Advertisement

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
Article

Abstract

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.

Keywords

Harmonic maps Extended solutions Canonical elements Finite uniton number Symmetric spaces 

Mathematics Subject Classification

58E20 53C43 53C35 

Notes

Acknowledgments

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).

References

  1. 1.
    Aithal, A.R.: Harmonic maps from \({\rm S}^2\) to \({\mathbb{H}}{\rm P}^{n-1}\). Osaka J. Math. 23, 255–270 (1986)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Baez, J.C.: The octonions. Bull. Am. Math. Soc. 39(2), 145–205 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bahy-El-Dien, A., Wood, J.C.: The explicit construction of all harmonic two-spheres in \({\rm G}_2({\mathbb{R}}^{n})\). J. Reine u. Angew. Math. 398, 36–66 (1989)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bahy-El-Dien, A., Wood, J.C.: The explicit construction of all harmonic two-spheres in quaternionic projective spaces. Proc. Lond. Math. Soc. 62, 202–224 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Burstall, F.E., Guest, M.A.: Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann. 309(4), 541–572 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Burstall, F.E., Rawnsley, J.H.: Twistor theory for Riemannian symmetric spaces with applications to harmonic maps of Riemann surfaces, lecture notes in mathematics, vol. 1424, Springer, Berlin (1990)Google Scholar
  7. 7.
    Borel, A., Siebenthal, J.: Les sous-groupes férmes de rang maximum des groupes de Lie clos. Comm. Math. Hel. 23, 200–221 (1949)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bryant, R.L.: Remarks on spinors in low dimensions. [Online]. http://www.math.duke.edu/bryant/Spinors
  9. 9.
    Correia, N., Pacheco, R.: Harmonic maps of finite uniton number into \(G_2\). Math. Z. 271(1–2), 13–32 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Correia, N., Pacheco, R.: Extended solutions of the Harmonic map equation in the special unitary group. Q. J. Math. 65(2), 637–654 (2014). doi: 10.1093/qmath/hat018
  11. 11.
    Correia, N., Pacheco, R.: Harmonic spheres and tori in the Cayley plane, in preparationGoogle Scholar
  12. 12.
    Eells, J., Wood, J.C.: Harmonic maps from surfaces to complex projective spaces. Adv. Math. 49(3), 217–263 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Eschenburg, J.-H., Mare, A.L., Quast, P.: Pluriharmonic maps into outer symmetric spaces and a subdivision of Weyl chambers. Bull. Lond. Math. Soc. 42(6), 1121–1133 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fulton, W., Harris, J.: Representation theory, graduate texts in mathematics, vol. 129, Springer, New York. A first course, readings in Mathematics (1991)Google Scholar
  15. 15.
    Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–152 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Pacheco, R.: Harmonic two-spheres in the symplectic group \({\rm Sp}(n)\). Int. J. Math. 17(3), 295–311 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Pressley, A., Segal, G.: Loop groups, oxford mathematical monographs. The Clarendon Press, Oxford University Press, Oxford Science Publications, New York (1986)Google Scholar
  18. 18.
    Segal, G.: Loop groups and harmonic maps, advances in homotopy theory. Cortona: London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge 1989, pp. 153–164 (1988)Google Scholar
  19. 19.
    Svensson, M., Wood, J.C.: Filtrations, factorizations and explicit formulae for harmonic maps. Comm. Math. Phys. 310(1), 99–134 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Svensson, M., Wood, J.C.: Harmonic maps into the exceptional symmetric space \(G_2/SO(4)\) (2013). arXiv:1303.7176
  21. 21.
    Uhlenbeck, K.: Harmonic maps into Lie groups: classical solutions of the chiral model. J. Differ. Geom. 30(1), 1–50 (1989)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Universidade da Beira InteriorCovilhaPortugal

Personalised recommendations