## Asian Journal of Mathematics

### $\mathcal{F}$-stability for self-shrinking solutions to mean curvature flow

#### Abstract

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. We show that the only $\mathcal{F}$-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 $\mathcal{F}$-stable self-shrinkers with parallel principal normal. In the codimension one case, our results reduce to those of Colding and Minicozzi.

#### Article information

Source
Asian J. Math., Volume 18, Number 5 (2014), 757-778.

Dates
First available in Project Euclid: 2 December 2014

Andrews, Ben; Li, Haizhong; Wei, Yong. $\mathcal{F}$-stability for self-shrinking solutions to mean curvature flow. Asian J. Math. 18 (2014), no. 5, 757--778. https://projecteuclid.org/euclid.ajm/1417489241