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Passing through nondegenerate singularities in mean curvature flows

Ao Sun, Zhihan Wang, Jinxin Xue

Abstract

In this paper, we study the properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing through the singularities. Particularly, the topology change agrees with the level sets change near a critical point of a Morse function, which is the same as performing surgery. The proof is based on a new $L^2$-distance monotonicity formula, which allows us to derive a discrete almost monotonicity of the ``decay order", a discrete mean curvature flow analog to Almgren's frequency function.

Passing through nondegenerate singularities in mean curvature flows

Abstract

In this paper, we study the properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing through the singularities. Particularly, the topology change agrees with the level sets change near a critical point of a Morse function, which is the same as performing surgery. The proof is based on a new -distance monotonicity formula, which allows us to derive a discrete almost monotonicity of the ``decay order", a discrete mean curvature flow analog to Almgren's frequency function.

Paper Structure

This paper contains 26 sections, 34 theorems, 214 equations, 1 table.

Table of Contents

  1. Introduction
  2. Why nondegenerate cylindrical singularities?
  3. Topological Implications
  4. Main Idea of the Proof
  5. Exponential growing modes out of nondegeneracy
  6. Nonconcentration at infinity
  7. Topology of the flow after passing through a nondegenerate singularity
  8. Organization of the paper
  9. Preliminary
  10. Geometry of generalized cylinders as shrinkers
  11. Brakke flow and weak set flow
  12. Nondegenerate singularity and its property before singular time.
  13. Classification of noncollapsing ancient solutions
  14. Use of constants.
  15. An $L^2$-distance monotonicity and applications
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $1\leq k\leq n-1$, $t\mapsto \mathbf{M}(t)$ be a mean curvature flow in $\mathbb{R}^{n+1}$ over $(-1, 1)$ with a nondegenerate cylindrical singularity modeled by $\mathcal{C}_{n,k}$ at $(\mathbf{0}, 0)$. Let $Q_r:= \mathbb{B}_{r}^{n-k+1}(0)\times\mathbb{B}_{r}^k(0)\subset \mathbb{R}^{n+1}$. Then

Theorems & Definitions (75)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • proof
  • Conjecture 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7: Lower bound on numbers of singularities
  • proof
  • Corollary 1.8
  • ...and 65 more