Type II smoothing in Mean curvature flow

Abstract: In 1994 Velázquez constructed a smooth $O(4)\times O(4)$ invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Recently, Stolarski showed that the mean curvature on this solution is uniformly bounded. Earlier, Velázquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Jointly with S. Angenent and N. Sesum we establish the short time existence of Velázquez' formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation. Combined with the earlier results of Velázquez–Stolarski we therefore show that there exists a solution $\left\{ M_t^7\subset \mathbb{R}^8 | -t_0 < t < t_0\right\}$ that has an isolated singularity at the origin $0$ in $\mathbb{R}^8$, 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.

differential geometry

Audience: researchers in the topic

B.O.W.L Geometry Seminar