Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions

TL;DR

This paper explicitly characterizes large isoperimetric regions in asymptotically flat manifolds across all dimensions, linking geometric properties to physical quantities like mass and center of mass in general relativity.

Contribution

It provides explicit descriptions of isoperimetric regions in a broad class of asymptotically flat manifolds and connects these regions to physical invariants such as mass and center of mass.

Findings

Explicit large volume isoperimetric regions in all dimensions.

Detection of mass and center of mass via isoperimetric regions.

Existence of large isoperimetric regions in asymptotically flat 3-manifolds with non-negative scalar curvature.

Abstract

We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivity of the isoperimetric mass established by G. Huisken we prove an existence result for large isoperimetric regions in general asymptotically flat three manifolds with non-negative scalar curvature.