Characterization of Large Isoperimetric Regions in Asymptotically Hyperbolic Initial Data

Abstract

Let (M, g) be a complete Riemannian 3-manifold asymptotic to Schwarzschild-anti-deSitter and with scalar curvature R ≥ −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 ≥ −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.

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

“…The question of existence, uniqueness, and characterization of isoperimetric sets for large volumes, and more in general of constant mean curvature hypersurfaces on complete Riemannian manifolds under various types of curvature constraints and asymptotic conditions at infinity has received a great deal of attention in the last thirty years, see for instance [37,55,13,14,17,16,29,28,22,21,23,33], and the references therein. We mention in particular [24] where the authors obtain the uniqueness of isoperimetric regions for each sufficiently large volume in complete Riemannian manifolds (M n , g) that are C 2,α -asymptotic to a fixed cone C(N n−1 ) whose link (N n−1 , g N ) satisfies (1.4) Ric N ≥ n − 2 , vol(N) < vol(S n−1 ) .…”

Isoperimetry for asymptotically flat 3-manifolds with positive ADM mass

“…Chodosh [4] has shown that large isoperimetric surfaces are centered coordinate spheres in the special case where the metric is isometric to Schwarzschildanti-de Sitter outside of a compact set. Under the assumption that the manifold (M 3 , g) is asymptotic to Schwarzschild-anti-de Sitter with scalar curvature R ≥ −6, Chodosh, Eichmair, Shi and Zhu [8] showed that the leaves of the canonical foliation constructed by Rigger [24] are unique isoperimetric surfaces for the volume they enclose. In their case, the scalar curvature assumption is necessary.…”