Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions

TL;DR

This paper establishes the existence of isoperimetric regions in certain non-compact Riemannian manifolds with Ricci or scalar curvature bounds, including cases with asymptotic flatness and small volume constraints.

Contribution

It proves the existence of isoperimetric regions under Ricci curvature bounds and asymptotic conditions, and also for small volumes under scalar curvature assumptions, extending previous results.

Findings

Existence of isoperimetric regions in manifolds with Ricci curvature bounds.

Indecomposability of isoperimetric regions when $k_0=0$.

Existence of small volume isoperimetric regions under scalar curvature conditions.

Abstract

We prove existence of isoperimetric regions for every volume in non-compact Riemannian $n$-manifolds $(M,g)$, $n\geq 2$, having Ricci curvature $Ric_g\geq (n-1) k_0 g$ and being locally asymptotic to the simply connected space form of constant sectional curvature $k_0$; moreover in case $k_0=0$ we show that the isoperimetric regions are indecomposable. We also discuss some physically and geometrically relevant examples. Finally, under assumptions on the scalar curvature we prove existence of isoperimetric regions of small volume.