On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

TL;DR

This paper establishes new existence results for large-volume isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth, using geometric and analytic techniques to analyze their asymptotic behavior.

Contribution

It provides sufficient conditions for the existence of large isoperimetric regions based on the manifold's geometry at infinity, extending previous results to broader classes of manifolds.

Findings

Existence of large isoperimetric regions under certain geometric conditions

Isoperimetric sets of large volume always exist on manifolds with nonnegative sectional curvature

Concavity of the isoperimetric profile established for noncollapsed manifolds with Ricci curvature bounded below

Abstract

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.