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SELF-SHRINKERS WITH SECOND FUNDAMENTAL FORM OF CONSTANT LENGTH

Part of: Global differential geometry

Published online by Cambridge University Press:  02 March 2017

QIANG GUANG
QIANG GUANG*
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, USA email guang@math.ucsb.edu
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Abstract

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We give a new and simple proof of a result of Ding and Xin, which states that any smooth complete self-shrinker in $\mathbb{R}^{3}$ with the second fundamental form of constant length must be a generalised cylinder $\mathbb{S}^{k}\times \mathbb{R}^{2-k}$ for some $k\leq 2$. Moreover, we prove a gap theorem for smooth self-shrinkers in all dimensions.

Keywords

self-shrinkersecond fundamental formmean curvature flow

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 326 - 332
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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