F-stability for self-shrinking solutions to mean curvature flow

Abstract

In this paper, we formulate the notion of the F-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the F-stable selfshrinkers in arbitrary codimension. We show that the only F-stable self-shrinking solution which is a closed minimal submanifold in a sphere must be the shrinking sphere. We also prove that the spheres and planes are the only F-stable self-shrinkers with parallel principal normal. In the codimension one case, our results reduce to those of Colding and Minicozzi [6].