Universality in mean curvature flow neckpinches

TL;DR

This paper proves that noncompact surfaces evolving under mean curvature flow that are initially close to a standard neck will develop a neckpinch singularity, become asymptotically rotationally symmetric, and have a unique tangent flow.

Contribution

It establishes the universality of neckpinch singularities and asymptotic symmetry without symmetry assumptions for mean curvature flow.

Findings

Neckpinch singularities occur in finite time.

Solutions become asymptotically rotationally symmetric near singularities.

Unique tangent flow at the singularity.

Abstract

We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is $C^3$-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singular set, and will have a unique tangent flow.