Skip to main content
SpringerLink
Log in

Biharmonic Maps from Tori into a 2-Sphere

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alias, L., Garcia-Martinez, S. and Rigoli, M., Biharmonic hypersurfaces in complete Riemannian manifolds, Pacific J. of Math, 263(1), (2013), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  2. Akutagawa, K. and Maeta, S., Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata, 164, (2013), 351–355.

    Article  MathSciNet  MATH  Google Scholar 

  3. Baird, P. and Kamissoko, D., On constructing biharmonic maps and metrics, Ann. Global Anal. Geom., 23(1), (2003), 65–75.

    Article  MathSciNet  MATH  Google Scholar 

  4. Baird, P., Fardoun, A. and Ouakkas, S., Conformal and semi-conformal biharmonic maps, Ann. Glob. Anal. Geom., 34, (2008), 403–414.

    Article  MathSciNet  MATH  Google Scholar 

  5. Baird, P., Fardoun, A. and Ouakkas, S., Liouville-type theorems for biharmonic maps between Riemannian manifolds, Adv. Calc. Var., 3, (2010), 49–68.

    Article  MathSciNet  MATH  Google Scholar 

  6. Baird, P. and Wood, J. C., Harmonic Morphisms Between Riemannian Manifolds, London Math. Soc. Monogr. (N. S.), 29, Oxford Univ. Press, 2003.

  7. Balmus, A., Montaldo, S. and Oniciuc, C., Classification results for biharmonic submanifolds in spheres, Israel J. Math., 168, (2008), 201–220.

    Article  MathSciNet  MATH  Google Scholar 

  8. Balmus, A., Montaldo, S. and Oniciuc, C., Biharmonic maps between warped product manifolds, J. Geom. Phys., 57(2), (2007), 449–466.

    Article  MathSciNet  MATH  Google Scholar 

  9. Balmus, A., Montaldo, S. and Oniciuc, C., Biharmonic PNMC Submanifolds in Spheres, preprint, Ark. Mat., 51, (2013), 197–221.

    Article  MathSciNet  MATH  Google Scholar 

  10. Brendle, S., Minimal surfaces in S 3: A survey of recent results, Bull. Math. Sci., 3, (2013), 133–171.

    Article  MathSciNet  MATH  Google Scholar 

  11. Caddeo, R., Montaldo, S. and Oniciuc, C., Biharmonic submanifolds of S3, Internat. J. Math., 12(8), (2001), 867–876.

    Article  MathSciNet  MATH  Google Scholar 

  12. Caddeo, R., Montaldo, S. and Oniciuc, C., Biharmonic submanifolds in spheres, Israel J. Math., 130, (2002), 109–123.

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, B. Y., Some open problems and conjectures on submanifolds of finite type, Soochow J. Math., 17(2), (1991), 169–188.

    MathSciNet  MATH  Google Scholar 

  14. Chen, B. Y., Pseudo-Riemannian Geometry, d-Invariants and Applications, World Scientific Publishing, 2011.

    Book  Google Scholar 

  15. Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, 2nd ed., World Scientific Publishing, Hackensack, NJ, 2014.

    Book  Google Scholar 

  16. Chen, B. Y. and Ishikawa, S., Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math., 52(1), (1998), 167–185.

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, B. Y. and Munteanu, M., Biharmonic ideal hypersurfaces in Euclidean spaces, Diff. Geom. Appl., 31(1), (2013), 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  18. Defever, F., Hypersurfaces of E 4 with harmonic mean curvature vector, Math. Nachr., 196, (1998), 61–69.

    Article  MathSciNet  MATH  Google Scholar 

  19. Dimitric, I., Submanifolds of E m with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica, 20(1), (1992), 53–65.

    MathSciNet  MATH  Google Scholar 

  20. Eells, J. and Sampson, J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86, (1964), 109–160.

    Article  MathSciNet  MATH  Google Scholar 

  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. Fu, Y., Biharmonic hypersurfaces with three distinct principal curvatures Euclidean spaces, Tohoku Math. J., 67(3), (2015), 465–479.

    Article  MathSciNet  MATH  Google Scholar 

  23. Hasanis, T. and Vlachos, T., Hypersurfaces in E 4 with harmonic mean curvature vector field, Math. Nachr., 172, (1995), 145–169.

    Article  MathSciNet  MATH  Google Scholar 

  24. Jiang, G. Y., 2-Harmonic maps and their first and second variational formulas, Chin. Ann. Math., Ser. A, 7, (1986), 389–402.

    MathSciNet  MATH  Google Scholar 

  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. Jiang, G. Y., 2-Harmonic isometric immersions between Riemannian manifolds. Chin. Ann. Math., Ser. A, 7(2), (1986), 130–144.

    MathSciNet  MATH  Google Scholar 

  27. Lawson, H. B., Complete minimal surfaces in S3, Annals of Math, Second Series, 92(3), (1970), 335–374.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  29. Loubeau, E. and Ou, Y.-L., Biharmonic maps and morphisms from conformal mappings, Tôhoku Math. J., 62(1), (2010), 55–73.

