Generic mean curvature flow I; generic singularities

TL;DR

This paper proves that in mean curvature flow of generic smooth closed surfaces in R^3, the only stable singularities are spherical or cylindrical, supporting the conjecture that other singularities can be perturbed away.

Contribution

It establishes that shrinking spheres, cylinders, and planes are the only stable self-shrinkers in all dimensions, advancing understanding of generic singularities in mean curvature flow.

Findings

Shrinking spheres, cylinders, and planes are the only stable self-shrinkers.

Other singularities can be perturbed away in generic flows.

Supports the conjecture that generic singularities are spherical or cylindrical.

Abstract

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every other singularity than spheres and cylinders can be perturbed away.