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Non-holomorphic harmonic morphisms from Kähler manifolds

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Abstract

We give a method for constructing non-holomorphic harmonic morphisms from Kähler manifolds. The method is then used to obtain many such examples defined locally on the complex Grassmannians.

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References

  1. P. Baird, J. Eells A conservation law for harmonic maps, inGeometry Symposium Utrecht 1980, Lecture Notes in Mathematics894, 1–25, Springer (1981).

  2. P. Baird, J. C. Wood, Bernstein theorems for harmonic morphisms from ℝ3 andS 3,Math. Ann. 280 (1988), 579–603

    Article  MATH  MathSciNet  Google Scholar 

  3. F. E. Burstall, D. Ferus, F. Pedit, U. Pinkall, Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras,Ann. of Math. 138 (1993), 173–212

    Article  MATH  MathSciNet  Google Scholar 

  4. F. E. Burstall, J. H. Rawnsley,Twistor theory for Riemannian symmetric spaces with applications to harmonic maps of Riemann surfaces, Lecture Notes in Mathematics1424, Springer (1990)

  5. F. E. Burstall, J. C. Wood, The construction of harmonic maps into complex Grassmannians,J. Diff. Geom. 23 (1986), 255–297

    MATH  MathSciNet  Google Scholar 

  6. D. Cox, J. Little, D. O’Shea,Ideals, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra, Undergraduate Text in Mathematics, Springer (1992)

  7. R. W. R. Darling, Martingales in manifolds-definition, examples and behaviour under maps, inSéminaire de Probabilités XVI 1980/81. Supplément: Géométrie Differentielle Stochastique, Lecture Notes in Mathematics921, Springer (1982)

  8. T. t. Dieck, Transformation groups, de Gruyter (1987)

  9. J. Eells, J. C. Wood, Harmonic maps from surfaces to complex projective spaces.,Advances in Mathematics 49 (1983), 217–263

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Eells, J. H. Sampson, Harmonic mappings of Riemannian manifolds,Amer. J. Math. 86 (1964), 109–160

    Article  MATH  MathSciNet  Google Scholar 

  11. A. P. Fordy, J. C. Wood (eds.),Harmonic maps and integrable systems, Aspects of Mathematics E23, Viehweg, Braunschweig, Wiesbaden (1994)

    MATH  Google Scholar 

  12. B. Fuglede, Harmonic morphisms between Riemannian manifoldsAnn. Inst. Fourier 28 (1978), 107–144

    MATH  MathSciNet  Google Scholar 

  13. S. Gudmundsson, Harmonic morphisms from complex projective spaces,Geom. Dedicata (to appear)

  14. S. Gudmundsson, J. C. Wood, Multivalued harmonic morphisms,Math. Scand. 73 (1993), 127–155

    MATH  MathSciNet  Google Scholar 

  15. T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions,J. Math. Kyoto Univ. 19 (1979), 215–229

    MATH  MathSciNet  Google Scholar 

  16. R. Remmert, T. van de Ven, Über holomorphe Abbildungen projektiv-algebraischer Mannigfaltigkeiten auf komplexe Räume,Math. Ann. 142 (1961), 453–486

    Article  MATH  MathSciNet  Google Scholar 

  17. K. Uhlenbeck, Harmonic maps into Lie groups (classical solutions of the chiral model),J. Diff. Geom. 30 (1989), 1–50

    MATH  MathSciNet  Google Scholar 

  18. J. G. Wolfson, Harmonic sequences and harmonic maps of surfaces into complex Grassmannian manifolds,J. Diff. Geom. 27 (1988), 161–178

    MATH  MathSciNet  Google Scholar 

  19. J. C. Wood, The explicit construction and parametrization of all harmonic maps from the two-sphere to a complex Grassmannian,J. Reine Angew. Math. 386 (1988), 1–31

    MATH  MathSciNet  Google Scholar 

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The research leading to this paper was done at the Mathematics Institute of the University of Copenhagen and supported by the Danish Science Research Council

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Gudmundsson, S. Non-holomorphic harmonic morphisms from Kähler manifolds. Manuscripta Math 85, 67–78 (1994). https://doi.org/10.1007/BF02568184

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  • DOI: https://doi.org/10.1007/BF02568184

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