On the existence of harmonic maps from a surface into the real projective plane

Jürgen Jost

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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...

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