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Non-fattening of mean curvature flow at singularities of mean convex type

Or Hershkovits, Brian White

Abstract

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. This generalizes the well known fact that the level set flow of a mean convex initial hypersurface M does not fatten. This also provides the first instance where non-fattening is concluded from local information around the singular set.

Non-fattening of mean curvature flow at singularities of mean convex type

Abstract

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. This generalizes the well known fact that the level set flow of a mean convex initial hypersurface M does not fatten. This also provides the first instance where non-fattening is concluded from local information around the singular set.

Paper Structure

This paper contains 6 sections, 16 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Mean convex blow up type implies mean convex type
  4. Weak set flows
  5. The outermost Brakke flow
  6. Additional Results about Inner and Outer Flows

Key Result

Theorem 7

Let $M\subseteq\mathbf{R}^{n+1}$ be a compact, smoothly embedded hypersurface. If $T_{\textnormal{disc}}<\infty$, then there exists a backwardly singular point $x\in M(T_{\textnormal{disc}})$ that is neither of mean convex nor of mean concave type. Equivalently, suppose that $0<T\le T_{\textnormal{d

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Remark 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Theorem 7
  • Theorem 8
  • Remark 9
  • Definition 10
  • ...and 41 more