Harmonic morphisms onto Riemann surfaces and generalized analytic functions
Annales de l'Institut Fourier, Volume 37 (1987) no. 1, p. 135-173

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.

On étudie les morphismes harmoniques définis sur les domaines de R 3 et S 3 et à valeurs dans une surface de Riemann N. Alors on obtient la classification en fonction des applications holomorphes d’un espace de recouvrement de N dans certaines variétés grassmanniennes. On montre que le seul morphisme harmonique, non constant et submersif, défini sur toute la sphère S 3 à valeurs dans une surface de Riemann est essentiellement l’application de Hopf.

On fait la comparaison avec la théorie des fonctions analytiques. En particulier on considère les morphismes harmoniques multivoques définis sur les domaines de R 3 . On montre donc comment on peut appliquer une procédure de découpage et collage pour obtenir un morphisme harmonique univoque défini sur une certaine variété à 3 dimensions. De la même façon, on construit une surface de Riemann associée à une fonction analytique multivoque.

@article{AIF_1987__37_1_135_0,
     author = {Baird, Paul},
     title = {Harmonic morphisms onto Riemann surfaces and generalized analytic functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {37},
     number = {1},
     year = {1987},
     pages = {135-173},
     doi = {10.5802/aif.1080},
     zbl = {0608.58015},
     mrnumber = {88h:31009},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1987__37_1_135_0}
}
Baird, Paul. Harmonic morphisms onto Riemann surfaces and generalized analytic functions. Annales de l'Institut Fourier, Volume 37 (1987) no. 1, pp. 135-173. doi : 10.5802/aif.1080. http://www.numdam.org/item/AIF_1987__37_1_135_0/

[1] P. Baird & J. Eells, A conservation law for harmonic maps, Geometry Symp. Utrecht 1980, Springer Notes, 894 (1981), 1-25. | MR 655417 | MR 83i:58031 | Zbl 0485.58008

[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. | Numdam | MR 526785 | MR 81b:30088 | Zbl 0386.30029

[3] M. Brelot, Lectures on potential theory, Tata Institute of Fundamental Research, Bombay (1960). | MR 118980 | MR 22 #9749 | Zbl 0098.06903

[4] E. Calabi, Minimal immersions of surfaces in Euclidan spheres, J. Diff. Geom., 1 (1967), 111-125. | MR 233294 | MR 38 #1616 | Zbl 0171.20504

[5] C. Constantinescu & A. Cornea, Compactifications of harmonic spaces, Nagoya Math. J., 25 (1965), 1-57. | MR 174760 | MR 30 #4960 | Zbl 0138.36701

[6] A.M. Din & W.J. Zakrzewki, General classical solutions in the CPn - 1 model, Nucl. Phys. B., 174 (1980), 397-406. | MR 591620

[7] A.M. Din & W.J. Zakrzewski, Properties of the general classical CPn - 1 model, Phys. Lett., 95 B (1980), 419-422. | MR 590024

[8] J. Eells, Gauss maps of surfaces, Perspectives in Mathematics, Birkhäuser Verlag, Basel (1984), 111-129. | MR 779673 | MR 86j:53090 | Zbl 0581.58013

[9] J. Eells & L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., 10 (1978), 1-68. | MR 495450 | MR 82b:58033 | Zbl 0401.58003

[10] J. Eells & L. Lemaire, On the construction of harmonic and holomorphic maps between surfaces, Math. Ann., 252 (1980), 27-52. | MR 81k:58030 | Zbl 0424.31009

[11] J. Eells & J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. | MR 29 #1603 | Zbl 0122.40102

[12] J. Eells & J.C. Wood, Harmonic maps from surfaces to complex projective spaces, Advances in Math., 49 (1983), 217-263. | MR 85f:58029 | Zbl 0528.58007

[13] O. Forster, Lectures on Riemann Surfaces, Springer (1981). | Zbl 0475.30002

[14] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier, 28-2 (1978), 107-144. | Numdam | MR 80h:58023 | Zbl 0339.53026

[15] V. Glaser & R. Stora, Regular solutions of the CPn models and further generalizations, CERN (1980).

[16] R. Hartshorne, Algebraic Geometry, Springer (1980).

[17] T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19 (1979), 215-229. | MR 80k:58045 | Zbl 0421.31006

[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] C.L. Siegel, Topics in complex function theory I, Wiley (1969). | Zbl 0184.11201

[20] J.C. Wood, Harmonic morphisms, foliations and Gauss maps, Contemporary Mathematics, Vol. 49 (1986), 145-183. | MR 87i:58045 | Zbl 0592.53020