A global pinching theorem for surfaces with constant mean curvature in $S^3$
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- by Yi-Jung Hsu and Tai-Ho Wang PDF
- Proc. Amer. Math. Soc. 130 (2002), 157-161 Request permission
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 $||\Phi ||_{L^2}\leq C$, then $M$ is either totally umbilical or an $H(r)$-torus, where $C$ is a constant depending only on the mean curvature $H$.References
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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
- MR Author ID: 674146
- Email: teich@math.sinica.edu.tw
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 157-161
- MSC (2000): Primary 53C40, 53C42
- DOI: https://doi.org/10.1090/S0002-9939-01-06030-0
- MathSciNet review: 1855633