A note on continuity of Green's functions on Riemann surfaces
type: Item
instance of: scholarly article
title: A note on continuity of Green's functions on Riemann surfaces
zbMATH Open document ID: 0633.30046
author: Masahiko Taniguchi
published in: Kodai Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let R be an arbitrary Riemann surface admitting Green's functions, and denote by g(\(\cdot,r)\) the Green function with the pole \(r\in R\). Set \(U(r,M)=\{s\in R:\) \(g(s,r)>M\}\) where M is positive. The main results of this paper are as follows: (a) Let q be a point on R. Take M so large that U(q,M) is simply connected. Then it holds that \[ \| dg(\cdot,q')- dg(\cdot,q)\|_{R-U(q,M)}<6\cdot e^{M+4} \exp (-g(q',q)) \] for every \(q'\in U(q,M+4).\) (b) Let p and q be two distinct points on R. Take M so large that U(q,M) is contained in R-\(\{\) \(p\}\) and simply connected. Then it holds that \[ | g(q',p)-g(q,p)| <5M^{1/2} e^{M+4} \exp (-g(q',q)) \] for every \(q'\in U(q,M+4)\).
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 4029927
zbMATH Keywords: Green's functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037411
OpenAlex ID: W2068538605
cites work: Variations of Abelian differentials under quasiconformal deformations
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cites work: Q3749304
cites work: Continuity of certain differentials on finitely augmented Teichmüller spaces and variational formulas of Schiffer-Spencer's type
DOI: 10.2996/KMJ/1138037411
description: scientific article; zbMATH DE number 4029927