Harmonic morphisms onto Riemann surfaces and generalized analytic functions

Baird, Paul

Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 135-173.

On étudie les morphismes harmoniques définis sur les domaines de ${\mathbf{R}}^3$ 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 ${\mathbf{R}}^3$. 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.

We study harmonic morphisms from domains in ${\mathbf{R}}^3$ 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 ${\mathbf{R}}^3$ 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.

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