    Article  MathSciNet  MATH  Google Scholar 

  30. Luo, Y., On biharmonic submanifolds in non-positively curved manifolds, J. Geom. Phys., 88, (2015), 76–87.

    Article  MathSciNet  MATH  Google Scholar 

  31. Maeta, S., Properly immersed submanifolds in complete Riemannian manifolds, Adv. Math., 253, (2014), 139–151.

    Article  MathSciNet  MATH  Google Scholar 

  32. Montaldo, S. and Oniciuc, C., A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina, 47(2), (2006), 1–22.

    MathSciNet  MATH  Google Scholar 

  33. Montaldo, S. and Ratto, A., A general approach to equivariant biharmonic maps, Mediterr. J. Math., 10, (2013), 1127–1139.

    Article  MathSciNet  MATH  Google Scholar 

  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. Nakauchi, N., Urakawa, H. and Gudmundsson, S., Biharmonic maps into a Riemannian manifold of nonpositive curvature, preprint, 2012, arXiv: 1201.6457.

    MATH  Google Scholar 

  36. Ou, Y.-L., p-Harmonic morphisms, biharmonic morphisms, and nonharmonic biharmonic maps, J. of Geo. Phy., 56(3), (2006), 358–374.

    Article  MathSciNet  MATH  Google Scholar 

  37. Ou, Y.-L., On conformal biharmonic immersions, Ann. Global Analysis and Geometry, 36(2), (2009), 133–142.

    Article  MathSciNet  MATH  Google Scholar 

  38. Ou, Y.-L., Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. of Math., 248(1), (2010), 217–232.

    Article  MathSciNet  MATH  Google Scholar 

  39. Ou, Y.-L., Some constructions of biharmonic maps and Chen’s conjecture on biharmonic hypersurfaces, J. Geom. Phys., 62, (2012), 751–762.

    Article  MathSciNet  MATH  Google Scholar 

  40. Ou, Y.-L., Biharmonic conformal immersions into 3-dimensional manifolds, Medierranean J. Math., 12(2), (2015), 541–554.

    Article  MathSciNet  MATH  Google Scholar 

  41. Ou, Y.-L., Some recent progress of biharmonic submanifolds, Contemporary Math. AMS, to appear, 2016.

    Google Scholar 

  42. Ou, Y.-L., f-Biharmonic maps and f-biharmonic submanifolds II, J. Math. Anal. Appl., 455(2), (2017), 1285–1296.

    Article  MathSciNet  MATH  Google Scholar 

  43. Ou, Y.-L. and Lu, S., Biharmonic maps in two dimensions, Annali di Matematica Pura ed Applicata, 192, (2013), 127–144.

    Article  MathSciNet  MATH  Google Scholar 

  44. Ou, Y.-L. and Tang, L., On the generalized Chen’s conjecture on biharmonic submanifolds, Michigan Math. J., 61, (2012), 531–542.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  46. Ouakkas, S., Biharmonic maps, conformal deformations and the Hopf maps, Diff. Geom. Appl., 26(5), (2008), 495–502.

    Article  MathSciNet  MATH  Google Scholar 

  47. Smith, R. T., Harmonic mappings of spheres, Amer. J. Math., 97, (1975), 364–385.

    Article  MathSciNet  MATH  Google Scholar 

  48. Tang, L. and Ou, Y.-L., Biharmonic hypersurfaces in a conformally flat space, Results Math., 64(1), (2013), 91–104.

    MathSciNet  MATH  Google Scholar 

  49. Wang, Z.-P. and Ou, Y.-L. Biharmonic Riemannian submersions from 3-manifolds, Math. Z., 269, (2011), 917–925.

    Article  MathSciNet  MATH  Google Scholar 

  50. Wang, Z.-P., Ou, Y.-L. and Yang, H.-C., Biharmonic maps from a 2-sphere, J. Geom. Phys., 77, (2014), 86–96.

    Article  MathSciNet  MATH  Google Scholar 

  51. Wheeler, G., Chen’s conjecture and o-superbiharmonic submanifolds of Riemannian manifolds, Internat. J. Math., 24(4), 2013, ID:1350028, 6pp.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yunnan University, Kunming, 650091, China

    Zeping Wang & Hanchun Yang

  2. Department of Mathematics, Guizhou Normal University, Guiyang, 550001, China

    Zeping Wang & Hanchun Yang

  3. Department of Mathematics, Texas A & M University-Commerce, Commerce, TX, 75429, USA

    Ye-Lin Ou

Authors
  1. Zeping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ye-Lin Ou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hanchun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zeping Wang.

Additional information

The third author was supported by the Natural Science Foundation of China (No. 11361073) and the second author was supported by the Natural Science Foundation of Guangxi Province of China (No. 2011GXNSFA018127).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Z., Ou, YL. & Yang, H. Biharmonic Maps from Tori into a 2-Sphere. Chin. Ann. Math. Ser. B 39, 861–878 (2018). https://doi.org/10.1007/s11401-018-0101-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-018-0101-9

Keywords

2000 MR Subject Classification

Search

Navigation