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A gap theorem of self-shrinkers


Authors: Qing-Ming Cheng and Guoxin Wei
Journal: Trans. Amer. Math. Soc. 367 (2015), 4895-4915
MSC (2010): Primary 53C44, 53C42
DOI: https://doi.org/10.1090/S0002-9947-2015-06161-3
Published electronically: March 13, 2015
MathSciNet review: 3335404
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Abstract: In this paper, we study complete self-shrinkers in Euclidean space and prove that an $ n$-dimensional complete self-shrinker with polynomial volume growth in Euclidean space $ \mathbb{R}^{n+1}$ is isometric to either $ \mathbb{R}^{n}$, $ S^{n}(\sqrt {n})$, or $ \mathbb{R}^{n-m}\times S^m (\sqrt {m})$, $ 1\leq m\leq n-1$, if the squared norm $ S$ of the second fundamental form is constant and satisfies $ S<\frac {10}{7}$.


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Additional Information

Qing-Ming Cheng
Affiliation: Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, 814-0180, Fukuoka, Japan
Email: cheng@fukuoka-u.ac.jp

Guoxin Wei
Affiliation: School of Mathematical Sciences, South China Normal University, 510631, Guangzhou, People’s Republic of China
Email: weiguoxin@tsinghua.org.cn

DOI: https://doi.org/10.1090/S0002-9947-2015-06161-3
Keywords: The second fundamental form, elliptic operator, self-shrinkers
Received by editor(s): December 8, 2012
Received by editor(s) in revised form: April 13, 2013
Published electronically: March 13, 2015
Additional Notes: The first author was partially supported by JSPS Grant-in-Aid for Scientific Research (B): No. 24340013 and Challenging Exploratory Research No. 25610016
The second author was partly supported by NSFC No. 11001087 and the project of Pear River New Star of Guangzhou No. 2012J2200028
Article copyright: © Copyright 2015 American Mathematical Society