  We show that a mean curvature flow starting from a compact, smoothly embedded
hypersurface M remains unique past singularities, provided the singularities
are of mean convex type, i.e., if around each singular point, the surface moves
in one direction. Specifically, the level set flow of M does not fatten if all
singularities are of mean convex type. This generalizes the well known fact
that the level set flow of a mean convex initial hypersurface M does not
fatten. This also provides the first instance where non-fattening is concluded
from local information around the singular set.
