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Harmonic Diffeomorphisms of Noncompact Surfaces and Teichmüller Spaces
Author(s) -
Markovic Vladimir
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070100268x
Subject(s) - diffeomorphism , unit disk , mathematics , corollary , pure mathematics , unit circle , harmonic , riemann surface , extension (predicate logic) , harmonic map , unit (ring theory) , mathematical analysis , riemann hypothesis , physics , computer science , mathematics education , quantum mechanics , programming language
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.