Type II Smoothing in Mean Curvature Flow

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.