Isoperimetric sets in spaces with lower bounds on the Ricci curvature. (July 2022) - Antonelli, Gioacchino; Pasqualetto, Enrico; Pozzetta, Marco
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, H N ), 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.