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

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

Keywords

Harmonic map Biharmonic map Isometric immersion Minimal Ricci curvature 

Mathematics Subject Classification (2000)

58E20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Caddeo R., Montaldo S., Oniciuc C.: Biharmonic submanifolds of \({{\mathbb S}^3}\) . Intern. J. Math. 12, 867–876 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Caddeo R., Montaldo S., Piu P.: On biharmonic maps. Contemp. Math. 288, 286–290 (2001)MathSciNetGoogle Scholar
  3. 3.
    Chen B.Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17, 169–188 (1991)MATHMathSciNetGoogle Scholar
  4. 4.
    Chen B.Y.: A report on submanifolds of finite type. Soochow J. Math. 22, 117–337 (1996)MATHMathSciNetGoogle Scholar
  5. 5.
    Eells, J., Lemaire, L.: Selected Topic in Harmonic Maps, C.M.M.S. Regional Conf. Series Math., vol. 50. American Mathematical Society, Providence (1983)Google Scholar
  6. 6.
    Helgason S.: Differential Geometry and Symmetric Spaces. Academic Press, New York and London (1962)MATHGoogle Scholar
  7. 7.
    Jiang G.Y.: 2-harmonic maps and their first and second variation formula. Chin. Ann. Math. 7A, 388–402 (1986)Google Scholar
  8. 8.
    Kobayashi, S., Nomizu, K.: Foundation of Differential Geometry, Vols. I, II. Wiley, New York (1963–1969)Google Scholar
  9. 9.
    Oniciuc C.: Biharmonic maps between Riemannian manifolds. An. St. Al. Univ. “Al. I. Cuza”, Iasi 68, 237–248 (2002)MathSciNetGoogle Scholar
  10. 10.
    Ou Y.-L.: Biharmonic hypersurfaces in Riemannian manifolds. Pac. J. Math. 248, 217–232 (2010) arXiv: 0901.1507v.3MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ou, Y.-L., Tang, L.: The generalized Chen’s conjecture on biharmonic submanifolds is false, a preprint, arXiv: 1006.1838v.1, (2010)Google Scholar

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

Personalised recommendations