On clusters and the multi-isoperimetric profile in Riemannian manifolds with bounded geometry

TL;DR

This paper establishes the existence, compactness, and boundedness of isoperimetric clusters in Riemannian manifolds with bounded geometry, introduces the multi-isoperimetric profile, and generalizes classical results to this broader setting.

Contribution

It proves the existence and properties of isoperimetric clusters in manifolds with bounded geometry and introduces the multi-isoperimetric profile with Holder continuity, extending previous Euclidean and space form results.

Findings

Existence of isoperimetric clusters in bounded geometry manifolds

Compactness theorem for sequences of clusters

Holder continuity of the multi-isoperimetric profile

Abstract

For a complete Riemannian manifold with bounded geometry, we prove the existence of isoperimetric clusters and also the compactness theorem for sequence of clusters in a larger space obtained by adding finitely many limit manifolds at infinity. Moreover, we show that isoperimetric clusters are bounded. We introduce and prove the Holder continuity of the multi-isoperimetric profile which has been explored by Emanuel Milman and Joe Neeman with a Gaussian-weighted notion of perimeter. We yield a proof of classical existence theorem, e.g. in space forms, for isoperimetric cluster using the results presented here. The results in this work generalize previous works of Stefano Nardulli, Andrea Mondino, Frank Morgan, Matteo Galli and Manuel Ritor\'e from the classical Riemannian and sub-Riemannian isoperimetric problem to the context of Riemannian isoperimetric clusters and also Frank Morgan and Francesco Maggi works on the clusters theory in the Euclidean setting.