Chinese Annals of Mathematics, Series B

, Volume 39, Issue 5, pp 861–878 | Cite as

Biharmonic Maps from Tori into a 2-Sphere

  • Zeping WangEmail author
  • Ye-Lin Ou
  • Hanchun Yang


Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree ±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree ±1 in a large family of maps from a torus into a sphere.


Biharmonic maps Biharmonic tori Harmonic maps Gauss maps Maps into a sphere 

2000 MR Subject Classification

58E20 53C12 


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  1. [1]
    Alias, L., Garcia-Martinez, S. and Rigoli, M., Biharmonic hypersurfaces in complete Riemannian manifolds, Pacific J. of Math, 263(1), (2013), 1–12.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Akutagawa, K. and Maeta, S., Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata, 164, (2013), 351–355.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Baird, P. and Kamissoko, D., On constructing biharmonic maps and metrics, Ann. Global Anal. Geom., 23(1), (2003), 65–75.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Baird, P., Fardoun, A. and Ouakkas, S., Conformal and semi-conformal biharmonic maps, Ann. Glob. Anal. Geom., 34, (2008), 403–414.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Baird, P., Fardoun, A. and Ouakkas, S., Liouville-type theorems for biharmonic maps between Riemannian manifolds, Adv. Calc. Var., 3, (2010), 49–68.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Baird, P. and Wood, J. C., Harmonic Morphisms Between Riemannian Manifolds, London Math. Soc. Monogr. (N. S.), 29, Oxford Univ. Press, 2003.Google Scholar
  7. [7]
    Balmus, A., Montaldo, S. and Oniciuc, C., Classification results for biharmonic submanifolds in spheres, Israel J. Math., 168, (2008), 201–220.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Balmus, A., Montaldo, S. and Oniciuc, C., Biharmonic maps between warped product manifolds, J. Geom. Phys., 57(2), (2007), 449–466.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Balmus, A., Montaldo, S. and Oniciuc, C., Biharmonic PNMC Submanifolds in Spheres, preprint, Ark. Mat., 51, (2013), 197–221.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Brendle, S., Minimal surfaces in S 3: A survey of recent results, Bull. Math. Sci., 3, (2013), 133–171.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Caddeo, R., Montaldo, S. and Oniciuc, C., Biharmonic submanifolds of S3, Internat. J. Math., 12(8), (2001), 867–876.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Caddeo, R., Montaldo, S. and Oniciuc, C., Biharmonic submanifolds in spheres, Israel J. Math., 130, (2002), 109–123.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Chen, B. Y., Some open problems and conjectures on submanifolds of finite type, Soochow J. Math., 17(2), (1991), 169–188.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Chen, B. Y., Pseudo-Riemannian Geometry, d-Invariants and Applications, World Scientific Publishing, 2011.CrossRefGoogle Scholar
  15. [15]
    Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, 2nd ed., World Scientific Publishing, Hackensack, NJ, 2014.CrossRefGoogle Scholar
  16. [16]
    Chen, B. Y. and Ishikawa, S., Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math., 52(1), (1998), 167–185.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Chen, B. Y. and Munteanu, M., Biharmonic ideal hypersurfaces in Euclidean spaces, Diff. Geom. Appl., 31(1), (2013), 1–16.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Defever, F., Hypersurfaces of E 4 with harmonic mean curvature vector, Math. Nachr., 196, (1998), 61–69.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Dimitric, I., Submanifolds of E m with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica, 20(1), (1992), 53–65.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Eells, J. and Sampson, J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86, (1964), 109–160.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Eells, J. and Wood, J. C., The existence and construction of certain harmonic maps, Symposia Mathematica, Vol. XXVI (Rome, 1980), Academic Press, London, New York, 1982, 123–138.Google Scholar
  22. [22]
    Fu, Y., Biharmonic hypersurfaces with three distinct principal curvatures Euclidean spaces, Tohoku Math. J., 67(3), (2015), 465–479.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Hasanis, T. and Vlachos, T., Hypersurfaces in E 4 with harmonic mean curvature vector field, Math. Nachr., 172, (1995), 145–169.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Jiang, G. Y., 2-Harmonic maps and their first and second variational formulas, Chin. Ann. Math., Ser. A, 7, (1986), 389–402.MathSciNetzbMATHGoogle Scholar
  25. [25]
