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

Ben Andrews (Mathematical Sciences Institute, Australia National University, Canberra, Australia; and Mathematical Sciences Center, Tsinghua University, Beijing, China)

Haizhong Li (Department of Mathematical Sciences, and Mathematical Sciences Center, Tsinghua University, Beijing, China)

Yong Wei (Department of Mathematical Sciences, Tsinghua University, Beijing, China; and Department of Mathematics, University College London, United Kingdom)

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.

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