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Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature

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Abstract

In this article, we show that, for a biharmonic hypersurface (M, g) of a Riemannian manifold (N, h) of non-positive Ricci curvature, if \({\int_M\vert H\vert^2 v_g<\infty}\), where H is the mean curvature of (M, g) in (N, h), then (M, g) is minimal in (N, h). Thus, for a counter example (M, g) in the case of hypersurfaces to the generalized Chen’s conjecture (cf. Sect. 1), it holds that \({\int_M\vert H\vert^2 v_g=\infty}\) .

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Authors and Affiliations

  1. Graduate School of Science and Engineering, Yamaguchi University, Yamaguchi, 753-8512, Japan

    Nobumitsu Nakauchi

  2. Institute for International Education, Tohoku University, Kawauchi 41, Sendai, 980-8576, Japan

    Hajime Urakawa

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  1. Nobumitsu Nakauchi
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  2. Hajime Urakawa
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Correspondence to Hajime Urakawa.

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Nakauchi, N., Urakawa, H. Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature. Ann Glob Anal Geom 40, 125–131 (2011). https://doi.org/10.1007/s10455-011-9249-1

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  • DOI: https://doi.org/10.1007/s10455-011-9249-1

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