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Mean curvature flow with generic initial data

Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, Felix Schulze

Abstract

We show that the mean curvature flow of generic closed surfaces in $\mathbb{R}^{3}$ avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in $\mathbb{R}^{4}$ is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

Mean curvature flow with generic initial data

Abstract

We show that the mean curvature flow of generic closed surfaces in avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

Paper Structure

This paper contains 46 sections, 97 theorems, 427 equations.

Table of Contents

  1. Introduction
  2. Overview of results
  3. Singularities in mean curvature flow
  4. Entropy and stability of shrinkers
  5. Our perturbative statements
  6. Our perturbative method: ancient one-sided flows
  7. Other results
  8. Organization of the paper
  9. Acknowledgements
  10. Preliminaries
  11. Spacetime
  12. The spacetime track of a classical flow
  13. Weak set flows and level set flows
  14. Integral Brakke flows
  15. Density and Huisken's monotonicity
  16. ...and 31 more sections

Key Result

Theorem 1.1

Let $M^3\subset \mathbb{R}^{4}$ be any closed connected hypersurface with $\lambda(M) \leq \lambda(\mathbb{S}^{2}\times \mathbb{R})$. There exist arbitrarily small $C^{\infty}$ graphs $M'$ over $M$ so that the mean curvature flow starting from $M'$ is smooth until it disappears in a round point.

Theorems & Definitions (201)

  • Theorem 1.1
  • Corollary 1.2: Bernstein--Wang BernsteinWang:schoenflies
  • Theorem 1.3
  • Theorem 1.4
  • Remark
  • Remark
  • Theorem 1.5
  • Remark
  • Remark
  • Lemma 3.1
  • ...and 191 more