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A note on Laplacian bounds, deformation properties and isoperimetric sets in metric measure spaces

Enrico Pasqualetto, Tapio Rajala

TL;DR

This paper investigates regularity of isoperimetric sets in non-smooth metric measure spaces. It first constructs a length PI space without the deformation property where an isoperimetric set has no interior in any $ rak m$-a.e. representative, illustrating sharp limits of topological regularity. It then establishes a sufficient condition for deformation based on an upper Laplacian bound for the squared distance function $ riangle d_x^2\nleq C rak m$, which yields perimeter control and, in PI spaces, the deformation property. When applied to essentially non-branching ${ m MCP}(K,N)$ spaces (encompassing many ${ m CD}(K,N)$ spaces and several sub-Riemannian settings), this framework ensures that every isoperimetric set has an open essential interior $E^{(1)}$, with boundary and density properties, and, under a unit-ball volume lower bound, the open representative is bounded. The results unify a counterexample with a broad deformation criterion, clarifying when isoperimetric sets exhibit strong topological regularity in non-smooth geometric contexts.

Abstract

In the setting of length PI spaces satisfying a suitable deformation property, it is known that each isoperimetric set has an open representative. In this paper, we construct an example of a length PI space (without the deformation property) where an isoperimetric set does not have any representative whose topological interior is non-empty. Moreover, we provide a sufficient condition for the validity of the deformation property, consisting in an upper Laplacian bound for the squared distance functions from a point. Our result applies to essentially non-branching ${\sf MCP}(K,N)$ spaces, thus in particular to essentially non-branching ${\sf CD}(K,N)$ spaces and to many Carnot groups and sub-Riemannian manifolds. As a consequence, every isoperimetric set in an essentially non-branching ${\sf MCP}(K,N)$ space has an open representative, which is also bounded whenever a uniform lower bound on the volumes of unit balls is assumed.

A note on Laplacian bounds, deformation properties and isoperimetric sets in metric measure spaces

TL;DR

This paper investigates regularity of isoperimetric sets in non-smooth metric measure spaces. It first constructs a length PI space without the deformation property where an isoperimetric set has no interior in any -a.e. representative, illustrating sharp limits of topological regularity. It then establishes a sufficient condition for deformation based on an upper Laplacian bound for the squared distance function , which yields perimeter control and, in PI spaces, the deformation property. When applied to essentially non-branching spaces (encompassing many spaces and several sub-Riemannian settings), this framework ensures that every isoperimetric set has an open essential interior , with boundary and density properties, and, under a unit-ball volume lower bound, the open representative is bounded. The results unify a counterexample with a broad deformation criterion, clarifying when isoperimetric sets exhibit strong topological regularity in non-smooth geometric contexts.

Abstract

In the setting of length PI spaces satisfying a suitable deformation property, it is known that each isoperimetric set has an open representative. In this paper, we construct an example of a length PI space (without the deformation property) where an isoperimetric set does not have any representative whose topological interior is non-empty. Moreover, we provide a sufficient condition for the validity of the deformation property, consisting in an upper Laplacian bound for the squared distance functions from a point. Our result applies to essentially non-branching spaces, thus in particular to essentially non-branching spaces and to many Carnot groups and sub-Riemannian manifolds. As a consequence, every isoperimetric set in an essentially non-branching space has an open representative, which is also bounded whenever a uniform lower bound on the volumes of unit balls is assumed.

Paper Structure

This paper contains 9 sections, 15 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. General overview
  3. Statement of results
  4. Preliminaries
  5. Metric spaces and Lipschitz functions
  6. Metric measure spaces and upper Laplacian bounds
  7. Sets of finite perimeter
  8. Proof of Theorem \ref{['thm:main_1']}
  9. Proof of Theorems \ref{['thm:main_2']} and \ref{['thm:MCP']}

Key Result

Theorem 1.1

There exist a length PI space $({\rm X},{\sf d},\mathfrak{m})$ and an isoperimetric set $E\subseteq{\rm X}$ such that every $\mathfrak{m}$-a.e. representative of $E$ has empty topological interior.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 24 more