# Type II Smoothing in Mean Curvature Flow

## [Virtual] Hot Topics: Regularity Theory for Minimal Surfaces and Mean Curvature Flow March 21, 2022 - March 24, 2022

**Speaker(s):**Panagiota Daskalopoulos (Columbia University)

**Location:**SLMath: Online/Virtual

**Primary Mathematics Subject Classification**No Primary AMS MSC

**Secondary Mathematics Subject Classification**No Secondary AMS MSC

In 1994 Vel\'azquez constructed a family of smooth \( O(4)\times O(4)\) invariant solutions to Mean Curvature Flow that form a type-II singularity at the origin. Stolarski has recently shown that the Vel\'azquez solutions have bounded Mean curvature at the singularity. Earlier, Vel\'azquez 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 Vel\'azquez--Stolarski we therefore show the existence of a Mean curvature flow solution \( \{ M_t^7\subset \R^8 \}_{-t_0 < t < t_0 } \), that has an {\em isolated singularity} at the origin \( 0\in\R^8\) at time \(t=0\). Moreover, the {\em Mean curvature is uniformly bounded} on this solution, even though the second fundamental form is unbounded near the singularity.