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Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces
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Markovic, V. (Vladimir). (2002) Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces. Journal of the London Mathematical Society, Vol.65 (No.1). pp. 103114. ISSN 00246107

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Official URL: http://dx.doi.org/10.1112/S002461070100268X
Abstract
Let g : M > N be a quasiconformal harmonic diffeomorphism between noncompact Riemann surfaces M and N. In this paper we study the relation between the map g and the complex structures given on M and N. In the case when M and N are of finite analytic type we derive a precise estimate which relates the map g and the Teichmüller distance between complex structures given on M and N. As a corollary we derive a result that every two quasiconformally related finitely generated Kleinian groups are also related by a harmonic diffeomorphism. In addition, we study the question of whether every quasisymmetric selfmap of the unit circle has a quasiconformal harmonic extension to the unit disk. We give a partial answer to this problem. We show the existence of the harmonic quasiconformal extensions for a large class of quasisymmetric maps. In particular it is proved that all symmetric selfmaps of the unit circle have a unique quasiconformal harmonic extension to the unit disk.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Teichmüller spaces, Riemann surfaces, Functions of several complex variables, Diffeomorphisms, Mappings (Mathematics) 
Journal or Publication Title:  Journal of the London Mathematical Society 
Publisher:  Cambridge University Press 
ISSN:  00246107 
Official Date:  February 2002 
Volume:  Vol.65 
Number:  No.1 
Page Range:  pp. 103114 
Identification Number:  10.1112/S002461070100268X 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  1. L. Ahlfors, Lectures on quasiconformal mappings (Van Nostrand, Princeton, NJ, 1966). 
URI:  http://wrap.warwick.ac.uk/id/eprint/783 
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