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Volume estimates for the singular sets of mean curvature flows

Hanbing Fang, Yu Li

TL;DR

The paper addresses sharp Minkowski-type bounds for the singular set of mean curvature flow (via level-set flows) initiated from smooth mean-convex hypersurfaces. It develops a quantitative framework for cylindrical singularities, proves a unique cylinder structure across scales, and introduces a neck decomposition to partition space into neck regions and higher-dimensional pieces with controlled packing. These ingredients yield uniform bounds on the parabolic volume and parabolic Hausdorff measure of quantitative strata and reveal that the top stratum lies on Lipschitz graphs, with refinements under uniform k-convexity. The results advance quantitative stratification in mean curvature flow, providing precise geometric and measure-theoretic control with explicit dependencies on entropy and noncollapsing constants, and have implications for regularity and rectifiability of singular sets in geometric flows.

Abstract

In this paper, we establish uniform and sharp volume estimates for the singular set and the quantitative singular strata of mean curvature flows starting from a smooth, closed, mean-convex hypersurface in $\mathbb R^{n+1}$.

Volume estimates for the singular sets of mean curvature flows

TL;DR

The paper addresses sharp Minkowski-type bounds for the singular set of mean curvature flow (via level-set flows) initiated from smooth mean-convex hypersurfaces. It develops a quantitative framework for cylindrical singularities, proves a unique cylinder structure across scales, and introduces a neck decomposition to partition space into neck regions and higher-dimensional pieces with controlled packing. These ingredients yield uniform bounds on the parabolic volume and parabolic Hausdorff measure of quantitative strata and reveal that the top stratum lies on Lipschitz graphs, with refinements under uniform k-convexity. The results advance quantitative stratification in mean curvature flow, providing precise geometric and measure-theoretic control with explicit dependencies on entropy and noncollapsing constants, and have implications for regularity and rectifiability of singular sets in geometric flows.

Abstract

In this paper, we establish uniform and sharp volume estimates for the singular set and the quantitative singular strata of mean curvature flows starting from a smooth, closed, mean-convex hypersurface in .

Paper Structure

This paper contains 9 sections, 21 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. Integral Brakke flows
  5. Level set flow from a mean-convex hypersurface
  6. Quantitative cylindrical singularities
  7. Quantitative uniqueness of cylindrical singularities
  8. Neck decomposition theorem
  9. Proof of Theorem \ref{['volumeestimate1']}

Key Result

Theorem 1.2

Let $\mathcal{M}=\{M_t\}_{t \ge 0}$ be a level set flow starting from a closed, smooth, mean-convex, $\alpha$-noncollapsed hypersurface $M_0 \subset \mathbb{R}^{n+1}$ with $\lambda(M_0) \le \Lambda$. Then, for any $\epsilon>0$, there exists a constant $C=C(n, \alpha, \Lambda, \epsilon)>0$ such that for $j\in\{0,1,\ldots,n-1\}$ and any $r \in (0,1)$. In particular, for $\mathcal{S}^j_{\epsilon}:=\

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1: Integral Brakke flow
  • Proposition 2.2
  • Definition 2.3: $\alpha$-noncollapsed
  • Theorem 2.4
  • Definition 2.5: $\alpha$-Andrews flows
  • ...and 34 more