Radial projection and the Poincaré metric
type: Item
instance of: scholarly article
title: Radial projection and the Poincaré metric
zbMATH Open document ID: 0936.30033
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-11-22T00:00:00Z
review text: Let us consider domains on the Riemann sphere \(\overline C\) which omit at least three points. The unit disk \(U=\{z:|z|<1\}\) is the universal covering surface of any such domain \(D\). If \(\pi:U\to D\) is the universal covering, then the Poincaré metric \(\lambda_D\) for \(D\) is given by \[ \lambda_D \bigl(\pi(z) \bigr)\bigl|\pi'(z)\bigr|= {1\over 1-|z |^2}, \quad z\in U,\;w=\pi(z) \neq\infty \] and if \(w=\infty\), then one sets \(\lambda_D(\infty)= \lambda_D^{-1}(0)\), where \(D^{-1}=\{z:z^{-1}\in D\}\). Let \(E\subset\overline C\) and let \(E_{\text{cire}}\) denote radial projection of \(E\) into the unit circle \(T=T_1\), where \(\text{Tr}= \{z:|z|=r\}\), \(r>0\). ``One of the main purposes of this note is to derive some new estimates for the Poincaré metric \(\lambda_D\). These estimates are given in terms of the radial projection \(E_{\text{cire}}\) of the set \(E=\overline C\setminus D\) onto the circle \(T\).
Mathematics Subject Classification ID: 30F45
zbMATH DE Number: 1364721
zbMATH Keywords: Poincaré metric
zbMATH Keywords: Riemann surface
zbMATH Keywords: radial projection
author: A. Yu. Solynin
reviewed by: Leon Mikołajczyk
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1364721