The tension field of the conformal Gauss map

Jianbo Fang, Lin Liang Colloquium Mathematicum MSC: Primary 53A10; Secondary 53B30. DOI: 10.4064/cm7955-11-2020 Published online: 28 May 2021

Abstract

Let $x:M^{m}\rightarrow S^{m+1}$ be an $m$-dimensional hypersurface isometrically immersed in an $(m+1)$-dimensional unit sphere. The smooth map $$\varphi =(H,Hx+e_{m+1}):M^{m}\rightarrow S^{m+2}_{1}$$ is called the conformal Gauss map of $x$, where $S^{m+2}_{1}$ is the $(m+2)$-dimensional de Sitter space, $H$ the mean curvature and $e_{m+1}$ the local normal frame field of $x$. Given the Möbius metric on $M^{m}$, the harmonicity of $ \varphi $ has some connection with the fact that the immersion $x$ is Willmore. In this paper, by pulling the metric back via $x$, we study the tension field of the conformal Gauss map, and prove that the conformal Gauss map is harmonic if and only if $x$ is a minimal immersion. Finally, we give some examples for which the conformal Gauss map is harmonic in $S^{3}$.

Authors

  • Jianbo FangSchool of Mathematics and Statistics
    Guizhou University of Finance and Economics
    Guiyang, 550025, China
    e-mail
  • Lin LiangPersonnel Division
    Chuxiong Normal University
    Chuxiong, 675000, China
    e-mail

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