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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A global pinching theorem for surfaces with constant mean curvature in $S^3$


Authors: Yi-Jung Hsu and Tai-Ho Wang
Journal: Proc. Amer. Math. Soc. 130 (2002), 157-161
MSC (2000): Primary 53C40, 53C42
Published electronically: May 3, 2001
MathSciNet review: 1855633
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Abstract:

Let $M$ be a compact immersed surface in the unit sphere $S^3$ with constant mean curvature $H$. Denote by $\phi$ the linear map from $T_p(M)$ into $T_p(M)$, $\phi=A-\frac H2I$, where $A$ is the linear map associated to the second fundamental form and $I$ is the identity map. Let $\Phi$ denote the square of the length of $\phi$. We prove that if $\vert\vert\Phi\vert\vert _{L^2}\leq C$, then $M$ is either totally umbilical or an $H(r)$-torus, where $C$ is a constant depending only on the mean curvature $H$.


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Additional Information

Yi-Jung Hsu
Affiliation: Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan
Email: yjhsu@math.nctu.edu.tw

Tai-Ho Wang
Affiliation: Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan
Email: teich@math.sinica.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06030-0
PII: S 0002-9939(01)06030-0
Keywords: Mean curvature, sphere, totally umbilical
Received by editor(s): April 17, 1997
Received by editor(s) in revised form: May 10, 2000
Published electronically: May 3, 2001
Communicated by: Christopher Croke
Article copyright: © Copyright 2001 American Mathematical Society