Two remarks on Riemann surfaces
type: Item
instance of: scholarly article
title: Two remarks on Riemann surfaces
zbMATH Open document ID: 0829.30027
DOI: 10.5565/PUBLMAT_38294_15
published in: Publicacions Matemàtiques
publication date: 1995-08-27T00:00:00Z
full work available at URL: https://eudml.org/doc/41175
review text: Let \(M\) be a complete Riemannian manifold of dimension 2 with a metric of constant curvature \(-1\). One says \(M\) satisfies a linear isoperimetric inequality (LII) if there exists a finite constant \(h(M)\) so that for every bounded open set \(G\) with smooth boundary, \(A(G) \leq h (M) L(\partial G)\). The author discusses connections between LII and some conformal invariants. Let \({\mathcal B}\) be the set of Riemann surfaces which do not satisfy LII, let \({\mathcal O}_G\) be the set of Riemann surfaces which do not have a Green's function, let \({\mathcal O}_{HP}\) be the set of Riemann surfaces which do not have a nonconstant positive harmonic function; \({\mathcal O}_G \subset {\mathcal O}_{HP}\). It is known \({\mathcal O}_G \subset {\mathcal B}\); the author proves that \({\mathcal B}\) is not contained in \({\mathcal O}_{HP}\), even with the hypothesis that the injectivity radius \(\iota (M)\) is positive: Theorem 1: There exists a Riemann surface \({\mathcal M}\) with LII, \(\iota (M) > 0\), and such that every positive harmonic function on \({\mathcal M}\) is constant. Let \(L_M (p,t)\) be the length of the boundary of the ball of radius \(t\) centered at \(t \in M\). The author also proves: Theorem 2: There exists a Riemann surface \({\mathcal N}\) such that \({\mathcal N}\) does not have a Green's function and so \(L_{\mathcal N} (p,t) \geq ce^{t/2}\) for \(t \geq t_0\) and every \(p \in {\mathcal N}\).
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 30F15
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 791195
zbMATH Keywords: linear isoperimetric inequality
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: Green's function
author: José M. Rodríguez
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 791195