Continuous deformations of harmonic maps and their unitons


It is known that any harmonic map of finite uniton number from a Riemann surface into \(\mathrm {U}(n)\) can be deformed into a new harmonic map with an associated \(S^1\)-invariant extended solution. We study this deformation in detail using operator-theoretic methods. In particular, we show that the corresponding unitons are real analytic functions of the deformation parameter, and that the deformation is closely related to the Bruhat decomposition of the corresponding extended solution.

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


  1. 1.

    Burstall, F.E., Guest, M.A.: Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann. 309, 541–572 (1997)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Burstall, F.E., Rawnsley, J.H.: Twistor theory for Riemannian symmetric spaces. Lecture Notes in Math, vol. 1424. Springer, Berlin (1990)

    Google Scholar 

  3. 3.

    Grauert, H.: Analytische Faserungen über holomorph-vollständigen Räumen. Math. Ann. 135, 263–273 (1958)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Nagy, B.S., Foias, C., Bercovici, H., Kérchy, L.: Harmonic analysis of operators on Hilbert space. 2nd edn. Universitext. Springer, New York (2010)

  5. 5.

    Ohnita, Y., Valli, G.: Pluriharmonic maps into compact Lie groups and factorization into unitons. Proc. Lond. Math. Soc. 6(1), 546–570 (1990)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Peller, V.: Hankel Operators and their Applications. Springer, New York (2003)

    Google Scholar 

  7. 7.

    Pressley, A., Segal, G.: Loop Groups. Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1986)

    Google Scholar 

  8. 8.

    Segal, G.: Loop groups and harmonic maps. Advances in homotopy theory (Cortona, 1988). London Math. Soc. Lecture Notes Ser. 139, pp. 153–164. Cambridge Univ. Press, Cambridge (1989)

  9. 9.

    Svensson, M., Wood, J.C.: Filtrations, factorizations and explicit formulae for harmonic maps. Commun. Math. Phys. 310, 99–134 (2012)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Uhlenbeck, K.: Harmonic maps into Lie groups: classical solutions of the chiral model. J. Differ. Geom. 30, 1–50 (1989)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to María J. Martín.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first and second authors are partially supported by MINECO/FEDER-EU research project MTM2015-65792-P, Spain. The second author is supported by UAM and EU funding through the InterTalentum Programme (COFUND 713366).

Communicated by A. Constantin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aleman, A., Martín, M.J., Persson, A. et al. Continuous deformations of harmonic maps and their unitons. Monatsh Math 190, 599–614 (2019).

Download citation


  • Harmonic maps
  • Bruhat decomposition
  • Extended solutions
  • Unitons
  • Shift-invariant subspaces
  • Blaschke–Potapov products

Mathematics Subject Classification

  • 58E20
  • 47A56