Mean curvature flow of pinched submanifolds to spheres

Abstract

We consider compact submanifolds of dimension n ≥ 2 in ℝn+k, with nonzero mean curvature vector everywhere, and such that the full norm of the second fundamental form is bounded by a fixed multiple (depending on n) of the length of the mean curvature vector at every point. We prove that the mean curvature flow deforms such a submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in some (n + 1)-dimensional affine subspace of ℝn+k © 2010 Journal of Differential Geometry.