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Acta Mathematica Vietnamica

, Volume 42, Issue 3, pp 471–490 | Cite as

L k -biharmonic Hypersurfaces in Space Forms

  • M. Aminian
  • S. M. B. KashaniEmail author
Article
  • 174 Downloads

Abstract

In this paper, we introduce L k -biharmonic hypersurfaces M in simply connected space forms R n+1(c) and propose L k -conjecture for them. For c=0,−1, we prove the conjecture when hypersurface M has two principal curvatures with multiplicities 1,n−1, or M is weakly convex, or M is complete with some constraints on it and on L k . We also show that neither there is any L k -biharmonic hypersurface M n in \( \mathbb {H}^{n+1} \) with two principal curvatures of multiplicities greater than one, nor any L k -biharmonic compact hypersurface M n in \( \mathbb {R}^{n+1} \) or in \( \mathbb {H}^{n+1} \). As a by-product, we get two useful, important variational formulas. The paper is a sequel to our previous paper, (Taiwan. J. Math., 19, 861–874, 5) in this context.

Keywords

Lk operator (bi)energy functionals (bi)harmonic maps Chen conjecture 

Mathematics Subject Classification (2010)

53C40 53C42 

References

  1. 1.
    Akutagawa, K., Maeta, S.: Biharmonic properly immersed submanifolds in Euclidean spaces. Geom. Ded. 164, 351–355 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alías, L.J., García-Martínez, S.C., Rigoli, M.: Biharmonic hypersurfaces in complete Riemannian manifolds. Pacific. J. Math. 263, 1–12 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alías, L. J., Gürbüz, N.: An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures. Geom. Ded. 121, 113–127 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alías, L.J., Kashani, S.M.B.: Hypersurfaces in space forms satisfying the condition L k x=A x+b. Taiwan. J. Math. 14, 1957–1977 (2010)zbMATHGoogle Scholar
  5. 5.
    Aminian, M., Kashani, S.M.B.: L k-biharmonic hypersurfaces in the Euclidean space. Taiwan. J. Math. 19, 861–874 (2015)CrossRefzbMATHGoogle Scholar
  6. 6.
    Alías, L.J., Kurose, T., Solanes, G.: Hadamard-type theorems for hypersurfaces in hyperbolic spaces Differ. Geom. Appl. 24, 492–502 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Balmuş, A.: Biharmonic Maps and Submanifolds. Geometry Balkan Press. Bucharest, Romania (2009)zbMATHGoogle Scholar
  8. 8.
    Balmuş, A., Montaldo, S., Oniciuc, C.: Biharmonic hypersurfaces in 4-dimensional space forms. Math. Nachr. 283, 1696–1705 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Balmuş, A., Montaldo, S., Oniciuc, C.: Classification results for biharmonic submanifolds in spheres. arXiv:0701155v1[math.DG] (2007)
  10. 10.
    Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of S 3. Inter. J. Math. 12, 867–876 (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cartan, É.: Familles de surfaces isoparamétriques dans les espaces à courbure constante. Ann. di Mat. 17, 177–191 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, B.Y.: Recent developments of biharmonic conjecture and modified biharmonic conjectures. arXiv:1307.0245v3[math.DG] (2013)
  13. 13.
    Chen, B.Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17, 169–188 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chen, B.Y.: Some open problems and conjectures on submanifolds of finite type: recent development. Tamkang J. Math. 45, 87–108 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, B.Y.: Total Mean Curvature and Submanifold of Finite Type, 2nd ed. Series in Pure Math, vol. 27. World Scientific, New Jersey (2014)Google Scholar
  16. 16.
    Chen, B.Y., Ishikawa, S.: Biharmonic surfaces in pseudo-Euclidean spaces. Mem. Fac. Sci. Kyushu Univ. 45A, 323–347 (1991)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chen, B.Y., Munteanu, M.I.: Biharmonic ideal hypersurfaces in Euclidean spaces. Differ. Geom. Appl. 31, 1–16 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Defever, F.: Hypersurfaces of \( \mathbb {E}^{4} \) with harmonic mean curvature vector. Math. Nachr. 196, 61–69 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dimitrić, I.: Quadratic representation and submanifolds of finite type, Ph.D. thesis. Michigan State Univ., Lansing MI (1989)Google Scholar
  20. 20.
    Dimitrić, I.: Submanifolds of \( \mathbb {E}^{m} \) with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sinica 20, 53–65 (1992)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hasanis, T., Vlachos, T.: Hypersurfaces in \( \mathbb {E}^{4} \) with harmonic mean curvature vector field. Math. Nachr. 172, 145–169 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jiang, G.Y.: 2-harmonic maps and their first and second variational formulas. Chinese Ann. Math. 7A, 388–402 (1986) ; the English translation, Note di Mathematica 28, 209–232 (2008)MathSciNetGoogle Scholar
  24. 24.
    Luo, Y.: Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal. Results. Math. 65, 49–56 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lawson, H.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. 89, 187–197 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nakauchi, N., Urakawa, H.: Biharmonic submanifolds in a Riemannian manifold with nonpositive curvature. Results. Math. 63, 467–471 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    O’Neill, B.: Semi-Riemannian Geometry: with Applications to Relativity. Pure Appl. Math. Acad Press, New York (1983)zbMATHGoogle Scholar
  28. 28.
    Ou, Y.L.: Biharmonic hypersurfaces in Riemannian manifolds. Pac. J. Math. 248, 217–232 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ou, Y.L., Tang, L.: The generalized Chen’s conjecture on biharmonic submanifolds is false. Mich. Math. J. 61, 531–542 (2012)CrossRefzbMATHGoogle Scholar
  30. 30.
    Reilly, R. C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8, 465–477 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rosenberg, H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117, 211–239 (1993)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Ryan, P.J.: Homogeneity and some curvature conditions for hypersurfaces. Tohoku Math. J. 21, 363–388 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of Pure MathematicsTarbiat Modares UniversityTehranIran
  2. 2.Department of Pure Mathematics Faculty of Mathematics SciencesTarbiat Modares UniversityTehranIran

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