Abstract
We consider pluriharmonic maps φ from simply connected complexmanifolds M to compact symmetric spaces G/K. Following [11]we introduce the loop parameter – and thus the associated familyof φ – in a geometric fashion. We define an extended framingglobally on M, but with values in Λ/K, whereΛ denotes a loop group associated with G. Locally we obtain adescription of pluriharmonic maps via normalized potentials similar tothat of Dorfmeister, Petit and Wu. For dimM > 1 thesepotentials satisfy a `curved flat' condition and they characterizepluriharmonic maps. We also briefly discuss the dressing action on theset of pluriharmonic maps. Finally, as special cases of the generaltheory, we discuss the isotropic case and pluriharmonic maps into Liegroups.
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References
Bryant, R.: Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Differential Geom. 17 (1982), 45–473.
Burstall, F. E.: Private communication.
Burstall, F. E., Eschenburg, J.-H., Ferreira, M. J. and Tribuzy, R.: Kähler submanifolds with parallel pluri-mean curvature, Preprint, math.DG/0111217, 2001.
Burstall, F. E. and Guest, M. A.: Harmonic two-spheres in compact symmetric spaces, Math. Ann. 309 (1997) 54–557.
Burstall, F. E. and Rawnsley, J. H.: Twistor Theory for Riemannian Symmetric Spaces, Lecture Notes in Math. 1424, Springer-Verlag, Berlin, 1990.
E. Calabi, E.: Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom. 1 (1967), 11–125.
Chi, Q.-S., Fernandez, L. and Wu, H.: Normalized potentials of minimal surfaces in spheres, Nagoya Math. J. 156 (1999), 18–214.
Dorfmeister, J., Pedit, F. and Wu, H.: Weierstrass type representations of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), 63–668.
Erdem, S. and Wood, J. C.: On the construction of harmonic maps into a Grassmannian, J. London Math. Soc. (2) 28 (1983), 16–174.
Eschenburg, J.-H. and Tribuzy, R.: Existence and uniqueness of maps into affine homogeneous spaces, Rend. Sem. Mat. Univ. Padova 89, 1–18.
Eschenburg, J.-H. and Tribuzy, R.: Associated families of pluriharmonic maps and isotropy, Manuscripta Math. 95 (1998), 29–310.
Ferus, D. and Pedit, F.: Curved flats in symmetric spaces, Manuscripta Math. 91 (1996), 44–454.
Guest, M.: Harmonic Maps, Loop Groups, and Integrable Systems, Cambridge Univ. Press, Cambridge, 1997.
Guest, M.: An update on harmonic maps of finite uniton number, via the zero curvature equation, in: M. Guest, R. Miyaoka and Y. Ohnita (eds), Integrable Systems, Topology, and Physics, Contemp. Math. 309, Amer. Math. Soc., Providence, RI, 2002.
Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.
McIntosh, I.: Global solutions of the elliptic 2D periodic Toda lattice, Nonlinearity 7 (1994), 8–108.
Ohnita, Y.: Harmonic maps into symmetric spaces and integrable system theory, in: Proceedings of the Conference ‘Theory of Lie Groups and Manifolds’, Sophia Kokyuroku in Mathematics 45, 2002, pp. 8–98.
Ohnita, Y. and Udagawa, S.: Harmonic maps of finite type into generalized flag manifolds and twistor fibrations, in: Differential Geometry and Integrable Systems, Proceedings of the 9th MSJ-IRI Tokyo 2000, Contemp. Math. 308, Amer. Math. Soc., Providence, NJ, 2002, pp. 24–270.
Ohnita, Y. and Valli, G.: Pluriharmonic maps into compact Lie groups and factorization into unitons, Proc. London Math. Soc. 61 (1990), 54–570.
Pohlmeyer, K.: Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), 20–221.
Pressley, A. and Segal, G.: Loop Groups, Oxford Science Publications, Clarendon Press, Oxford, 1986.
Uhlenbeck, K.: Harmonic maps into Lie groups (classical solutions of the chiral model), J. Differential Geom. 30 (1989), –50.
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Dorfmeister, J., Eschenburg, JH. Pluriharmonic Maps, Loop Groups and Twistor Theory. Annals of Global Analysis and Geometry 24, 301–321 (2003). https://doi.org/10.1023/A:1026225029745
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DOI: https://doi.org/10.1023/A:1026225029745