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Surfaces expanding by the power of the Gauss curvature flow


Author: Qi-Rui Li
Journal: Proc. Amer. Math. Soc. 138 (2010), 4089-4102
MSC (2010): Primary 53C44, 35K55
DOI: https://doi.org/10.1090/S0002-9939-2010-10431-8
Published electronically: June 16, 2010
MathSciNet review: 2679630
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Abstract: In this paper, we describe the flow of 2-surfaces in $ \mathbb{R}^{3}$ for some negative power of the Gauss curvature. We show that strictly convex surfaces expanding with normal velocity $ K^{-\alpha }$, when $ \frac{1}{2}<\alpha \leq 1$, converge to infinity in finite time. After appropriate rescaling, they converge to spheres. In the 2-dimensional case, our results close an apparent gap in the powers considered by previous authors, that is, for $ \alpha \in (0,\frac{1}{2}]$ by Urbas and Huisken and for $ \alpha =1$ by Schnürer.


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Additional Information

Qi-Rui Li
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: 85lqr@163.com

DOI: https://doi.org/10.1090/S0002-9939-2010-10431-8
Keywords: Surface, expanding curvature flow, velocity function $K^{-\alpha}$
Received by editor(s): June 2, 2009
Received by editor(s) in revised form: October 9, 2009, November 26, 2009, December 19, 2009, December 31, 2009, and February 6, 2010
Published electronically: June 16, 2010
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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