Regularity of the level set flow
  We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed $C^1$ manifold with cylindrical singularities. 
Tobias Holck Colding and William P. Minicozzi II
math.DG math.AP