Harmonic morphisms onto Riemann surfaces and generalized analytic functions

Paul Baird

Annales de l'institut Fourier (1987)

  • Volume: 37, Issue: 1, page 135-173
  • ISSN: 0373-0956

Abstract

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We study harmonic morphisms from domains in R 3 and S 3 to a Riemann surface N , obtaining the classification of such in terms of holomorphic mappings from a covering space of N into certain Grassmannians. We show that the only non-constant submersive harmonic morphism defined on the whole of S 3 to a Riemann surface is essentially the Hopf map.Comparison is made with the theory of analytic functions. In particular we consider multiple-valued harmonic morphisms defined on domains in R 3 and show how a cutting and glueing procedure may be applied to obtain a single-valued harmonic morphism from a certain 3-manifold. This is similar to the way in which the Riemann surface of a multiple-valued analytic function is constructed.

How to cite

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Baird, Paul. "Harmonic morphisms onto Riemann surfaces and generalized analytic functions." Annales de l'institut Fourier 37.1 (1987): 135-173. <http://eudml.org/doc/74741>.

@article{Baird1987,
abstract = {We study harmonic morphisms from domains in $\{\bf R\}^3$ and $S^3$ to a Riemann surface $N$, obtaining the classification of such in terms of holomorphic mappings from a covering space of $N$ into certain Grassmannians. We show that the only non-constant submersive harmonic morphism defined on the whole of $S^3$ to a Riemann surface is essentially the Hopf map.Comparison is made with the theory of analytic functions. In particular we consider multiple-valued harmonic morphisms defined on domains in $\{\bf R\}^3$ and show how a cutting and glueing procedure may be applied to obtain a single-valued harmonic morphism from a certain 3-manifold. This is similar to the way in which the Riemann surface of a multiple-valued analytic function is constructed.},
author = {Baird, Paul},
journal = {Annales de l'institut Fourier},
keywords = {harmonic morphisms; Hopf map},
language = {eng},
number = {1},
pages = {135-173},
publisher = {Association des Annales de l'Institut Fourier},
title = {Harmonic morphisms onto Riemann surfaces and generalized analytic functions},
url = {http://eudml.org/doc/74741},
volume = {37},
year = {1987},
}

TY - JOUR
AU - Baird, Paul
TI - Harmonic morphisms onto Riemann surfaces and generalized analytic functions
JO - Annales de l'institut Fourier
PY - 1987
PB - Association des Annales de l'Institut Fourier
VL - 37
IS - 1
SP - 135
EP - 173
AB - We study harmonic morphisms from domains in ${\bf R}^3$ and $S^3$ to a Riemann surface $N$, obtaining the classification of such in terms of holomorphic mappings from a covering space of $N$ into certain Grassmannians. We show that the only non-constant submersive harmonic morphism defined on the whole of $S^3$ to a Riemann surface is essentially the Hopf map.Comparison is made with the theory of analytic functions. In particular we consider multiple-valued harmonic morphisms defined on domains in ${\bf R}^3$ and show how a cutting and glueing procedure may be applied to obtain a single-valued harmonic morphism from a certain 3-manifold. This is similar to the way in which the Riemann surface of a multiple-valued analytic function is constructed.
LA - eng
KW - harmonic morphisms; Hopf map
UR - http://eudml.org/doc/74741
ER -

References

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  1. [1] P. BAIRD & J. EELLS, A conservation law for harmonic maps, Geometry Symp. Utrecht 1980, Springer Notes, 894 (1981), 1-25. Zbl0485.58008MR83i:58031
  2. [2] A. BERNARD, E.A. CAMPBELL & A.M. DAVIE, Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier, 29-1 (1979), 207-228. Zbl0386.30029MR81b:30088
  3. [3] M. BRELOT, Lectures on potential theory, Tata Institute of Fundamental Research, Bombay (1960). Zbl0098.06903MR22 #9749
  4. [4] E. CALABI, Minimal immersions of surfaces in Euclidan spheres, J. Diff. Geom., 1 (1967), 111-125. Zbl0171.20504MR38 #1616
  5. [5] C. CONSTANTINESCU & A. CORNEA, Compactifications of harmonic spaces, Nagoya Math. J., 25 (1965), 1-57. Zbl0138.36701MR30 #4960
  6. [6] A.M. DIN & W.J. ZAKRZEWKI, General classical solutions in the CPn - 1 model, Nucl. Phys. B., 174 (1980), 397-406. 
  7. [7] A.M. DIN & W.J. ZAKRZEWSKI, Properties of the general classical CPn - 1 model, Phys. Lett., 95 B (1980), 419-422. 
  8. [8] J. EELLS, Gauss maps of surfaces, Perspectives in Mathematics, Birkhäuser Verlag, Basel (1984), 111-129. Zbl0581.58013MR86j:53090
  9. [9] J. EELLS & L. LEMAIRE, A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68. Zbl0401.58003MR82b:58033
  10. [10] J. EELLS & L. LEMAIRE, On the construction of harmonic and holomorphic maps between surfaces, Math. Ann., 252 (1980), 27-52. Zbl0424.31009MR81k:58030
  11. [11] J. EELLS & J.H. SAMPSON, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. Zbl0122.40102MR29 #1603
  12. [12] J. EELLS & J.C. WOOD, Harmonic maps from surfaces to complex projective spaces, Advances in Math., 49 (1983), 217-263. Zbl0528.58007MR85f:58029
  13. [13] O. FORSTER, Lectures on Riemann Surfaces, Springer (1981). Zbl0475.30002
  14. [14] B. FUGLEDE, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier, 28-2 (1978), 107-144. Zbl0339.53026MR80h:58023
  15. [15] V. GLASER & R. STORA, Regular solutions of the CPn models and further generalizations, CERN (1980). 
  16. [16] R. HARTSHORNE, Algebraic Geometry, Springer (1980). 
  17. [17] T. ISHIHARA, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19 (1979), 215-229. Zbl0421.31006MR80k:58045
  18. [18] C.G.J. JACOBI, Uber Eine Particuläre Lösung der Partiellen Differential Gleichung ∂2V/∂x2 + ∂2V/∂y2 + ∂2V/∂z2 = 0, Crelle Journal für die reine und angewandte Mathematik, 36 (1847), 113-134. 
  19. [19] C.L. SIEGEL, Topics in complex function theory I, Wiley (1969). Zbl0184.11201
  20. [20] J.C. WOOD, Harmonic morphisms, foliations and Gauss maps, Contemporary Mathematics, Vol. 49 (1986), 145-183. Zbl0592.53020MR87i:58045

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