Potential Analysis

, Volume 2, Issue 3, pp 295–298 | Cite as

A note on the classification of holomorphic harmonic morphisms

  • Sigmundur Gudmundsson
  • Ragnar Sigurdsson


In this note we give a complete classification of those holomorphic maps φ:U→ℂ n defined on open and connected subsets of ℂ m which are harmonic morphisms.

Mathematics Subject Classifications (1991)

58E20 58G32 32A10 

Key words

Harmonic morphisms Brownian motions holomorphic maps 


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  1. 1.
    Baird, P.:Harmonic Maps with Symmetry, Harmonic Morphisms and Deformation of Metrics, Research Notes in Mathematics87, Pitman (1983).Google Scholar
  2. 2.
    Baird, P.: Harmonic morphisms and circle actions on 3- and 4-manifolds,Ann. Inst. Fourier, Grenoble 40 (1990), 177–212.Google Scholar
  3. 3.
    Baird, P.: Riemannian twistors and Hermitian structures on low-dimensional space forms,J. Math. Phys. 33 (1992), 3340–3350.Google Scholar
  4. 4.
    Baird, P. and Eells, J.: A conservation law for harmonic maps, inGeometry Symposium Utrecht 1980, Lecture Notes in Mathematics894, Springer (1981), pp. 1–25.Google Scholar
  5. 5.
    Baird, P. and Wood, J. C.: Harmonic morphisms and conformal foliation by geodesics of three-dimensional space forms,J. Australian Math. Soc. (A)51 (1991), 118–153.Google Scholar
  6. 6.
    Darling, R. W. R.: 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), 217–236.Google Scholar
  7. 7.
    Gudmundsson, S.: Harmonic morphisms between spaces of constant curvature,Proc. Edinburgh Math. Soc. 36 (1992), 133–143.Google Scholar
  8. 8.
    Gudmundsson, S. and Wood, J. C.: Multivalued harmonic morphisms,Math. Scand. 72 (1994), (to appear).Google Scholar
  9. 9.
    Fuglede, B.: Harmonic morphisms between Riemannian manifolds,Ann. Inst. Fourier 28 (1978), 107–144.Google Scholar
  10. 10.
    Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions,J. Math. Kyoto Univ. 19 (1979), 215–229.Google Scholar
  11. 11.
    Wood, J. C.: Harmonic morphisms and Hermitian structures on Einstein 4-manifolds,Intern. J. Math. 3 (1992), 415–439.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Sigmundur Gudmundsson
    • 1
  • Ragnar Sigurdsson
    • 1
  1. 1.Science InstituteUniversity of IcelandReykjavikIceland

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