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    • Rigidity theorem forWillmore surfaces in a sphere

      HONGWEI XU DENGYUN YANG

      Research Article Volume 126 Issue 2 May 2016 pp 253-260

    • Fulltext

       

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      Permanent link:
      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

       

      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

    • Dates

       
      Published
      1 May 2016
      Manuscript received
      31 July 2014
      Manuscript revised
      7 November 2014
      Accepted
      24 November 2014

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