Existence of singular harmonic functions

Nakai, Mitsuru ; Segawa, Shigeo

Kodai Math. J., Tome 33 (2010) no. 1, p. 99-115 / Harvested from Project Euclid

Résumé

An afforested surface W := 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 ${\mathcal O}_s$ be the family of hyperbolic Riemann surfaces on which there are no nonzero singular harmonic functions. One might feel that any afforested surface W := n)_{n $\in$ N}, (σ_n)_{n $\in$ N}> belongs to the family ${\mathcal O}_s$ as far as its plantation P and all its trees T_n belong to ${\mathcal O}_s$ . The aim of this paper is, contrary to this feeling, to maintain that this is not the case.

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Publié le : 2010-03-15
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