Bifurcation of perturbations of non-generic closed self-shrinkers

TL;DR

This paper investigates how small perturbations of non-generic closed self-shrinkers influence the nature and direction of subsequent singularities in mean curvature flow, revealing bifurcation phenomena based on perturbation direction.

Contribution

It introduces a bifurcation analysis of perturbations of non-generic closed self-shrinkers, linking perturbation direction to the type and collapse side of ensuing singularities.

Findings

Outward perturbations lead to cylindrical singularities collapsing from outside.

Inward perturbations lead to cylindrical singularities collapsing from inside.

Bifurcation behavior depends on the perturbation direction of the self-shrinkers.

Abstract

We discover a bifurcation of the perturbations of non-generic closed self-shrinkers. If the generic perturbation is outward, then the next mean curvature flow singularity is cylindrical and collapsing from outside; if the generic perturbation is inward, then the next mean curvature flow singularity is cylindrical and collapsing from inside.