Length of curves under conformal mappings
type: Item
instance of: scholarly article
title: Length of curves under conformal mappings
zbMATH Open document ID: 0611.30011
DOI: 10.1007/BF02564441
published in: Commentarii Mathematici Helvetici
publication date: 1987-01-01T00:00:00Z
review text: Let f be a conformal map of the unit disk D onto a region \(\Omega\). Hayman and Wu proved that there is an absolute constant A such that for any line L it follows that \[ (*)\quad \lambda (f^{-1}(\Omega \cap L))<A. \] (\(\lambda\) (E) denotes the Hausdorff linear measure of the set E.) Garnett, Gehring and Jones gave a simpler proof of this result and considered conditions on the rectifiable Jordan curve in order that (*) continues to hold. The authors prove that (*) remains valid for quasismooth, that is, chord-arc, curves with a constant A that depends only on the chord-arc constant. In addition, they prove that \(\lambda (f^{-1}(\Omega \cap L))<\infty\) if \(\Omega\) is a hyperbolic plane region, f:D\(\to \Omega\) is a universal covering map, L is a quasismooth arc and \(\Omega\) is of finite length type. This latter condition means that \(\sum \lambda (\partial T{\mathcal D})<\infty\), where \({\mathcal D}\) denotes the Dirichlet fundamental region for the covering projection and the sum is over all T in the group of cover transformations. In particular, this result holds for any finitely connected hyperbolic region without point boundary components. The tools used in the proofs include harmonic measure, quasiconformal mappings and Carleson measures.
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 3987622
zbMATH Keywords: chord-arc curve
zbMATH Keywords: quasismooth curve
zbMATH Keywords: finite length type
author: J. L. Fernández
author: David . H. Hamilton
reviewed by: D. Minda
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3987622