Calculus of Variations and Geometric Measure Theory

G. Antonelli - E. Pasqualetto - M. Pozzetta - I. Y. Violo

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

created by pasqualetto on 03 Mar 2023
modified by pozzetta1 on 09 Sep 2023


Accepted Paper

Inserted: 3 mar 2023
Last Updated: 9 sep 2023

Journal: Proc. R. Soc. Edinb., Sect. A, Math.
Year: 2023

ArXiv: 2303.01280 PDF


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.