On the range of $${\mathbb{R}}^2$$ or $${\mathbb{R}}^3$$-valued harmonic morphisms.(English)Zbl 0933.31006

Summary: The author proves that, under some general assumptions, the range of any nonconstant harmonic morphism from a simply connected open set $$U$$ in $$\mathbb{R}^n$$ to $$\mathbb{R}^3$$, $$n>3$$, cannot avoid three concurrent half-lines, which is an extension of Picard’s little theorem. To this end, he proves two results concerning the windings of Brownian motion around three concurrent half-lines in $$\mathbb{R}^3$$ and the recurrence of some domains linked with the harmonic morphism.

MSC:

 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 60J45 Probabilistic potential theory 60J65 Brownian motion
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References:

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