Advertisement

manuscripta mathematica

, Volume 152, Issue 3–4, pp 399–432 | Cite as

Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass-type representations

  • N. Correia
  • R. Pacheco
Article
  • 57 Downloads

Abstract

Making use of Murakami’s classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such maps. Several examples of harmonic maps into classical outer symmetric spaces are given in terms of meromorphic functions on \(S^2\).

Mathematics Subject Classification

58E20 53C35 53C43 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bahy-El-Dien, A., Wood, J.C.: The explicit construction of all harmonic two-spheres in \(G_2({\mathbb{R}}^n)\). J. Reine Angew. Math. 398, 36–66 (1989)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Burstall, F.E., Guest, M.A.: Harmonic two-spheres in compact symmetric spaces, revisited. Math. Ann. 309(4), 541–572 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burstall, F.E., Rawnsley, J.H.: Twistor Theory for Riemannian Symmetric Spaces. In: Dols, A., Eckmann, B., Takens, F. (eds.) Lecture Notes in Mathematics, vol. 1424. Springer, Berlin, Heidelberg, New York (1990)Google Scholar
  4. 4.
    Calabi, E.: Minimal immersions of surfaces in Euclidean spheres. J. Diff. Geom. 1, 111–125 (1967)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Correia, N., Pacheco, R.: Harmonic maps of finite uniton number into \(G_2\). Math. Z. 271(1–2), 13–32 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Correia, N., Pacheco, R.: Extended Solutions of the Harmonic Map Equation in the Special Unitary Group. Q. J. Math. 65(2), 637–654 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Correia, N., Pacheco, R.: Harmonic maps of finite uniton number and their canonical elements. Ann. Global Anal. Geom. 47(4), 335–358 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dorfmeister, J., Pedit, F., Wu, H.: Weiestrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6, 633–668 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eschenburg, J.-H., Mare, A.-L., Quast, P.: Pluriharmonic maps into outer symmetric spaces and a subdivision of Weyl chambers. Bull. London Math. Soc. 42(6), 1121–1133 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eells, J., Wood, J.C.: Harmonic maps from surfaces to complex projective spaces. Adv. in Math. 49(3), 217–263 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fulton, W., Harris, J.: Representation theory. A first course. In: Axler, S., Gehring, F.W., Ribet, K.A. (eds.) Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)Google Scholar
  12. 12.
    Guest, M.A., Ohnita, Y.: Loop group actions on harmonic maps and their applications. Harmonic maps and integrable systems, 273–292, Aspects Math., E23, Vieweg, Braunschweig (1994)Google Scholar
  13. 13.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978)zbMATHGoogle Scholar
  14. 14.
    Ma, H.: Explicit construction of harmonic two-spheres in \(SU(3)/SO(3)\). Kyushu J. Math. 55, 237–247 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Murakami, S.: Sur la classification des algèbres de Lie réelles et simples. Osaka J. Math. 2, 291–307 (1965)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Pressley, A.N., Segal, G.B.: Loop Groups. Oxford University Press, (1986)Google Scholar
  17. 17.
    Uhlenbeck, K.: Harmonic maps into Lie groups (classical solutions of the chiral model). J. Diff. Geom. 30, 1–50 (1989)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ziller, W.: Lie groups: representation theory and symmetric spaces. Notes for a course given in the fall of 2010 at the University of Pennsylvania and 2012 at IMPA. www.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf. Accessed 20May 2015

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Universidade da Beira InteriorCovilhãPortugal

Personalised recommendations