Geometriae Dedicata

, Volume 169, Issue 1, pp 263–272 | Cite as

Biharmonic maps into a Riemannian manifold of non-positive curvature

  • Nobumitsu Nakauchi
  • Hajime Urakawa
  • Sigmundur Gudmundsson
Original paper


We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to isometric immersions and horizontally conformal submersions.


Harmonic map Biharmonic map Chen’s conjecture   Generalized Chen’s conjecture 

Mathematics Subject Classification (2000)

Primary 58E20 Secondary 53C43 



The second author would like to express his sincere gratitude to Professor Sigmundur Gudmundsson for his hospitality and very intensive and helpful discussions during for the second author staying at Lund University, at 2012 May. Addition of part of Section Three to the original version of the first and second authors was based by this joint work with him during this period.


  1. 1.
    Baird, P., Eells, J.: A conservation law for harmonic maps. Lecture Notes in Mathematics, Springer 894, 1–25 (1981)Google Scholar
  2. 2.
    Baird, P., Fardoun, A., Ouakkas, S.: Liouville-type theorems for biharmonic maps between Riemannian manifolds. Adv. Calc. Var. 3, 49–68 (2010)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Baird, P., Wood, J.: Harmonic Morphisms Between Riemannian Manifolds. Oxford Science Publication, Oxford (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Caddeo, R., Montaldo, S., Piu, P.: On biharmonic maps. Contemp. Math. 288, 286–290 (2001)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, B.Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17, 169–188 (1991)MATHMathSciNetGoogle Scholar
  6. 6.
    Eells, J., Lemaire, L.: A report on harmonic maps. Bull. Lond. Math. Soc. 10, 1–68 (1978)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Eells, J., Lemaire, L.: Selected topics in harmonic maps, CBMS, Am. Math. Soc 50 (1983)Google Scholar
  8. 8.
    Eells, J., Lemaire, L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20, 385–524 (1988)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Gaffney, M.P.: A special Stokes’ theorem for complete Riemannian manifold. Ann. Math. 60, 140–145 (1954)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Gudmundsson, S.: The Bibliography of Harmonic Morphisms,
  12. 12.
    Ichiyama, T., Inoguchi, J., Urakawa, H.: Biharmonic maps and bi-Yang-Mills fields. Note di Matematica 28, 233–275 (2009)MATHMathSciNetGoogle Scholar
  13. 13.
    Ichiyama, T., Inoguchi, J., Urakawa, H.: Classifications and isolation phenomena of biharmonic maps and bi-Yang-Mills fields. Note di Matematica 30, 15–48 (2010)MathSciNetGoogle Scholar
  14. 14.
    Ishihara, S., Ishikawa, S.: Notes on relatively harmonic immersions. Hokkaido Math. J. 4, 234–246 (1975)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Jiang, G.Y.: 2-harmonic maps and their first and second variational formula. Chin. Ann. Math. 7A, 388–402 (1986)Google Scholar
  16. 16.
    Jiang, G.Y.: 2-harmonic maps and their first and second variational formula. Note di Matematica 28, 209–232 (2009)MATHMathSciNetGoogle Scholar
  17. 17.
    Kasue, A.: Riemannian Geometry, in Japanese. Baihu-kan, Tokyo (2001)Google Scholar
  18. 18.
    Lamm, T.: Biharmonic map heat flow into manifolds of nonpositive curvature. Calc. Var. 22, 421–445 (2005)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Loubeau, E., Oniciuc, C.: The index of biharmonic maps in spheres. Compositio Math. 141, 729–745 (2005)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Loubeau, E., Oniciuc, C.: On the biharmonic and harmonic indices of the Hopf map. Trans. Am. Math. Soc. 359, 5239–5256 (2007)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Loubeau, E., Ou, Y.-L.: Biharmonic maps and morphisms from conformal mappings. Tohoku Math. J. 62, 55–73 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Montaldo, S., Oniciuc, C.: A short survey on biharmonic maps between Riemannian manifolds. Rev. Un. Mat. Argentina 47, 1–22 (2006)MATHMathSciNetGoogle Scholar
  23. 23.
    Nakauchi, N., Urakawa, H.: Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature. Ann. Global Anal. Geom. 40, 125–131 (2011)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Nakauchi, N., Urakawa, H.: Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, to apper in results. Math. 63, 467–474 (2013) arXiv:1201.6455v1Google Scholar
  25. 25.
    Oniciuc, C.: On the second variation formula for biharmonic maps to a sphere. Publ. Math. Debrecen. 67, 285–303 (2005)MathSciNetGoogle Scholar
  26. 26.
    Ou, Ye-Lin, Tang, Liang: The generalized Chen’s conjecture on biharmonic submanifolds is false, arXiv: 1006.1838v1Google Scholar
  27. 27.
    Sasahara, T.: Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors. Publ. Math. Debrecen 67, 285–303 (2005)MATHMathSciNetGoogle Scholar
  28. 28.
    Schoen, R., Yau, S.T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Comment. Math. Helv. 51, 333–341 (1976)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Yau, S.T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Wang, Z.-P., Ou, Y.-L.: Biharmonic Riemannian submersions from 3-manifolds. Math. Z. 269, 917–925 (2011) arXiv: 1002.4439v1Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Nobumitsu Nakauchi
    • 1
  • Hajime Urakawa
    • 2
    • 3
  • Sigmundur Gudmundsson
    • 4
  1. 1.Graduate School of Science and EngineeringYamaguchi UniversityYamaguchiJapan
  2. 2.Division of Mathematics, Graduate School of Information SciencesTohoku UniversitySendaiJapan
  3. 3.Institute for International EducationTohoku UniversitySendaiJapan
  4. 4.Department of Mathematics, Faculty of ScienceLund UniversityLundSweden

Personalised recommendations