Mean curvature flow with generic initial data II

TL;DR

This paper proves that, for generic initial conditions, the mean curvature flow of closed surfaces in three-dimensional space avoids certain singularities, specifically non-round cylindrical tangent flows and non-cylindrical self-shrinkers with cylindrical ends.

Contribution

It establishes generic avoidance of specific singularities in mean curvature flow, extending understanding of flow behavior near singularities for closed surfaces in bc.

Findings

Generic mean curvature flow avoids non-round cylindrical tangent flows.

Non-cylindrical self-shrinkers with cylindrical ends do not occur generically.

Results improve understanding of singularity formation in mean curvature flow.

Abstract

We show that the mean curvature flow of a generic closed surface in $\mathbb{R}^3$ avoids multiplicity one tangent flows that are not round spheres/cylinders. In particular, we show that any non-cylindrical self-shrinker with a cylindrical end cannot arise generically.