$\mathcal{F}$-stability for self-shrinking solutions to mean curvature   flow
  In this paper, we formulate the notion of the $\mathcal{F}$-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the $\mathcal{F}$-stable self-shrinkers in arbitrary codimension, in codimension one case, our results reduce to Colding-Minicozzi's results. 
Ben Andrews, Haizhong Li and Yong Wei
math.DG