Area growth and Green's function of Riemann surfaces
type: Item
instance of: scholarly article
title: Area growth and Green's function of Riemann surfaces
zbMATH Open document ID: 0766.30038
DOI: 10.1007/BF02384863
published in: Arkiv för Matematik
publication date: 1993-01-16T00:00:00Z
review text: Let \(R\) be a Riemann surface of hyperbolic type endowed with the Poincaré metric (which has constant negative curvature). For \(p\in R\) let \(A(p,t)\) denote the (Poincaré) area of \(\{q\in R\): \(\text{dist}(p,q)\leq t\}\). The authors consider the condition \[ \inf_{p\in R} A(p,t)> ce^{\alpha t} \quad\text{for}\quad t>t_ 0\tag{*} \] where \(c>0\) and \(\alpha\) are constants. If \(R\) is a plane domain they show that (*) even with \(\alpha=0\) implies that \(R\) has a Green's function. On the other hand they construct a Riemann surface of infinite genus with (*) holding for \(\alpha=0.08\) that does not have a Green's function. They ask whether this is impossible for any \(\alpha<1\); for \(\alpha=1\) it is classical that (*) implies the existence of a Green's function.
Mathematics Subject Classification ID: 30F20
zbMATH DE Number: 91592
zbMATH Keywords: Poincaré metric
zbMATH Keywords: area
zbMATH Keywords: Green's function
author: J. L. Fernández
author: José M. Rodríguez
reviewed by: Christian Pommerenke
MaRDI profile type: Publication
cites work: Q5593095
cites work: Q3310101
cites work: Cubic graphs and the first eigenvalue of a Riemann surface
cites work: Q3344779
cites work: Random walk on the Speiser graph of a Riemann surface
cites work: Q3706275
cites work: Positive harmonic functions on abelian covers
cites work: On the Existence of Green's Function in Riemannian Manifolds
cites work: Rough isometries and the parabolicity of Riemannian manifolds
cites work: Q3948276
cites work: Q5666814
cites work: Function theory, random paths and covering spaces
cites work: Fundamental regions and the type problem for a Riemann surface
cites work: Uniformly perfect sets and the Poincaré metric
cites work: On Fuchsian groups of accessible type
cites work: Q3683227
cites work: Q3223122
cites work: Q3482602
Wikidata QID: Q126120827
full work available at URL: https://doi.org/10.1007/bf02384863
OpenAlex ID: W2070469167
description: scientific article; zbMATH DE number 91592