Isoperimetric sets in spaces with lower bounds on the Ricci curvature

In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in 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 RCD(K, N) space (X, d, 1.tN), with K E R, N > 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 RCD(K, N) spaces (X, d, m) and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters.(c) 2022 Elsevier Ltd. All rights reserved.

Automated summary

This paper studies the regularity and topological properties of volume constrained minimizers of quasi-perimeters in RCD spaces. The authors prove that under certain conditions, the minimizers of quasi-perimeters have specific geometric and topological properties, such as being open bounded sets with a specific boundary.