{\L}ojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers

TL;DR

This paper proves {}ojasiewicz inequalities for cylindrical self-shrinkers in mean curvature flow, establishing uniqueness and rigidity results, and confirming a conjecture for Abresch-Langer cylinders through novel perturbative analysis.

Contribution

It introduces new {}ojasiewicz inequalities for a broad class of cylindrical self-shrinkers, including Abresch-Langer cylinders, and proves their uniqueness and rigidity in the space of self-shrinkers.

Findings

Uniqueness of blowups modeled on these cylinders

Isolation of cylinders in the space of self-shrinkers

Confirmation of a conjecture for Abresch-Langer cylinders

Abstract

We establish {\L}ojasiewicz inequalities for a class of cylindrical self-shrinkers for the mean curvature flow, which includes round cylinders and cylinders over Abresch-Langer curves, in any codimension. We deduce the uniqueness of blowups at singularities modelled on this class of cylinders, and that any such cylinder is isolated in the space of self-shrinkers. The Abresch-Langer case answers a conjecture of Colding-Minicozzi. Our proof uses direct perturbative analysis of the shrinker mean curvature, so it is new even for round cylinders.