Moduli of doubly connected domains under univalent harmonic maps
type: Item
instance of: scholarly article
title: Moduli of doubly connected domains under univalent harmonic maps
zbMATH Open document ID: 1496.31001
author: Daoud Bshouty
author: Abdallah Lyzzaik
author: A. Rasila
author: A. Vasudevarao
published in: The Journal of Geometric Analysis
publication date: 2022-03-21T00:00:00Z
review text: Let \(\mathcal{T}(t)=\mathbb{C}\setminus([-1,1]\cup[t,\infty))\), \(t>1\), be a Teichmüller doubly connected domain. The Teichmüller-Nitsche problem is formulated as follows: For which values \(s,t\), \(1<s,t<\infty\), does a harmonic homeomorphism \(f:\mathcal{T}(s)\rightarrow\mathcal{T}(t)\) exist? In the paper, the Teichmüller-Nitsche problem is solved for symmetric harmonic homeomorphisms between \(\mathcal{T}(s)\) and \(\mathcal{T}(t)\). This problem is solved by using the method of extremal length. The following question suggested by \textit{T. Iwaniec} et al. [Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 5, 1017--1030 (2011; Zbl 1267.30059)] is also considered: Characterize pairs \((\Omega,\Omega^*)\) of doubly connected domains that admit a univalent harmonic mapping from \(\Omega\) onto \(\Omega^*\). This question is tested regarding the moduli of the doubly connected domains related by harmonic homeomorphisms. The paper concludes with relevant questions.
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 7493883
zbMATH Keywords: Teichmüller-Nitsche problem
zbMATH Keywords: univalent harmonic mappings
zbMATH Keywords: homeomorphism
zbMATH Keywords: analytic dilatation
zbMATH Keywords: modulus of doubly connected domains
zbMATH Keywords: affine capacity
zbMATH Keywords: Schwarz-Christoffel transformations
MaRDI profile type: Publication
full work available at URL: s12220-022-00895-2
OpenAlex ID: W4213146324
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DOI: 10.1007/S12220-022-00895-2
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description: scientific article; zbMATH DE number 7493883