Generalized existence of isoperimetric regions in non-compact Riemannian manifolds and applications to the isoperimetric profile

TL;DR

This paper proves the existence of isoperimetric regions in certain noncompact Riemannian manifolds by extending the space with limit manifolds, and applies this to analyze the isoperimetric profile.

Contribution

It establishes the existence of isoperimetric regions in noncompact manifolds with bounded geometry by enlarging the space with limit manifolds, extending isoperimetric profile properties.

Findings

Existence of isoperimetric regions in extended noncompact manifolds.

Extension of isoperimetric profile properties to noncompact manifolds.

Application of the theory to analyze geometric properties.

Abstract

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric profile from compact manifolds to such noncompact manifolds.