• Rigidity theorem forWillmore surfaces in a sphere

• Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/126/02/0253-0260

• Keywords

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

• Abstract

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.

• Author Affiliations

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

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