Nonconvex Surfaces which Flow to Round Points

Abstract

In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that exist on a uniform time interval, have uniformly bounded diameter, and shrink to round points, yet the sequence of initial surfaces has no subsequence converging in the Gromov-Hausdorff sense. Moreover, we find a sequence of flows which all shrink to round points, yet the initial surfaces converge to a space-filling surface. Also constructed are surfaces of arbitrarily large area which are close in Hausdorff distance to the round sphere yet shrink to round points.