Topological regularity of isoperimetric sets in PI spaces having a deformation property

TL;DR

This paper establishes topological regularity of isoperimetric sets in PI spaces with a deformation property, showing they are open, have boundary density estimates, and are bounded under certain conditions, with implications for RCD spaces.

Contribution

It proves new topological regularity results for isoperimetric sets in PI spaces with a deformation property, including applications to collapsed RCD spaces and simplification of classical Euclidean arguments.

Findings

Isoperimetric sets are open and satisfy boundary density estimates.

Under volume lower bounds, isoperimetric sets are bounded.

Rigidity in the isoperimetric inequality holds without boundedness assumptions.

Abstract

We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply, in particular, to the class of possibly collapsed $\mathrm{RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed $\mathrm{RCD}(0,N)$ spaces with Euclidean volume growth holds without the additional assumption on the boundedness of isoperimetric sets. Our strategy is of interest even in the Euclidean setting, as it simplifies some classical arguments.