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Nondegenerate neck pinches along the mean curvature flow

Gábor Székelyhidi

Abstract

We show that for generic smooth compact initial surfaces the mean curvature flow in $\mathbb{R}^3$ has spherical or nondegenerate neck pinch singularities at the first singular time. In particular the singularities at the first singular time are isolated in spacetime.

Nondegenerate neck pinches along the mean curvature flow

Abstract

We show that for generic smooth compact initial surfaces the mean curvature flow in has spherical or nondegenerate neck pinch singularities at the first singular time. In particular the singularities at the first singular time are isolated in spacetime.
Paper Structure (4 sections, 16 theorems, 87 equations)

This paper contains 4 sections, 16 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Perturbations of a degenerate cylindrical singularity
  4. Proof of the main result

Key Result

Theorem 1

Suppose that $S_0\subset \mathbb{R}^3$ is a smooth compact surface. There are arbitrarily small $C^2$ perturbations $\tilde{S}_0$ of $S_0$ such that the mean curvature flow $\tilde{S}_t$ with initial condition $\tilde{S}_0$ admits only spherical and non-degenerate cylindrical singularities at its fi

Theorems & Definitions (30)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7
  • ...and 20 more