Journal of Geometry

, 109:12 | Cite as

A note on the spectral deformation of harmonic maps from \({\varvec{S}}^{\varvec{2}}\) into the unitary group

  • Maria João Ferreira
  • Bruno Ascenso Simões


In Ferreira et al. (Math Z 266:953–978, 2010), together with J. C. Wood, the authors gave a completely explicit formula for all harmonic maps from 2-spheres to the unitary group U(n) in terms of freely chosen meromorphic functions on \(S^2\). The simplest harmonic maps are the isotropic ones. Using Morse theory Burstall and Guest (Math Ann 309:541–572, 1997) showed that the harmonic maps are partitioned into classes labeled by the isotropic ones. In this work, using the formula for harmonic maps aforementioned, we describe explicitly this procedure, showing how all harmonic maps can be built from the isotropic ones.


Harmonic map uniton loop group 

Mathematics Subject Classification

Primary 58E20 Secondary 53C43 



The authors would like to thank the referee for valuable corrections and suggestions. This work was partially supported by Fundação para a Ciência e Tecnologia, Portugal.


  1. 1.
    Burstall, F.E., Guest, M.A.: Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann. 309, 541–572 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Correia, N., Pacheco, R.: Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass type representations. arXiv:1412.8348v1 (2014)
  3. 3.
    Dorfmeister, J., Pedit, F., Wu, H.: Weierstrass representations of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6(4), 633–668 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Eschenburg, J.H., Quast, P.: The spectral parameter of pluriharmonic maps. Bull. Lond. Math. Soc. 42, 229–236 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eschenburg, J.H., Tribuzy, R.: Associated families of pluriharmonic maps and isotropy. Manuscr. Math. 95, 295–310 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ferreira, M.J., Simões, B.A., Wood, J.C.: All harmonic 2-spheres in the unitary group, completely explicitly. Math. Z. 266, 953–978 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ferreira, M.J., Simões, B.A.: Explicit construction of harmonic two-spheres into the complex Grassmannian. Math. Z. 272, 151–174 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Pressley, A., Segal, G.: Loop Groups. Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford, Oxford Mathematical Monographs, Oxford (1986)Google Scholar
  9. 9.
    Segal, G.: Loop groups and harmonic maps, Advances in homotopy theory (Cortona, 1988), 153–164, London Mathematical Society Lecture Note Series, 139. Cambridge University Press, Cambridge (1989)Google Scholar
  10. 10.
    Svensson, M., Wood, J.C.: Filtrations, factorizations and explicit formulae for harmonic maps. Comm. Math. Phys. 310(1), 99–134 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Svensson, M., Wood, J.C.: New constructions of twistor lifts for harmonic maps. Manuscr. Math. 144(3–4), 457–502 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Uhlenbeck, K.: Harmonic maps into Lie groups: classical solutions of the chiral model. J. Differ. Geom. 30, 1–50 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Maria João Ferreira
    • 1
    • 2
  • Bruno Ascenso Simões
    • 1
    • 2
  1. 1.Departamento de Matemática, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal
  2. 2.Centro de Matemática e Aplicações FundamentaisUniversidade de LisboaLisbonPortugal

Personalised recommendations