2006 / June Volume 1 No.2
A pinching theorem for conformal classes of Willmore surfaces in the unit $n$-sphere
| Published Date |
2006 / June
|
|---|---|
| Title | A pinching theorem for conformal classes of Willmore surfaces in the unit $n$-sphere |
| Author | |
| Keyword | |
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| Pagination | 231-261 |
| Abstract | Let $x: M \to S^n$ be a compact immersed Willmore surface in the $n$-dimensional unit sphere. In this paper, we consider the
case of $n \ge 4$. We prove that if
$\inf_{g \in G} \max_{g \circ x(M)} (\Phi_g - {\frac 18}H^2_g - \sqrt{\frac 49 + \frac16 H^2_g + \frac 1{96} H^4_g} \le \frac 23 $
where $G$ is the conformal group of the
ambient space $S^n$; $\Phi_g$ and $H_g$ are the square of the length of the trace free part of the second fundamental form and the length of the mean curvature vector of the immersion $g \circ x$ respectively, then
$x(M)$ is either a totally umbilical sphere or a conformal Veronese surface.
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| AMS Subject Classification |
53A10, 32J15
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| Received |
2004-12-13
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