Nondegenerate neck pinches along the mean curvature flow

TL;DR

This paper proves that for most smooth compact surfaces evolving under mean curvature flow in three-dimensional space, the initial singularities are well-behaved neck pinches that are isolated in spacetime.

Contribution

It establishes the generic occurrence of nondegenerate neck pinch singularities at the first singular time for mean curvature flow in b2, providing insight into the nature and isolation of these singularities.

Findings

First singularities are spherical or nondegenerate neck pinches.

Singularities at the first singular time are isolated in spacetime.

Results apply to generic smooth compact initial surfaces.

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.