Calculus of Variations and Geometric Measure Theory

G. Antonelli - M. Fogagnolo - M. Pozzetta

The isoperimetric problem on Riemannian manifolds via Gromov-Hausdorff asymptotic analysis

created by pozzetta1 on 01 Feb 2021
modified on 09 Dec 2023


Published Paper

Inserted: 1 feb 2021
Last Updated: 9 dec 2023

Journal: Communications in Contemporary Mathematics
Year: 2022
Doi: 10.1142/S0219199722500687
Links: arXiv, PDF


In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below and with a mild assumption at infinity, that is Gromov-Hausdorff asymptoticity to simply connected models of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some Gromov-Hausdorff limits at infinity. The Gromov-Hausdorff asymptotic analysis conducted allows us to provide, in low dimensions, a result of nonexistence of isoperimetric regions in Cartan-Hadamard manifolds that are Gromov-Hausdorff asymptotic to the Euclidean space. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.

Tags: GeoMeG