Generic regularity of Level Set Flows with spherical singularity

Sphere is known to be the only generic compact shrinker for mean curvature flow (MCF). In this paper, we characterize generic dynamics of MCFs with a spherical singularity. In terms of the level set flow formulation of MCF, this gives that generically the arrival time function of level set flow with spherical singularity has at most $C^2$ regularity.