Isoperimetric sets in spaces with lower bounds on the Ricci curvature

Abstract

In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $\mathsf{RCD}$ 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 $\mathsf{RCD}(K,N)$ space $(\mathrm{X},\mathsf{d},{\mathcal{H}}^N)$, with $K\in \mathbb{R}$, $N\ge 2$, 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 $\mathsf{RCD}(K,N)$ spaces $(\mathrm{X},\mathsf{d},\mathfrak{m})$ 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.

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• The Cheeger problem in abstract measure spaces
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• On clusters and the multi-isoperimetric profile in Riemannian manifolds with bounded geometry
• Topological regularity of isoperimetric sets in PI spaces having a deformation property
• Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature