Geometriae Dedicata

, Volume 163, Issue 1, pp 379–390 | Cite as

Immersed surfaces in Lie algebras associated to primitive harmonic maps

  • R. Pacheco
Original Paper


Sym and Bobenko gave a construction to recover a constant mean curvature surface in 3-dimensional euclidean space from the one-parameter family of harmonic maps associated to its Gauss map into the sphere. More recently, Eschenburg and Quast generalized this construction by replacing the sphere by a Kähler symmetric space of compact type. In this paper we shall take the generalization of Eschenburg and Quast a step further: our target space is now a generalized flag manifold N = G/K and we consider immersions of M in the Lie algebra \({\mathfrak{g}}\) of G associated to primitive harmonic maps.


Primitive harmonic maps Generalized flag manifolds Immersed surfaces 

Mathematics Subject Classification (2000)

58E20 53C43 53C35 


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  1. 1.
    Black, M.: Harmonic maps into homogeneous spaces. Pitman Res. Notes in Math. vol. 255. Longman, Harlow (1991)Google Scholar
  2. 2.
    Bobenko A.: Constant mean curvature surfaces and integrable equations. Russ. Math. Surv. 46, 1–45 (1991)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Burstall F.E.: Harmonic Tori in spheres and complex projectives spaces. J. reine u. angew. Math. 469, 149–177 (1995)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Burstall, F.E., Pedit, F.: Harmonic maps via Adler-Konstant-Symes theory, harmonic maps and integrable Systems. In: Fordy, A.P., Wood, J.C. (eds.) Aspects of Mathematics 23, pp. 221–272. Vieweg (1994)Google Scholar
  5. 5.
    Burstall, F.E., Rawnsley, J.H.: Twistor theory for Riemannian symmetric Spaces. Lectures Notes in Math. 1424. Berlin, Heidelberg (1990)Google Scholar
  6. 6.
    Dorfmeister J., Eschenburg J.: Pluriharmonic maps, loop groups and twistor theory. Ann. Glob. Anal. Geom. 24, 301–321 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eschenburg J., Quast P.: Pluriharmonic maps into Kähler symmetric spaces and Sym’s formula. Math. Z. 264(2), 469–481 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hélein, F.: Constant mean curvature surfaces, harmonic maps and integrable systems. Lectures in Mathematics. ETH Zürich, Birkhäuser (2001)Google Scholar
  9. 9.
    Ohnita Y., Udagawa S.: Harmonic maps of finite type into generalized flag manifolds and twistor fibrations. Contemp. Math. 308, 245–270 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pacheco R.: Twistor fibrations giving primitive harmonic maps of finite type. Int. J. Math. Math. Sci 2005(20), 3199–3212 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sym, A.: Soliton surfaces and their appliations (Soliton geometry from spectral problems). Geometric aspects of the Einstein equations and integrable systems. Lect. notes Phys. vol. 239, pp. 154–231. Springer, Berlin (1986)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal

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