theorem 5.1 in  that the reduced boundary of outside of is a smooth connected closed hypersurface with the properties asserted for the boundary of in the statement of the theorem. Assume that and let . The half-space theorem [67, Corollary 37.6] shows that consists of regular points. If is mean convex, this contradicts the maximum principle. Since all sufficiently large coordinate spheres are mean convex, we conclude that is bounded independently of . If it were non-zero, we could consider the smooth region and move its mean convex outer boundary inwards to adjust the (relatively small) increase in volume back to . The resulting region has less boundary area than , a contradiction. ∎
Theorem 4.2 (Cf. [21, Theorem 5.2]).
Let be an initial data set that is -asymptotic to Schwarzschild of mass . There exists so that for every volume there exists a smooth isoperimetric region with .
It follows from the argument in [7, Lemma 5] that a closed isoperimetric surface in a Riemannian manifold with non-negative Ricci-curvature is either connected or totally geodesic. It is tempting to impose a curvature condition and transplant Bray’s argument to minimizing sequences (as in Proposition E.3) to prevent them from splitting up into a part that stays behind and a part that diverges to infinity. Such arguments are investigated for various kinds of asymptotic geometries in recent work of A. Mondino and S. Nardulli. Note that the Ricci-tensor of the Schwarzschild manifold has a negative eigenvalue. Moreover, every complete one-ended asymptotically flat manifold that has non-negative Ricci-curvature is flat. (This follows from the Bishop-Gromov comparison theorem.)
5. Uniqueness of large isoperimetric regions and the existence of an isoperimetric foliation
Let and . We consider the Banach space of tuples , where is such that
where the derivatives and norms are those of , and where is a metric on on such that for some ,
Here, are the coefficients of the Schwarzschild metric of mass . Given , we will consider the surface and compute associated geometric quantities with respect to .
The classes and how we use them are closely related to the classes in the work of G. Huisken and S.-T. Yau [32, p. 286], cf. the proof of Theorem 5.1 and the remarks in the last paragraph on p. in their paper.
5.1. Curvature estimates for surfaces in
Given , there exist and such that for all and , we have the following estimates for geometric quantities of with respect to for all :
Here, and denote the Riemann and the Ricci curvature tensors of , the unit normal of with respect to , the trace free part of the second fundamental form of , the mean curvature, the embedding of into , the covariant derivative with respect to the induced metric on , and is a constant that only depends on and . Contractions are taken with respect to the first index. Our sign conventions are reviewed in Appendix C.
Lemma 5.2 (J. Simons’ identity).
Let be a hypersurface of a Riemannian manifold with induced metric , second fundamental form , and mean curvature . Then
The corollary below follows from separating into its trace free and pure trace part, .
Assumptions as in Lemma 5.2. Then
Here, denotes the inner product with respect to .
Given , there exist and such that for all and the following holds: If is such that has constant mean curvature with respect to , and if is a non-negative Lipschitz function on , then
Here, denotes a constant which only depends on and .
Using Proposition 5.1 we can estimate
provided that is small enough and is large enough. In conjunction with (5) we obtain the differential inequality
We multiply this inequality with and integrate over . Upon an integration by parts of the first term on the left and the two terms on the right, we obtain
The result below corresponds to [32, Lemma 5.6], where a variant of the iteration technique of  for volume preserving stable constant mean curvature surfaces is applied to obtain curvature estimates. Here, we use only that the surfaces have constant mean curvature, along with J. Simons’ identity and a standard Stampacchia iteration. Another ingredient in our proof is an insight from  related to an integration by parts on certain covariant derivatives of curvature that appear contracted with the tracefree part of the second fundamental form in J. Simons’ identity. This is applied to the effect that we get by assuming that is -asymptotic to Schwarzschild of mass , rather than -asymptotic as in .
Given , there exist and such that for all and the following holds: If is such that has constant mean curvature with respect to , then
where is a constant that only depends on and .
We let and . Let , where is a constant. Then satisfies . On we have that , on its complement we have almost everywhere. Let .
Using as a test function in Lemma 5.4, we obtain that