Harmonic Diffeomorphisms of Noncompact Surfaces and Teichmüller Spaces

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.