Stability of conformal distortion
type: Item
instance of: scholarly article
title: Stability of conformal distortion
zbMATH Open document ID: 0822.30010
DOI: 10.1007/BF02564489
author: Alfred Huber
author: Reiner Kühnau
published in: Commentarii Mathematici Helvetici
publication date: 1995-10-08T00:00:00Z
full work available at URL: https://eudml.org/doc/140341
review text: Let \(e^{i \theta} \mapsto e^{i \varphi (\theta)}\) be an homeomorphisms of the unit circle \(C = \{| z | = 1)\}\) such that \(\varphi (\theta)\) is a differentiable function with \(\varphi' (\theta) > 0\). Then it is well known that there exists a Jordan curve \(\Gamma\) together with conformal mappings \(F : \text{int} C \to \text{int} \Gamma, G : \text{ext} C \to \text{ext} \Gamma\) satisfying \(F(e^{i \theta)}) = G(e^{i \varphi (\theta)})\). The conformal welding (consisting of \(\Gamma\) and the pair \((F,G))\) is uniquely determined if \(G\) is normalized by \(G(z) = z + O(1/z)\) \((z \to \infty)\). The first result of the authors is that if \(\varphi\) is close to the identity in the sense that \(\varphi' (\theta)\) is close to 1 then the curve \(\Gamma\) is close to a circle in the sense that \(\Gamma\) admits some quasiconformal reflection with small maximal dilatation. This implies also that \(\Gamma\) is contained in some circular ring domain of small modulus. The main result of the paper is that under the assumptions above the problem of conformal welding is ``well posed'' which means that the solution of the problem is stable in the following sense: If \(\varphi (\theta)\) differs only a little from \(\varphi_ 0 (\theta)\) such that the ratio \(\varphi_ 0' (\theta)/ \varphi' (\theta)\) is again close to 1 then \(\Gamma\) is also close to the curve \(\Gamma_ 0\) corresponding to \(\varphi_ 0\). More precisely the distance in the Hausdorff metric is small. For the proof quasiconformal distortion theorems are used. Furthermore examples are studied showing that the quantitative estimates given in the results above are best possible in a sense. The paper closes with a result on convergence with regard to the Dirichlet norm in the special case that \(\varphi_ 0 (\theta) = \theta\), \(\Gamma_ 0 = C\).
reviewed by: Jochen Becker
Mathematics Subject Classification ID: 30C20
zbMATH DE Number: 720099
zbMATH Keywords: conformal welding
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 720099