The Isoperimetric Profile of a Noncompact Riemannian Manifold for Small Volumes

TL;DR

This paper investigates the existence of isoperimetric minimizers in noncompact Riemannian manifolds for small volumes and explores their asymptotic behavior.

Contribution

It provides a main theorem establishing conditions for minimizer existence at small volumes and applies it to asymptotic expansions.

Findings

Existence of isoperimetric minimizers for small volumes in noncompact manifolds

Asymptotic expansion formulas for small-volume isoperimetric problems

Theoretical framework for analyzing isoperimetric profiles in noncompact settings

Abstract

In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are given.