Skip to main content
SpringerLink
Log in

Regularity theorems and energy identities for Dirac-harmonic maps

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study Dirac-harmonic maps from a Riemann surface to a sphere We show that a weakly Dirac-harmonic map is in fact smooth, and prove that the energy identity holds during the blow-up process.

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. Bär, C.: Metrics with harmonic spinors. Geom. Funct. Anal. 6, 899–942 (1996)

    Google Scholar 

  2. Bär, C., Schmutz, P.: Harmonic spinors on Riemann surfaces. Ann. Glob. Anal. Geom. 10, 263–273 (1992)

    Article  Google Scholar 

  3. Chen, Q., Jost, J., Li, J. Y., Wang, G.: Dirac-harmonic maps. preprint 2004

  4. Ding, W. Y., Tian, G.: Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom. 3, 543–554 (1996)

    Google Scholar 

  5. Eells, J., James, Lemaire, L.: Two reports on harmonic maps. World Scientific Publishing Co., Inc., River Edge, NJ, 1995

  6. Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C.R. Acad. Sci. Paris Sé r. I Math. 312, 591–596 (1991)

    Google Scholar 

  7. Hélein, F.: Harmonic maps, conservation laws and moving frames. 2nd edtion, Cambridge University Press, 2002

  8. Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    Article  Google Scholar 

  9. Jost, J.: Riemannian Geometry and geometric analysis. 3rd edition, Springer-Verlag, 2002

  10. Jost, J.: Two-dimensional geometric variational problems. Wiley, 1991

  11. Lamm, T.: Fourth order approximation of harmonic maps from surfaces. preprint, 2004

  12. Lawson, H., Michelsohn, M.L.: Spin geometry. Princeton University Press, 1989

  13. Parker, T., Wolfson, J.G.: Pseudo-holomorphic maps and bubble trees. J. Geom. Anal. 3, 63–98 (1996)

    Google Scholar 

  14. Parker, T.: Bubble tree convergence for harmonic maps. J. Differential Geom. 44, 595–633 (1996)

    Google Scholar 

  15. Qing, J., Tian, G.: Bubbling of the heat flows for harmonic maps from surfaces. Comm. Pure Appl. Math. 50, 295–310 (1997)

    Article  Google Scholar 

  16. Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. of Math. 113, 1–24 (1981)

    Google Scholar 

  17. Wente, H.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318–344 (1969)

    Article  Google Scholar 

  18. Ye, R.: Gromov’s compactness theorem for pseudo holomorphic curves. Trans. Amer. Math. Soc. 342, 671–694 (1994)

    Google Scholar 

Download references

Author information

Author notes

  1. Qun Chen

    Present address: School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China

  2. Jiayu Li

    Present address: Mathematics Group, The Abdus Salam ICTP, 34100, Trieste, Italy

Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, P.R. China

    Qun Chen

  2. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103, Leipzig, Germany

    Jürgen Jost & Guofang Wang

  3. Partner Group of Max Planck Institute for Mathematics in the Sciences, Institute of Mathematics, Chinese Academy of Sciences, Beijing, 100080, P. R. China

    Jiayu Li

Authors
  1. Qun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürgen Jost
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiayu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guofang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jürgen Jost.

Additional information

The research of QC and JYL was partially supported by NSFC. QC was also partially supported by the FOK Yingtung Education Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, Q., Jost, J., Li, J. et al. Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z. 251, 61–84 (2005). https://doi.org/10.1007/s00209-005-0788-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-005-0788-7

Keywords

Search

Navigation