Isoperimetric sets in spaces with lower bounds on the Ricci curvature

TL;DR

This paper investigates the regularity and topological properties of volume-constrained minimizers of quasi-perimeters in spaces with Ricci curvature lower bounds, extending results to both RCD spaces and smooth Riemannian manifolds.

Contribution

It introduces a new Deformation Lemma for sets of finite perimeter in RCD spaces and establishes regularity and boundary properties of minimizers in these spaces and smooth manifolds.

Findings

Minimizers are open, bounded sets with regular boundaries.

Boundaries are $(N-1)$-Ahlfors regular and coincide with the essential boundary.

Results apply to smooth Riemannian manifolds, including those with boundary.

Abstract

In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $\sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an ${\sf RCD}(K,N)$ space $({\rm X},{\sf d},\mathcal{H}^N)$, with $K\in\mathbb R$, $N\geq 2$, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with $(N-1)$-Ahlfors regular topological boundary coinciding with the essential boundary. The proof is based on a new Deformation Lemma for sets of finite perimeter in ${\sf RCD}(K,N)$ spaces $({\rm X},{\sf d},\mathfrak m)$ and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters. The theory applies to volume constrained minimizers in smooth Riemannian manifolds, possibly with boundary, providing a general regularity result for such minimizers in the smooth setting.