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Willmore functional; Sobolev inequality; mean curvature; totally umbilical surface.

Let $M^{2}$ be a compact Willmore surface in the $(2 + p)$-dimensional unit sphere $S^{2+p}$. Denote by $H$ and $S$ the mean curvature and the squared length of the second fundamental form of $M^{2}$, respectively. Set $\rho^{2} = S − 2H^{2}$. In this note, we proved that there exists a universal positive constant $C$, such that if $\parallel \rho^{2}\parallel_{2}$ < $C$, then $\rho^{2} = 0$ and $M^{2}$ is a totally umbilical sphere.

HONGWEI XU¹ DENGYUN YANG²

1. Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
2. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, People’s Republic of China

Published

Manuscript received

31 July 2014

Manuscript revised

7 November 2014

Accepted

24 November 2014