Annals of Global Analysis and Geometry

, Volume 47, Issue 4, pp 335–358 | Cite as

Harmonic maps of finite uniton number and their canonical elements

  • Nuno Correia
  • Rui Pacheco


We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group \(G\), whether \(G\) has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group \(\Omega _\mathrm {alg}{G}\) of algebraic loops in \(G\) and corresponding canonical elements. This will allow us to give estimations for the minimal uniton number of the corresponding harmonic maps with respect to different representations and to make more explicit the relation between previous work by different authors on harmonic two-spheres in classical compact Lie groups and their inner symmetric spaces and the Morse theoretic approach to the study of such harmonic two-spheres introduced by Burstall and Guest. As an application, we will also give some explicit descriptions of harmonic spheres in low-dimensional spin groups making use of spinor representations.


Harmonic maps Extended solutions Canonical elements Finite uniton number Symmetric spaces 

Mathematics Subject Classification

58E20 53C43 53C35 



The second author would like to thank John Wood for helpful conversations. He also benefited from clarifying correspondence with Francis Burstall. This work was partially supported by FCT - Portugal through CMUBI (project PEst-OE/MAT/UI0212/2014).


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Universidade da Beira InteriorCovilhaPortugal

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