Mean curvature flow of pinched submanifolds to spheres

We consider compact submanifolds of dimension n 2 in R n+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 R n+k .