Large isoperimetric surfaces in initial data sets

TL;DR

This paper investigates the structure of large isoperimetric surfaces in asymptotically flat 3-manifolds, proving their existence and closeness to coordinate spheres, and confirming a conjecture related to stable constant mean curvature surfaces.

Contribution

It refines volume comparison in Schwarzschild space and proves the existence and shape of large isoperimetric regions in initial data sets, confirming a conjecture by Bray.

Findings

Existence of isoperimetric regions for large volumes

Isoperimetric regions are close to centered coordinate spheres

Confirms Bray's conjecture on stable constant mean curvature spheres

Abstract

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volume-preserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.