Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces
Markovic, V. (Vladimir). (2002) Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces. Journal of the London Mathematical Society, Vol.65 (No.1). pp. 103-114. ISSN 0024-6107
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Official URL: http://dx.doi.org/10.1112/S002461070100268X
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|
|Official Date:||February 2002|
|Page Range:||pp. 103-114|
|Access rights to Published version:||Open Access|
1. L. Ahlfors, Lectures on quasiconformal mappings (Van Nostrand, Princeton, NJ, 1966).
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