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

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.

43 pages. We divide the original version into two parts, and this is the first part. The second part will be posted in the future. There are more details included, and for the references of later work, some new statements and arguments are added. Comments are welcome!