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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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 $S<\frac {10}{7}$.
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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