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Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers

Jonathan J. Zhu

Abstract

We establish Łojasiewicz inequalities for a class of cylindrical self-shrinkers for the mean curvature flow, which includes round cylinders and cylinders over Abresch-Langer curves, in any codimension. We deduce the uniqueness of blowups at singularities modelled on this class of cylinders, and that any such cylinder is isolated in the space of self-shrinkers. The Abresch-Langer case answers a conjecture of Colding-Minicozzi. Our proof uses direct perturbative analysis of the shrinker mean curvature, so it is new even for round cylinders.

Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers

Abstract

We establish Łojasiewicz inequalities for a class of cylindrical self-shrinkers for the mean curvature flow, which includes round cylinders and cylinders over Abresch-Langer curves, in any codimension. We deduce the uniqueness of blowups at singularities modelled on this class of cylinders, and that any such cylinder is isolated in the space of self-shrinkers. The Abresch-Langer case answers a conjecture of Colding-Minicozzi. Our proof uses direct perturbative analysis of the shrinker mean curvature, so it is new even for round cylinders.

Paper Structure

This paper contains 33 sections, 40 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Simply non-integrable shrinkers
  3. Background
  4. Explicit Łojasiewicz inequalities and deformation theory
  5. Technical overview
  6. Outline of the paper
  7. Preliminaries
  8. Generalised cylinders
  9. Shrinking spheres
  10. Abresch-Langer curves
  11. Gaussian Poincaré inequality
  12. Cutoff lemma
  13. Rotation and Jacobi fields
  14. Variation of geometric quantities
  15. First variation
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $\mathring{\Gamma}^k \subset \mathbb{R}^{k+m}$ be a closed, simply non-integrable shrinker and let $M_t^n$ be a MCF in $\mathbb{R}^N$. Assume either that $\mathring{\Gamma}$ is embedded, or that $M_t$ has a type I singularity at $(x_0,t_0)$. If one tangent flow at $(x_0,t_0)$ is induced by $\Gam

Theorems & Definitions (75)

  • Theorem 1.1: Uniqueness of tangent flows
  • Theorem 1.2: Rigidity
  • Theorem 1.3: Łojasiewicz inequality of the first kind
  • Theorem 1.4: Łojasiewicz inequality of the second kind
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: CM19
  • Lemma 2.4
  • ...and 65 more