Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian $3$-manifolds

TL;DR

This paper proves that in asymptotically flat 3-manifolds with non-negative scalar curvature and positive mass, the canonical foliation's leaves are uniquely isoperimetric surfaces for their enclosed volume.

Contribution

It establishes a unique isoperimetric property for the canonical foliation in such manifolds, linking geometric analysis with scalar curvature and mass.

Findings

Leaves of the canonical foliation are uniquely isoperimetric surfaces.

The result connects scalar curvature, mass, and isoperimetric properties.

Provides new insights into the geometry of asymptotically flat manifolds.

Abstract

Let $(M, g)$ be an asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature and positive mass. We show that each leaf of the canonical foliation through stable constant mean curvature surfaces of the end of $(M, g)$ is uniquely isoperimetric for the volume it encloses.