Toti Daskalopoulos

Columbia

Type II smoothing in Mean curvature flow

In 1994 Velázquez constructed a family of smooth \(O(4) × O(4)\) invariant solutions to Mean Curvature Flow that form a type-II singularity at the origin. Stolarski has recently shown that the Velázquez solutions have bounded Mean curvature at the singularity. 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’s formal continuation, and we verify that the Mean curvature is also uniformly bounded. Combined with the earlier results of Velázquez–Stolarski we therefore show the existence of a Mean curvature flow solution \(\{M^7_t ⊂ \mathbb R^8\}_{−t_0<t<t_0}\), that has an isolated singularity at the origin \(0 ∈ \mathbb R^8\) at time \(t = 0\). Moreover, the Mean curvature is uniformly bounded on this solution, even though the second fundamental form is unbounded near the singularity.