MEAN CURVATURE FLOW OF SUBMANIFOLDS WITH SMALL TRACELESS SECOND FUNDAMENTAL FORM - Zhou, Zhe; WU, CHUANXI; LI, GUANGHAN (JOURNAL OF ADVANCES IN MATHEMATICS, 2015)
Consider a family of smooth immersions F(; t) : Mn Mn+k of submanifolds in Mn+k moving by mean curvature flow = , where is the mean curvature vector for the evolving submanifold. We prove that for any n >-2 and k>-1, the flow starting from a closed submanifold with small L2-norm of the traceless second fundamental form contracts to a round point in finite time, and the corresponding normalized flow converges exponentially in the C-topology, to an n-sphere in some subspace Mn+1 of Mn+k.