A pinching theorem for conformal classes of Willmore surfaces in the unit $n$-sphere
by
Yu-Chung Chang
Yi-Jung Hsu
Vol. 1 No. 2 (2006) P.231~P.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.
KEYWORDS
Willmore surface, totally umbilical, sphere
MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 53A10, 32J15
MILESTONES
Received: 2004-12-13
Revised :
Accepted:
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