Rigidities of Isoperimetric inequality under nonnegative Ricci curvature

TL;DR

This paper proves that in non-negative Ricci curvature spaces, the only isoperimetric equality cases are metric balls, leading to new rigidity results and characterizations of optimizers in various geometric settings.

Contribution

It establishes that equality in the isoperimetric inequality occurs only for metric balls, extending rigidity results to weighted Riemannian manifolds, RCD spaces, and Euclidean cones.

Findings

Equality cases are metric balls in non-negative Ricci curvature spaces.

Spaces attaining equality are measure-theoretically cones.

Optimizers in Euclidean cones are Wulff shapes.

Abstract

The sharp isoperimetric inequality for non-compact Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions [arXiv:1812.05022, arXiv:2012.09490, arXiv:2009.13717, arXiv:2103.08496] culminated by Balogh and Kristaly [arXiv:2012.11862] covering also m.m.s.'s verifying the non-negative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterisation of the equality cases is present in the literature. The scope of this note is to settle this problem by proving, in the same generality of [arXiv:2012.11862], that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extend to weighted Riemannian manifold the rigidity results of [arXiv:2009.13717], it extend to general $\mathsf{RCD}$ spaces the rigidity results of [arXiv:2201.04916] and finally applies also to the Euclidean setting by proving that that optimisers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.