We study harmonic morphisms from domains in and to a Riemann surface , obtaining the classification of such in terms of holomorphic mappings from a covering space of into certain Grassmannians. We show that the only non-constant submersive harmonic morphism defined on the whole of 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 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 et et à valeurs dans une surface de Riemann . Alors on obtient la classification en fonction des applications holomorphes d’un espace de recouvrement de dans certaines variétés grassmanniennes. On montre que le seul morphisme harmonique, non constant et submersif, défini sur toute la sphère 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 . 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}, pages = {135--173}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {37}, number = {1}, year = {1987}, doi = {10.5802/aif.1080}, mrnumber = {88h:31009}, zbl = {0608.58015}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1080/} }
TY - JOUR AU - Baird, Paul TI - Harmonic morphisms onto Riemann surfaces and generalized analytic functions JO - Annales de l'Institut Fourier PY - 1987 SP - 135 EP - 173 VL - 37 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.1080/ DO - 10.5802/aif.1080 LA - en ID - AIF_1987__37_1_135_0 ER -
%0 Journal Article %A Baird, Paul %T Harmonic morphisms onto Riemann surfaces and generalized analytic functions %J Annales de l'Institut Fourier %D 1987 %P 135-173 %V 37 %N 1 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/aif.1080/ %R 10.5802/aif.1080 %G en %F 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/articles/10.5802/aif.1080/
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