An example of an open Riemann surface not uniformly large with respect to Green's functions
type: Item
instance of: scholarly article
title: An example of an open Riemann surface not uniformly large with respect to Green's functions
zbMATH Open document ID: 0646.30037
author: Masahiko Taniguchi
published in: Kodai Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: In the present paper the author compares two properties, called (G) and (H), respectively, which would reflect certain uniformity of Riemann surfaces. A Riemann surface R is said to have the property (G) if it has the Green function g(p,q) and there exists a constant \(M>0\) such that \(\{\) \(p\in R:\) \(g(p,q)>M\}\) is simply connected for every \(q\in R\). R is said to have the property (H) if there is a constant \(\epsilon >0\) such that \(\{\) \(p\in R:\) \(d_ R(p,q)<\epsilon \}\) is simply connected for every \(q\in R\), where \(d_ R\) denotes the hyperbolic distance on R. These two properties have appeared in the study of BMO for Riemann surfaces. It is seen that (G) implies (H). The main objective of the paper under review is to prove that there exists a regular Riemann surface of Parreau-Widom type which satisfies (H) but not (G). In order to construct a surface of this property, the author takes two kinds of finite bordered surfaces. The one is a compact bordered surface \(R_ 1\) of genus one and with a single contour and its Nielsen kernel is denoted by \(S_ 1\), so that its contour \(d_ 1\) is a geodesic, whose length is denoted by c. The other is a compact bordered surface \(S_ 0\) of genus one and with two contours which are geodesics of the same length c. Let \(W=R_ 1-(S_ 1-d_ 1)\). Roughly speaking, one then constructs finite bordered surfaces \(S'_ j\) by pasting together \(S_ 1\), j-1 copies of \(S_ 0\), and W sequentially in this order. Then, a surface R having the desired property is obtained by pasting \(S'_ 1,S'_ 2,S'_ 3,..\). sequentially in this order along certain short geodesics. See the original paper for technical details.
Mathematics Subject Classification ID: 30F20
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 4054012
zbMATH Keywords: open Riemann surface
zbMATH Keywords: Green function
zbMATH Keywords: hyperbolic distance
zbMATH Keywords: Riemann surface of Parreau-Widom type
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037415
OpenAlex ID: W2042244418
cites work: Q5593095
cites work: Nielsen extensions of Riemann surfaces
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cites work: On some extension property for BMO functions on Riemann surfaces
cites work: Hardy classes on infinitely connected Riemann surfaces
cites work: Continuity of certain differentials on finitely augmented Teichmüller spaces and variational formulas of Schiffer-Spencer's type
cites work: A note on continuity of Green's functions on Riemann surfaces
DOI: 10.2996/KMJ/1138037415
description: scientific article; zbMATH DE number 4054012