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Precise asymptotics near a generic $\mathbb S^1\times\mathbb R^3$ singularity of mean curvature flow

Zhou Gang, Shengwen Wang

Abstract

In the present paper we study a type of generic singularity of mean curvature flow modelled on the bubble-sheet $\mathbb S^1\times\mathbb R^3$ , and we derive an asymptotic profile for a neighborhood of singularity.

Precise asymptotics near a generic $\mathbb S^1\times\mathbb R^3$ singularity of mean curvature flow

Abstract

In the present paper we study a type of generic singularity of mean curvature flow modelled on the bubble-sheet , and we derive an asymptotic profile for a neighborhood of singularity.

Paper Structure

This paper contains 3 sections, 4 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Extension of the asymptotic profile to the singular time
  3. Proof of the main theorem

Key Result

Theorem 1.1

[Theorem 2.1 of GZ] Let $M^4_t$ be a MCF in $\mathbb R^5$ that develops a generic singularity at the space-time $(0,0)\subset\mathbb R^5\times\mathbb R$ and is modeled on the self-shrinker $C_1$ defined above. Then there exist two independent positive constants $\tau_0\gg1$ and $C$ such that for $\t where $y\in\mathbb R^3$ and $\theta\in [0,2\pi)$, $B$ is a $3\times 3$ symmetric real matrix satisf

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • proof : Proof of the Main Theorem \ref{['main']}