Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds

Otis Chodosh,Yuguang Shi,Michael Eichmair,Haobin Yu

doi:10.1002/cpa.21981

Abstract

AbstractLet (M, g) be an asymptotically flat Riemannian 3‐manifold with nonnegative scalar curvature and positive mass. We show that each leaf of the canonical foliation of the end of (M, g) through stable constant mean curvature spheres encloses more volume than any other surface of the same area.Unlike all previous characterizations of large solutions of the isoperimetric problem, we need no asymptotic symmetry assumptions beyond the optimal conditions for the positive mass theorem. This generality includes examples where global uniqueness of the leaves of the canonical foliation as stable constant mean curvature spheres fails dramatically.Our results here resolve a question of G. Huisken on the isoperimetric content of the positive mass theorem. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Highlights

IntroductionA complete Riemannian 3-manifold (M, g) is said to be asymptotically flat if there is a compact subset K ⊂ M and a diffeomorphism (1)
A complete Riemannian 3-manifold (M, g) is said to be asymptotically flat if there is a compact subset K ⊂ M and a diffeomorphism (1)M \ K ∼= {x ∈ R3 : |x| > 1/2} with (2)gij = δij + σij where|x||α||(∂ασij)(x)| = O(|x|−τ ) as |x| → ∞for some τ > 1/2 and all multi-indices α with |α| = 0, 1, 2, 3
The main result of this paper is to show that if (M, g) is not Euclidean space and provided that the volume V > 0 is sufficiently large, ΩV is bounded by the horizon ∂M and a stable constant mean curvature surface that belongs to the canonical foliation of the end of M

Summary

Introduction

A complete Riemannian 3-manifold (M, g) is said to be asymptotically flat if there is a compact subset K ⊂ M and a diffeomorphism (1). In the proof of Theorem 1.1 we use results on the canonical foliation through stable constant mean curvature surfaces of the end of asymptotically flat manifolds with positive mass. The uniqueness question for large stable constant mean curvature surfaces in asymptotically flat 3-manifolds with non-negative scalar curvature that intersect the center of (M, g) has been developed in [15, 7, 8]. We compare with Schwarzschild of mass o(1)mADM until the surfaces have all but disappeared In this argument we only need a very weak characterization of the leaves of the canonical foliation as being unique among stable constant mean curvature spheres in (M, g). The analytic argument described above is likely to yield further information about stable constant mean curvature spheres and could potentially apply to the study of large isoperimetric regions in asymptotically hyperbolic 3-manifolds, where it is not possible to appeal to scaling

Divergent sequences of isoperimetric regions

Isoperimetric deficit of large outlying isoperimetric spheres

Findings

Hawking mass of outlying stable constant mean curvature tori