On the range of ${\bf R}\sp 2$ or ${\bf R}\sp 3$-valued harmonic morphisms

Abstract

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