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

Abstract

Let (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.