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Immersed surfaces in Lie algebras associated to primitive harmonic maps

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Abstract

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.

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References

  1. Black, M.: Harmonic maps into homogeneous spaces. Pitman Res. Notes in Math. vol. 255. Longman, Harlow (1991)

  2. Bobenko A.: Constant mean curvature surfaces and integrable equations. Russ. Math. Surv. 46, 1–45 (1991)

    Article  MathSciNet  Google Scholar 

  3. Burstall F.E.: Harmonic Tori in spheres and complex projectives spaces. J. reine u. angew. Math. 469, 149–177 (1995)

    MathSciNet  MATH  Google Scholar 

  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)

  5. Burstall, F.E., Rawnsley, J.H.: Twistor theory for Riemannian symmetric Spaces. Lectures Notes in Math. 1424. Berlin, Heidelberg (1990)

  6. Dorfmeister J., Eschenburg J.: Pluriharmonic maps, loop groups and twistor theory. Ann. Glob. Anal. Geom. 24, 301–321 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Eschenburg J., Quast P.: Pluriharmonic maps into Kähler symmetric spaces and Sym’s formula. Math. Z. 264(2), 469–481 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hélein, F.: Constant mean curvature surfaces, harmonic maps and integrable systems. Lectures in Mathematics. ETH Zürich, Birkhäuser (2001)

  9. Ohnita Y., Udagawa S.: Harmonic maps of finite type into generalized flag manifolds and twistor fibrations. Contemp. Math. 308, 245–270 (2002)

    Article  MathSciNet  Google Scholar 

  10. Pacheco R.: Twistor fibrations giving primitive harmonic maps of finite type. Int. J. Math. Math. Sci 2005(20), 3199–3212 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

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  1. Departamento de Matemática, Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama, 6201-001, Covilhã, Portugal

    R. Pacheco

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  1. R. Pacheco
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Pacheco, R. Immersed surfaces in Lie algebras associated to primitive harmonic maps. Geom Dedicata 163, 379–390 (2013). https://doi.org/10.1007/s10711-012-9755-8

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  • DOI: https://doi.org/10.1007/s10711-012-9755-8

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