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 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,HN), with K∈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.