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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

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