Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy

TL;DR

This paper analyzes the behavior of mean curvature flow near cylindrical singularities, providing a normal form for the asymptotics, defining nondegeneracy, and establishing key properties of such singularities.

Contribution

It introduces a normal form for the asymptotics of mean curvature flow near cylindrical singularities and defines nondegeneracy, revealing their isolated and mean convex nature.

Findings

Rescaled flow converges to a smooth generalized cylinder

Normal form of asymptotics over the cylinder

Nondegenerate singularities are isolated, mean convex, and type-I

Abstract

This paper studies the dynamics of mean curvature flow as it approaches a cylindrical singularity. We proved that the rescaled mean curvature flow converging to a smooth generalized cylinder can be written as a graph over the cylinder in a ball of radius $K\sqrt{t}$, and a normal form of the asymptotics. Using the normal form, we can define the nondegeneracy of cylindrical singularities, and we show that nondegenerate cylindrical singularities are isolated in space, have a mean convex neighborhood, and are type-I.