The index of shrinkers of the mean curvature flow

TL;DR

This paper introduces an index concept for mean curvature flow shrinkers, proves a gap theorem for rotationally symmetric shrinkers in three dimensions, and extends the results to higher dimensions.

Contribution

It defines a new index for shrinkers and establishes a lower bound for the index of rotationally symmetric shrinkers, identifying stable cases and generalizing to higher dimensions.

Findings

Rotationally symmetric shrinkers in R^3 have index at least 3.

The sphere, cylinder, and plane are the only stable shrinkers.

The results extend to higher-dimensional shrinkers.

Abstract

We introduce a notion of index for shrinkers of the mean curvature flow. We then prove a gap theorem for the index of rotationally symmetric immersed shrinkers in R^3, namely, that such shrinkers have index at least 3, unless they are one of the stable ones: the sphere, the cylinder, or the plane. We also provide a generalization of the result to higher dimensions.