Type-II smoothing in mean curvature flow

Velázquez (1994) constructed a smooth O(4) × O(4) invariant mean curvature flow that forms a type-II singularity at the origin in spacetime.Stolarski (2023) showed that the mean curvature on this solution is uniformly bounded.In a work in preparation, Angenent, Ilmanen, and Velázquez (Angenent et al.) also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity.Here we prove 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 and Stolarski we therefore show that there exists a solution {M 7 t ⊂ ‫ޒ‬ 8 | -t 0 < t < t 0 } that has an isolated singularity at the origin 0 ∈ ‫ޒ‬ 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. P. Daskalopoulos