• Hypersurfaces Satisfying $L_rx = Rx$ in Sphere $\mathbb{S}^{n+1}$ or Hyperbolic Space $\mathbb{H}^{n+1}$

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      https://www.ias.ac.in/article/fulltext/pmsc/119/04/0487-0499

    • Keywords

       

      𝑟-Minimal; linearized operator $L_r$; isoparametric hypersurface.

    • Abstract

       

      In this paper, using the method of moving frames, we consider hypersurfaces in Euclidean sphere $\mathbb{S}^{n+1}$ or hyperbolic space $\mathbb{H}^{n+1}$ whose position vector 𝑥 satisfies $L_r x=Rx$, where $L_r$ is the linearized operator of the $(r+1)$-th mean curvature of the hypersurfaces for a fixed $r=0,\ldots,n-1,R\in \mathbb{R}^{(n+2)\times(n+2)}$. If the 𝑟-th mean curvature $H_r$ is constant, we prove that the only hypersurfaces satisfying that condition are 𝑟-minimal $(H_{r+1}\equiv 0)$ or isoparametric. In particular, we locally classify such hypersurfaces which are not 𝑟-minimal.

    • Author Affiliations

       

      Biaogui Yang1 Ximin Liu2

      1. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350108, People’s Republic of China
      2. Department of Mathematics, South China University of Technology, Gangzhou 510641, People’s Republic of China
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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