Annals of Global Analysis and Geometry

, Volume 40, Issue 2, pp 125–131 | Cite as

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

Original Paper


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 \({\int_M\vert H\vert^2 v_g=\infty}\) .


Harmonic map Biharmonic map Isometric immersion Minimal Ricci curvature 

Mathematics Subject Classification (2000)



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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Graduate School of Science and EngineeringYamaguchi UniversityYamaguchiJapan
  2. 2.Institute for International EducationTohoku UniversitySendaiJapan

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