Generalized compactness for finite perimeter sets and applications to the isoperimetric problem

TL;DR

This paper establishes a generalized compactness theorem for finite perimeter sets in noncompact Riemannian manifolds with bounded geometry, extending previous results and applying to isoperimetric problems.

Contribution

It extends prior compactness results to include limit manifolds at infinity, providing a more general framework for the isoperimetric problem in noncompact manifolds.

Findings

Proves a compactness theorem for finite perimeter sets with bounded volume and perimeter.

Provides a multipointed version of a key isoperimetric theorem.

Offers a simple proof of the continuity of the isoperimetric profile function.

Abstract

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove a compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit manifolds at infinity. We extends previous results contained in [Nar14a], in such a way that the generalized existence theorem, Theorem 1 of [Nar14a] is actually a generalized compactness theorem. The suitable modifications to the arguments and statements of the results in [Nar14a] are non-trivial. As a consequence we give a multipointed version of Theorem 1.1 of [LW11], and a simple proof of the continuity of the isoperimetric profile function.