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Singularity of mean curvature flow with bounded mean curvature and Morse index

Yongheng Han

TL;DR

This work analyzes singularity formation in mean curvature flow for closed smooth embedded hypersurfaces in $\mathbb{R}^{n+1}$ under uniform bounds on the mean curvature and Morse index. It proves that in dimensions $3\le n\le 6$ the first singular time cannot feature uncontrolled blow-up: either $|H|$ or the index must diverge, and the flow converges to a limit with multiplicity one away from a small singular set; the singular set has Minkowski dimension at most $n-7$ and rescaled flows converge to stable minimal cones of multiplicity one. The authors develop a weak-compactness framework for hypersurfaces and flows, construct auxiliary functions to capture stability, and establish a multiplicity-one convergence for the rescaled flow via a detailed space-decomposition and $L$-stability arguments. These results sharpen the understanding of the local and global structure of singularities in mean curvature flow and connect to the broader theory of self-shrinkers and Brakke flows.

Abstract

We study the multiplicity of the singularity of mean curvature flow with bounded mean curvature and Morse index. For $3\leq n\leq 6$, we show that either the mean curvature or the Morse index blows up at the first singular time for a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$.

Singularity of mean curvature flow with bounded mean curvature and Morse index

TL;DR

This work analyzes singularity formation in mean curvature flow for closed smooth embedded hypersurfaces in under uniform bounds on the mean curvature and Morse index. It proves that in dimensions the first singular time cannot feature uncontrolled blow-up: either or the index must diverge, and the flow converges to a limit with multiplicity one away from a small singular set; the singular set has Minkowski dimension at most and rescaled flows converge to stable minimal cones of multiplicity one. The authors develop a weak-compactness framework for hypersurfaces and flows, construct auxiliary functions to capture stability, and establish a multiplicity-one convergence for the rescaled flow via a detailed space-decomposition and -stability arguments. These results sharpen the understanding of the local and global structure of singularities in mean curvature flow and connect to the broader theory of self-shrinkers and Brakke flows.

Abstract

We study the multiplicity of the singularity of mean curvature flow with bounded mean curvature and Morse index. For , we show that either the mean curvature or the Morse index blows up at the first singular time for a closed smooth embedded mean curvature flow in .
Paper Structure (17 sections, 43 theorems, 122 equations)

This paper contains 17 sections, 43 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integral Varifolds and Mean Curvature
  4. The second variation of stationary varifold
  5. F-stationary rectifiable varifolds
  6. mean curvature flow
  7. Two-sided pseudolocality theorem
  8. Weak compactness of the hypersurface
  9. Weak compactness of the mean curvature flow
  10. Multiplicity-one convergence of the rescaled mean curvature flow
  11. Convergence away from singularities
  12. Decomposition of spaces
  13. Construction of auxiliary functions
  14. Proof of the multiplicity-one convergence
  15. Proof of Theorem \ref{['thm0.1']}
  16. ...and 2 more sections

Key Result

Theorem 4

For $n\ge 3$, let $\mathbf{x}:M^n\to \mathbb{R}^{n+1}(t\in [0,T))$ be a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$. Suppose that and where $\mathrm{index}(M_t)$ is the number of negative eigenvalue of the operator $\Delta+|A|^2$ on $M_t$. Then, there exist a hypersurface $M_T$ and subset $\mathcal{S}\subset \bar{M}_T$ such that $M_t$ smoothly converges to $\bar{M}_T$ away f

Theorems & Definitions (80)

  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • ...and 70 more