Kodai Mathematical Journal

Existence of singular harmonic functions

Mitsuru Nakai and Shigeo Segawa

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An afforested surface W := <P, (Tn)n$\in$N, (σn)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 (Tn)n$\in$N of hyperbolic Riemann surfaces Tn each of which is called a tree, and a sequence (σn)n$\in$N of slits σn called the roots of Tn contained commonly in P and Tn which are mutually disjoint and not accumulating in P. Then the surface W is formed by foresting trees Tn on the plantation P at the roots for all n $\in$ N, or more precisely, by pasting surfaces Tn to P crosswise along slits σn 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 := <P, (Tn)n$\in$N, (σn)n$\in$N> belongs to the family ${\mathcal O}_s$ as far as its plantation P and all its trees Tn 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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Kodai Math. J., Volume 33, Number 1 (2010), 99-115.

First available in Project Euclid: 6 April 2010

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Nakai, Mitsuru; Segawa, Shigeo. Existence of singular harmonic functions. Kodai Math. J. 33 (2010), no. 1, 99--115. doi:10.2996/kmj/1270559160. https://projecteuclid.org/euclid.kmj/1270559160

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