{\L}ojasiewicz inequalities for mean convex self-shrinkers

TL;DR

This paper establishes {}ojasiewicz inequalities for specific self-shrinkers in geometric analysis, using a novel perturbative approach that bridges previous methods for analyzing mean convex self-shrinkers.

Contribution

Introduces a new perturbative method for proving {}ojasiewicz inequalities applicable to cylinders over Abresch-Langer curves and round cylinders, unifying previous approaches.

Findings

Proves {}ojasiewicz inequalities for certain self-shrinkers.

Develops a perturbative analysis based on a new auxiliary quantity.

Bridges techniques between higher order perturbation and differential geometry.

Abstract

We prove {\L}ojasiewicz inequalities for round cylinders and cylinders over Abresch-Langer curves, using perturbative analysis of a quantity introduced by Colding-Minicozzi. A feature is that this auxiliary quantity allows us to work essentially at first order. This new method interpolates between the higher order perturbative analysis used by the author for certain shrinking cylinders, and the differential geometric method used by Colding-Minicozzi for the round case.