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

Gioacchino Antonelli, Enrico Pasqualetto, Marco Pozzetta, Ivan Yuri Violo

TL;DR

The paper extends topological regularity for isoperimetric sets to length PI spaces endowed with a deformation property, a broad class including collapsed $\mathsf{RCD}(K,N)$ spaces. It proves that volume-constrained minimizers have open essential interiors and that their essential boundary coincides with the topological boundary, together with boundary-density estimates; it also proves boundedness of isoperimetric sets under a unit-ball volume lower bound. The deformation property enables a direct comparison framework that avoids flows and provides a streamlined Euclidean-analog proof path. As an application, the authors obtain unconditional rigidity results for the sharp isoperimetric inequality on ${\sf RCD}(0,N)$ spaces with Euclidean volume growth, broadening the scope of regularity and rigidity phenomena in nonsmooth geometric analysis.

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.

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

TL;DR

The paper extends topological regularity for isoperimetric sets to length PI spaces endowed with a deformation property, a broad class including collapsed spaces. It proves that volume-constrained minimizers have open essential interiors and that their essential boundary coincides with the topological boundary, together with boundary-density estimates; it also proves boundedness of isoperimetric sets under a unit-ball volume lower bound. The deformation property enables a direct comparison framework that avoids flows and provides a streamlined Euclidean-analog proof path. As an application, the authors obtain unconditional rigidity results for the sharp isoperimetric inequality on spaces with Euclidean volume growth, broadening the scope of regularity and rigidity phenomena in nonsmooth geometric analysis.

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 spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed 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.
Paper Structure (6 sections, 12 theorems, 57 equations)

This paper contains 6 sections, 12 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sets of finite perimeter in metric measure spaces
  4. PI spaces
  5. Topological regularity
  6. Boundedness of isoperimetric sets

Key Result

Theorem 1.1

Let $({\rm X},{\sf d},\mathfrak{m})$ be a length PI space having the deformation property. Let $E\subseteq{\rm X}$ be a volume-constrained minimizer of the perimeter. Then $E^{(1)}={\rm int}(E^{(1)})$ and $E^{(0)}={\rm int}(E^{(0)})$. In particular, it holds that $E^{(1)}$, $E^{(0)}$ are open sets a

Theorems & Definitions (31)

  • Theorem 1.1: Topological regularity of volume-constrained minimizers
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: Sharp and rigid isoperimetric inequality on ${\sf RCD}(0,N)$ spaces with Euclidean volume growth
  • Remark 2.1
  • Theorem 2.2: Coarea formula MIRANDA2003
  • Corollary 2.3
  • proof
  • Definition 2.4: PI space
  • Remark 2.5
  • ...and 21 more