# Harmonic morphisms onto Riemann surfaces and generalized analytic functions

Annales de l'institut Fourier (1987)

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

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topBaird, 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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