Type II smoothing in mean curvature flow

TL;DR

This paper proves the short-time existence of a smooth continuation of a mean curvature flow solution that develops a type-II singularity, showing the mean curvature remains bounded despite unbounded second fundamental form near the singularity.

Contribution

It establishes the short-time existence of Velazquez's formal continuation of a mean curvature flow with a type-II singularity and confirms bounded mean curvature on this continuation.

Findings

Existence of a smooth continuation after the singularity.

Mean curvature remains bounded on the continuation.

Second fundamental form is unbounded near the singularity.

Abstract

In 1994 Velazquez constructed a smooth \(O(4)\times O(4)\) invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly bounded. Earlier, Velazquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Here we prove short time existence of Velazquez formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation. Combined with the earlier results of Velazquez-Stolarski we therefore show that there exists a solution \(\{M_t^7\subset\R^8 \mid -t_0 <t<t_0\}\) that has an isolated singularity at the origin \(0\in\R^8\), and at \(t=0\); moreover, the mean curvature is uniformly bounded on this solution, even though the second fundamental form is unbounded near the singularity.