A gap theorem of self-shrinkers
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- by Qing-Ming Cheng and Guoxin Wei PDF
- Trans. Amer. Math. Soc. 367 (2015), 4895-4915 Request permission
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 $S<\frac {10}{7}$.References
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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
- ORCID: 0000-0003-3191-2013
- Email: weiguoxin@tsinghua.org.cn
- 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 - © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3335404