The isoperimetric problem of a complete Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends

TL;DR

This paper investigates the existence and geometric properties of large-volume isoperimetric regions in certain asymptotically Schwarzschild Riemannian manifolds, extending previous results and relevant to mathematical general relativity.

Contribution

It extends the existence and characterization of isoperimetric regions to manifolds with multiple asymptotically Schwarzschild ends, including a class with stronger convergence properties.

Findings

Existence of large-volume isoperimetric regions in asymptotically Schwarzschild manifolds.

Geometric characterization of these isoperimetric regions.

Extension of results to manifolds with stronger asymptotic conditions.

Abstract

We study the problem of existence of isoperimetric regions for large volumes, in $C^0$-locally asymptotically Euclidean Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends. Then we give a geometric characterization of these isoperimetric regions, extending previous results contained in [EM13b], [EM13a], and [BE13]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we obtain existence of isoperimetric regions for all volumes for a class of manifolds that we named $C^0$-strongly asymptotic Schwarzschild, extending results of [BE13]. Such results are of interest in the field of mathematical general relativity.