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 $sf RCD$ 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 $Kinmathbb R$, $Ngeq 2$, 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.