Deforming submanifolds of arbitrary codimension in a sphere
  In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\mathbb{S}^{n+d}$. 
Kefeng Liu, Hongwei Xu, Entao Zhao
math.DG