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Biharmonic Hypersurfaces in a Conformally Flat Space

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Abstract

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and used to determine the conformally flat metric \({f^{-2}\delta_{ij}}\) on the Euclidean space \({\mathbb{R}^{m+1}}\) so that a minimal hypersurface \({M^{m}\longrightarrow (\mathbb{R}^{m+1}, \delta_{ij})}\) in a Euclidean space becomes a biharmonic hypersurface \({M^m\longrightarrow (\mathbb{R}^{m+1}, f^{-2}\delta_{ij})}\) in the conformally flat space. Our examples include all biharmonic hypersurfaces found in Ou (Pac J Math 248(1):217–232, 2010) and Ou and Tang (Mich Math J 61:531–542, 2012) as special cases.

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Authors and Affiliations

  1. Department of Mathematics and Physics, Hunan University of Technology, Zhuzhou Hunan, People’s Republic of China

    Tang Liang

  2. Department of Mathematics, Guangxi University for Nationalities, Nanning, 530006, People’s Republic of China

    Ye-Lin Ou

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  1. Tang Liang
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  2. Ye-Lin Ou
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Correspondence to Ye-Lin Ou.

Additional information

Ye-Lin Ou Research supported by NSF of Guangxi (People’s Republic of China), 2011GXNSFA018127.

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Liang, T., Ou, YL. Biharmonic Hypersurfaces in a Conformally Flat Space. Results. Math. 64, 91–104 (2013). https://doi.org/10.1007/s00025-012-0299-x

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  • DOI: https://doi.org/10.1007/s00025-012-0299-x

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