Characterization of quasi-disks and Teichmüller spaces
type: Item
instance of: scholarly article
title: Characterization of quasi-disks and Teichmüller spaces
zbMATH Open document ID: 0583.32056
author: Hiroshige Shiga
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1985-01-01T00:00:00Z
review text: Suppose \(\Gamma\) is a finitely generated Fuchsian group of the first kind acting on the upper half-plane U. Let T(\(\Gamma)\) denote the Bers embedding of the Teichmüller space of \(\Gamma\) into the Banach space \(B_ 2(L,\Gamma)\) of holomorphic quadratic differentials on the lower half-plane L which are bounded when divided by the square of the hyperbolic metric on L. Let S(\(\Gamma)\) be the set of all Schwarzian derivatives of univalent meromorphic functions on L. It is known that S(\(\Gamma)\) is closed and \(T(\Gamma)\cup \partial T(\Gamma)\subset S(\Gamma).\) The author proves that the interior of S(\(\Gamma)\) in \(B_ 2(L,\Gamma)\) is connected and equal to T(\(\Gamma)\). Also, he shows that the boundary of T(\(\Gamma)\) in \(B_ 2(L,\Gamma)\) can be very complicated by presenting a negative answer to a conjecture of Bers. In addition, the author discusses quasi-disks. P. Jones proved that a simply connected region \(\Omega\) in the complex plane \({\mathbb{C}}\) is a quasi-disk if and only if \(\Omega\) has the BMO extension property (that is, there is a constant \(M>0\) such that for all \(f\in BMO(\Omega)\) there exists \(F\in BMO({\mathbb{C}})\) such that \(\| F\|_{BMO({\mathbb{C}})}\leq M \| f\|_{BMO(\Omega)}).\) The author proves that it is sufficient to verify the BMO extension property just for the subclass ABD(\(\Omega)\) of bounded holomorphic functions on \(\Omega\) having finite Dirichlet integral. He also establishes an analog for finitely connected Jordan regions.
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 3933478
zbMATH Keywords: Fuchsian group
zbMATH Keywords: Bers embedding of the Teichmüller space
zbMATH Keywords: Schwarzian derivatives
zbMATH Keywords: quasi-disk
zbMATH Keywords: BMO extension property
reviewed by: D. Minda
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228594
OpenAlex ID: W1968926633
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DOI: 10.2748/TMJ/1178228594
description: scientific article; zbMATH DE number 3933478