    Jiang, G. Y., Some non-existence theorems of 2-harmonic isometric immersions into Euclidean spaces, Chin. Ann. Math., Ser. A, 8(3), (1987), 376–383.Google Scholar
  26. [26]
    Jiang, G. Y., 2-Harmonic isometric immersions between Riemannian manifolds. Chin. Ann. Math., Ser. A, 7(2), (1986), 130–144.MathSciNetzbMATHGoogle Scholar
  27. [27]
    Lawson, H. B., Complete minimal surfaces in S3, Annals of Math, Second Series, 92(3), (1970), 335–374.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Loubeau, E. and Oniciuc, C., On the biharmonic and harmonic indices of the Hopf map, Trans. Amer. Math. Soc., 359(11), (2007), 5239–5256.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Loubeau, E. and Ou, Y.-L., Biharmonic maps and morphisms from conformal mappings, Tôhoku Math. J., 62(1), (2010), 55–73.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Luo, Y., On biharmonic submanifolds in non-positively curved manifolds, J. Geom. Phys., 88, (2015), 76–87.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Maeta, S., Properly immersed submanifolds in complete Riemannian manifolds, Adv. Math., 253, (2014), 139–151.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Montaldo, S. and Oniciuc, C., A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina, 47(2), (2006), 1–22.MathSciNetzbMATHGoogle Scholar
  33. [33]
    Montaldo, S. and Ratto, A., A general approach to equivariant biharmonic maps, Mediterr. J. Math., 10, (2013), 1127–1139.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Nakauchi, N. and Urakawa, H., Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results. Math., 2011, DOI: 10.1007/s00025-011-0209-7.Google Scholar
  35. [35]
    Nakauchi, N., Urakawa, H. and Gudmundsson, S., Biharmonic maps into a Riemannian manifold of nonpositive curvature, preprint, 2012, arXiv: 1201.6457.zbMATHGoogle Scholar
  36. [36]
    Ou, Y.-L., p-Harmonic morphisms, biharmonic morphisms, and nonharmonic biharmonic maps, J. of Geo. Phy., 56(3), (2006), 358–374.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    Ou, Y.-L., On conformal biharmonic immersions, Ann. Global Analysis and Geometry, 36(2), (2009), 133–142.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    Ou, Y.-L., Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. of Math., 248(1), (2010), 217–232.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    Ou, Y.-L., Some constructions of biharmonic maps and Chen’s conjecture on biharmonic hypersurfaces, J. Geom. Phys., 62, (2012), 751–762.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    Ou, Y.-L., Biharmonic conformal immersions into 3-dimensional manifolds, Medierranean J. Math., 12(2), (2015), 541–554.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    Ou, Y.-L., Some recent progress of biharmonic submanifolds, Contemporary Math. AMS, to appear, 2016.Google Scholar
  42. [42]
    Ou, Y.-L., f-Biharmonic maps and f-biharmonic submanifolds II, J. Math. Anal. Appl., 455(2), (2017), 1285–1296.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    Ou, Y.-L. and Lu, S., Biharmonic maps in two dimensions, Annali di Matematica Pura ed Applicata, 192, (2013), 127–144.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    Ou, Y.-L. and Tang, L., On the generalized Chen’s conjecture on biharmonic submanifolds, Michigan Math. J., 61, (2012), 531–542.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    Ou, Y.-L. and Wang, Z.-P., Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries, J. Gem. Phys., 61, (2011), 1845–1853.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    Ouakkas, S., Biharmonic maps, conformal deformations and the Hopf maps, Diff. Geom. Appl., 26(5), (2008), 495–502.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    Smith, R. T., Harmonic mappings of spheres, Amer. J. Math., 97, (1975), 364–385.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    Tang, L. and Ou, Y.-L., Biharmonic hypersurfaces in a conformally flat space, Results Math., 64(1), (2013), 91–104.MathSciNetzbMATHGoogle Scholar
  49. [49]
    Wang, Z.-P. and Ou, Y.-L. Biharmonic Riemannian submersions from 3-manifolds, Math. Z., 269, (2011), 917–925.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    Wang, Z.-P., Ou, Y.-L. and Yang, H.-C., Biharmonic maps from a 2-sphere, J. Geom. Phys., 77, (2014), 86–96.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    Wheeler, G., Chen’s conjecture and o-superbiharmonic submanifolds of Riemannian manifolds, Internat. J. Math., 24(4), 2013, ID:1350028, 6pp.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsYunnan UniversityKunmingChina
  2. 2.Department of MathematicsGuizhou Normal UniversityGuiyangChina
  3. 3.Department of MathematicsTexas A & M University-CommerceCommerceUSA

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