A structure theorem of Dirac-harmonic maps between spheres

  • Ling YangEmail author
Open Access


For an arbitrary Dirac-harmonic map (φ,ψ) between compact oriented Riemannian surfaces, we shall study the zeros of |ψ|. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of |ψ| and the genus of M and N. On the basis, we could clarify all of non-trivial Dirac-harmonic maps from S 2 to S 2.

Mathematics Subject Classification (2000)

58E20 53C27 



The author wishes to express his sincere gratitude to Professor Y.L. Xin in Fudan University, for his inspiring suggestions.

Open Access

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  1. 1.
    Chen Q., Jost J., Li J., Wang G.: Dirac-harmonic maps. Math. Z. 254, 409–432 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chen Q., Jost J., Li J., Wang G.: Regularity and energy identities for Dirac-harmonic maps. Math. Z. 251, 61–84 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen Q., Jost J., Wang G.: Liouville theorems for Dirac-harmonic maps. J. Math. Phys. 48(113517), 13 (2007)MathSciNetGoogle Scholar
  4. 4.
    Friedrich, T.: Dirac operators in Riemannian geometry. Graduate Studies in Mathematics, vol. 25. American Mathematical Society, Providence, pp. xvi+195 (2000)Google Scholar
  5. 5.
    Hijazi, O.: Spectral properties of the Dirac operator and geometrical structures. In: Proceedings of the Summer School on Geometric Methods in Quantum Field Theory. 12–30 July 1999, Villa de Leyva, Colombia, World Scientific, Physics (2001)Google Scholar
  6. 6.
    Lawson H.B., Michelsohn M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)zbMATHGoogle Scholar
  7. 7.
    Schoen, R., Yau, S.T.: Lectures on harmonic maps. Conference Proceedings and Lecture Notes in Geometry and Topology, II. International Press, Cambridge (1997)Google Scholar
  8. 8.
    Xin Y.L.: Geometry of harmonic maps. Birkhäuser, Cambridge (1996)zbMATHGoogle Scholar
  9. 9.
    Zhao L.: Energy identities for Dirac-harmonic maps. Calc. Var. PDE 28, 121–138 (2007)zbMATHCrossRefGoogle Scholar
  10. 10.
    Zhu, M.: Dirac-harmonic maps from degenerating spin surfaces I: the Neveu–Schwarz case. arxiv: 0803. 3723.Google Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Max-Planck Institute for Mathematics in the SciencesLeipzigGermany

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