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Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds

Gioacchino Antonelli, Enrico Pasqualetto, Marco Pozzetta, Daniele Semola

TL;DR

The paper develops a framework to study sharp isoperimetric comparison on non-smooth, non-collapsed spaces with lower Ricci bounds by combining a sharp Laplacian analysis of the distance to isoperimetric sets with a viscosity-interpretation of second-order inequalities for the isoperimetric profile $I$. A key ingredient is the Laplacian comparison for the signed distance from an isoperimetric region, yielding mean curvature barrier information that replaces classical second variation arguments in low regularity. The authors also prove an asymptotic mass decomposition for perimeter-minimizing sequences and leverage localization techniques to obtain one-dimensional reductions and stability results under pmGH convergence. Consequently, they obtain sharp second-order differential inequalities for $I$, uniform regularity and diameter bounds for small volumes, and a robust stability and convergence theory for isoperimetric regions in ${ m RCD}(K,N)$ spaces, extending known compact, smooth, and Alexandrov results to a broad non-smooth non-compact setting.

Abstract

This paper studies sharp isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called $N$-dimensional ${\rm RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences. Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies.

Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds

TL;DR

The paper develops a framework to study sharp isoperimetric comparison on non-smooth, non-collapsed spaces with lower Ricci bounds by combining a sharp Laplacian analysis of the distance to isoperimetric sets with a viscosity-interpretation of second-order inequalities for the isoperimetric profile . A key ingredient is the Laplacian comparison for the signed distance from an isoperimetric region, yielding mean curvature barrier information that replaces classical second variation arguments in low regularity. The authors also prove an asymptotic mass decomposition for perimeter-minimizing sequences and leverage localization techniques to obtain one-dimensional reductions and stability results under pmGH convergence. Consequently, they obtain sharp second-order differential inequalities for , uniform regularity and diameter bounds for small volumes, and a robust stability and convergence theory for isoperimetric regions in spaces, extending known compact, smooth, and Alexandrov results to a broad non-smooth non-compact setting.

Abstract

This paper studies sharp isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called -dimensional spaces . The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences. Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies.
Paper Structure (12 sections, 31 theorems, 158 equations)

This paper contains 12 sections, 31 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence and stability results
  4. BV functions and sets of finite perimeter in metric measure spaces
  5. Sobolev functions, Laplacians and vector fields in metric measure spaces
  6. Geometric analysis on RCD spaces
  7. Localization of the curvature-dimension condition
  8. The distance function from isoperimetric sets
  9. Concavity properties of the isoperimetric profile function and consequences
  10. Sharp concavity inequalities for the isoperimetric profile
  11. Fine properties of the isoperimetric profile
  12. Consequences

Key Result

Theorem 1.1

Let $(X,\mathsf{d},\mathscr{H}^N)$ be an $\mathop{\mathrm{RCD}}\nolimits(K,N)$ space. Assume that there exists $v_0>0$ such that $\mathscr{H}^N(B_1(x))\geq v_0$ for every $x\in X$. Let $I:(0,\mathscr{H}^N(X))\to (0,\infty)$ be the isoperimetric profile of $X$. Then:

Theorems & Definitions (69)

  • Theorem 1.1: cf. with \ref{['thm:BavardPansu']}
  • Theorem 1.2: cf. with \ref{['thm:Isoperimetrici']}
  • Conjecture 1
  • Definition 1: pGH and pmGH convergence
  • Definition 2: $L^1$-strong and $L^1_{\mathrm{loc}}$ convergence
  • Definition 3: Hausdorff convergence
  • Definition 4: $\rm BV$ functions and perimeter on m.m.s.
  • Theorem 2.1: Coarea formula
  • Definition 5: Local Laplacian
  • Definition 6: Measure-valued Laplacian
  • ...and 59 more