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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

1 May 2016

Manuscript received

31 July 2014

Manuscript revised

7 November 2014

Accepted

24 November 2014