Isoperimetry for asymptotically flat 3-manifolds with positive ADM mass

TL;DR

This paper proves that in asymptotically flat 3-manifolds with positive ADM mass, each leaf of the canonical foliation uniquely minimizes surface area for its enclosed volume, using a fill-in argument and sharp isoperimetric inequalities.

Contribution

It establishes the uniqueness of isoperimetric surfaces in such manifolds, extending understanding of geometric properties related to mass and curvature.

Findings

Each leaf of the canonical foliation is the unique isoperimetric surface.

The proof employs a fill-in argument and sharp isoperimetric inequality.

Results connect geometric analysis with concepts of mass and curvature in 3-manifolds.

Abstract

Let $(M^3, g)$ be an asymptotically flat 3-manifold with positive ADM mass. In this paper, we show that each leaf of the canonical foliation is the unique isoperimetric surface for the volume it encloses. Our proof is based on the "fill-in" argument and sharp isoperimetric inequality on asymptotically flat 3-manifold with nonnegative scalar curvature.