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# Asian Journal of Mathematics

## Volume 18 (2014)

### Number 5

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

Pages: 757 – 778

DOI: http://dx.doi.org/10.4310/AJM.2014.v18.n5.a1

#### Authors

#### 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.

#### Keywords

mean curvature flow, $\mathcal{F}$-stability, self-shrinker

#### 2010 Mathematics Subject Classification

Primary 53C44. Secondary 53C42.