Calculus of Variations and Geometric Measure Theory

M. Novaga - E. Paolini - E. Stepanov - V. M. Tortorelli

Isoperimetric clusters in homogeneous spaces via concentration compactness

created by novaga on 15 Dec 2021
modified on 19 Aug 2022


Published Paper

Inserted: 15 dec 2021
Last Updated: 19 aug 2022

Journal: J. Geom. Anal.
Volume: 32
Number: 11
Pages: Article n. 263
Year: 2022
Doi: 10.1007/s12220-022-01009-8

ArXiv: 2112.08170 PDF


We show the existence of generalized clusters with 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.