Isoperimetric clusters in homogeneous spaces via concentration compactness

TL;DR

This paper proves the existence of generalized clusters with minimal perimeter and specified masses in various homogeneous metric measure spaces, extending classical isoperimetric results to more complex settings.

Contribution

It introduces a framework for generalized clusters in homogeneous spaces, accommodating a wide class of perimeter functionals, including anisotropic and Riemannian perimeters.

Findings

Existence of minimal perimeter clusters in Euclidean, hyperbolic, and Heisenberg spaces.

Generalized clusters can be infinite and are minimal among all with the same total masses.

Applicable to a broad range of perimeter functionals in various geometric contexts.

Abstract

We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms, for a quite wide range of perimeter functionals. Such generalized clusters are a natural "relaxed'' version of a cluster and can be thought of as ``albums'' with possibly infinite pages, having a minimal cluster drawn on each page, the total perimeter and the vector of masses being calculated by summation over all pages, the total perimeter being minimal among all generalized clusters with the same masses. The examples include any anisotropic perimeter in a Euclidean space, as well as a hyperbolic plane with the Riemannian perimeter and Heisenberg groups with a canonical left invariant perimeter or its equivalent versions.