Platonic surfaces
type: Item
instance of: scholarly article
title: Platonic surfaces
zbMATH Open document ID: 0920.30037
author: Robert Brooks
published in: Commentarii Mathematici Helvetici
publication date: 1999-04-26T00:00:00Z
review text: The main purpose of this paper is to show that it is possible to compare the hyperbolic metrics on two Riemann surfaces one of which is compact and the other is a punctured version of the same Riemann surface. Surprisingly this is possible outside a neighbourhood of the cusps, and it can be proved using the Ahlfors-Schwarz ``Lemma''. The author uses this to prove that if \(G_k\) uniformizes the compactification of \(\Gamma(k)\setminus{\mathbf H}\), \(k>6\), then the first non-trivial eigenvalue \(\lambda_1\) of the Laplace operator on \(G_k \setminus{\mathbf H}\) satisfies \(\lambda_1\geq 5/36\). This makes use of Selberg's 3/16 estimate in the case of the \(\Gamma(k)\). The author also gives a comparison of the Cheeger constants in the general situation.
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1278817
zbMATH Keywords: Ahlfors-Schwarz lemma
zbMATH Keywords: Selberg estimates
zbMATH Keywords: Cheeger constant
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
full work available at URL: s000140050082
OpenAlex ID: W4246987810
DOI: 10.1007/S000140050082
description: scientific article; zbMATH DE number 1278817