Existence of singular harmonic functions

Abstract

An afforested surface W := <P, (T_n)_n$\in$N, (σ_n)_n$\in$N>, N being the set of positive integers, is an open Riemann surface consisting of three ingredients: a hyperbolic Riemann surface P called a plantation, a sequence (T_n)_n$\in$N of hyperbolic Riemann surfaces T_n each of which is called a tree, and a sequence (σ_n)_n$\in$N of slits σ_n called the roots of T_n contained commonly in P and T_n which are mutually disjoint and not accumulating in P. Then the surface W is formed by foresting trees T_n on the plantation P at the roots for all n $\in$ N, or more precisely, by pasting surfaces T_n to P crosswise along slits σ_n for all n $\in$ N. Let ${\mathscr O}_s$ be the family of hyperbolic Riemann surfaces on which there are no nonzero singular harmonic functions. One might feel that any afforested surface W := <P, (T_n)_n$\in$N, (σ_n)_n$\in$N> belongs to the family ${\mathscr O}_s$ as far as its plantation P and all its trees T_n belong to ${\mathscr O}_s$. The aim of this paper is, contrary to this feeling, to maintain that this is not the case.