Asian Journal of Mathematics

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

Ben Andrews, Haizhong Li, and Yong Wei

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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

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1417489241

Mathematical Reviews number (MathSciNet)
MR3287002

Zentralblatt MATH identifier
1306.53059

Subjects
Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Keywords
Mean curvature flow $\mathcal{F}$-stability self-shrinker

Citation

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


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