Mean Curvature Flow of Submanifolds With Small Traceless Second Fundamental Form

Abstract

Consider a family of smooth immersions F(; t) : Mn ! Rn+k of submanifolds in Rn+k moving by mean curvature ow @F @t = ~H, where ~H is the mean curvature vector for the evolving submanifold. We prove that for any n 2 and k 1, the ow starting from a closed submanifold with small L2-norm of the traceless second fundamental form contracts to a round point in nite time, and the corresponding normalized ow converges exponentially in the C1-topology, to an n-sphere in some subspace Rn+1 of Rn+k.