The isoperimetric inequality on asymptotically flat manifolds with nonnegative scalar curvature

TL;DR

This paper improves the classical isoperimetric inequality on asymptotically flat manifolds with nonnegative scalar curvature by utilizing Hawking mass and establishes a rigidity result when equality is achieved.

Contribution

It introduces an improved isoperimetric inequality using Hawking mass and proves a rigidity theorem for cases of equality on such manifolds.

Findings

Enhanced isoperimetric inequality with Hawking mass

Rigidity result for equality cases

Better understanding of geometric properties of asymptotically flat manifolds

Abstract

In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature.