## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 33, Number 1 (2010), 99-115.

### Existence of singular harmonic functions

Mitsuru Nakai and Shigeo Segawa

#### Abstract

An afforested surface *W* := <*P*, (*T*_{n})_{nN}, (σ_{n})_{nN}>, **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})_{nN} of hyperbolic Riemann surfaces *T*_{n} each of which is called a tree, and a sequence (σ_{n})_{nN} 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* **N**, or more precisely, by pasting surfaces *T*_{n} to *P* crosswise along slits σ_{n} for all *n* **N**. Let 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})_{nN}, (σ_{n})_{nN}> belongs to the family as far as its plantation *P* and all its trees *T*_{n} belong to . The aim of this paper is, contrary to this feeling, to maintain that this is not the case.

#### Article information

**Source**

Kodai Math. J., Volume 33, Number 1 (2010), 99-115.

**Dates**

First available in Project Euclid: 6 April 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1270559160

**Digital Object Identifier**

doi:10.2996/kmj/1270559160

**Mathematical Reviews number (MathSciNet)**

MR2732233

**Zentralblatt MATH identifier**

1192.30014

#### Citation

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