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Monotonicity formula and stratification of the singular set of perimeter minimizers in RCD spaces

Francesco Fiorani, Andrea Mondino, Daniele Semola

TL;DR

The paper extends a fundamental Euclidean-geometry tool—the monotonicity formula for perimeter-minimizing sets—to non-smooth settings provided by $ ext{RCD}$ spaces, specifically cones over $ ext{RCD}(N-2,N-1)$ bases. It establishes a nondecreasing density function $\Phi(r)=\mathrm{Per}(E;B_r(O))/r^{N-1}$ for global minimizers in these cones and a rigidity statement that equality forces conical structure, with a complete proof that carefully adapts BV and Gauss-Green techniques to the metric-measure context. Building on this, the authors derive sharp Hausdorff-dimension bounds for the singular strata of perimeter minimizers in $ ext{RCD}(K,N)$ spaces and show that blow-down limits of spaces with Euclidean volume growth inherit perimeter-minimizing cones, thereby connecting tangent/asymptotic cone analysis to variational structures. The results significantly extend geometric measure theory to non-smooth Ricci-curvature–bounded spaces, with potential consequences for Ricci-limit spaces and non-collapsed theories, via a mix of BV approximations, blow-up analysis, and dimension-reduction arguments.

Abstract

The goal of this paper is to establish a monotonicity formula for perimeter minimizing sets in RCD(0,N) metric measure cones, together with the associated rigidity statement. The applications include sharp Hausdorff dimension estimates for the singular strata of perimeter minimizing sets in non collapsed RCD spaces and the existence of blow-down cones for global perimeter minimizers in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth.

Monotonicity formula and stratification of the singular set of perimeter minimizers in RCD spaces

TL;DR

The paper extends a fundamental Euclidean-geometry tool—the monotonicity formula for perimeter-minimizing sets—to non-smooth settings provided by spaces, specifically cones over bases. It establishes a nondecreasing density function for global minimizers in these cones and a rigidity statement that equality forces conical structure, with a complete proof that carefully adapts BV and Gauss-Green techniques to the metric-measure context. Building on this, the authors derive sharp Hausdorff-dimension bounds for the singular strata of perimeter minimizers in spaces and show that blow-down limits of spaces with Euclidean volume growth inherit perimeter-minimizing cones, thereby connecting tangent/asymptotic cone analysis to variational structures. The results significantly extend geometric measure theory to non-smooth Ricci-curvature–bounded spaces, with potential consequences for Ricci-limit spaces and non-collapsed theories, via a mix of BV approximations, blow-up analysis, and dimension-reduction arguments.

Abstract

The goal of this paper is to establish a monotonicity formula for perimeter minimizing sets in RCD(0,N) metric measure cones, together with the associated rigidity statement. The applications include sharp Hausdorff dimension estimates for the singular strata of perimeter minimizing sets in non collapsed RCD spaces and the existence of blow-down cones for global perimeter minimizers in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth.
Paper Structure (8 sections, 27 theorems, 173 equations)

This paper contains 8 sections, 27 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Monotonicity Formula
  3. Stratification of the singular set and other applications
  4. Preliminaries
  5. Metric-measure cones
  6. Finite perimeter sets in $\mathrm{RCD}$ spaces
  7. Monotonicity Formula
  8. Stratification of the Singular Set and further applications

Key Result

Theorem 1.2

Let $N\ge 2$ and let $(X,\mathsf{d},\mathfrak{m})$ be an $\mathrm{RCD}(N-2,N-1)$ space (with ${\rm diam}(X)\le \pi$, if $N=2$). Let $C(X)$ be the metric measure cone over $(X,\mathsf{d},\mathfrak{m})$ and let $O$ denote its tip. Let $E \subset C(X)$ be a global perimeter minimizer. Then the function is non-decreasing. Moreover, if there exist $0<r_1<r_2<\infty$ such that $\Phi(r_1)=\Phi(r_2)$, the

Theorems & Definitions (53)

  • Definition 1.1: Local and Global Perimeter Minimizer
  • Theorem 1.2: Monotonicity Formula
  • Definition 1.3: Singular Strata
  • Definition 1.4: Interior and Boundary Regularity Points
  • Theorem 1.5
  • Theorem 1.6: Stratification of the singular set
  • Theorem 1.7
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • ...and 43 more