Criterion for Cannon's Conjecture

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon's Conjecture: A hyperbolic group $G$ (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic to a Kleinian group if and only if every two points in the boundary of $G$ are separated by a quasi-convex surface subgroup. Thus, the Cannon's conjecture is reduced to showing that such a group contains "enough" quasi-convex surface subgroups.Comment: Revised versio

Harmonic maps between 3-dimensional hyperbolic spaces

We prove that a quasiconformal map of the 2-sphere admits a harmonic quasi-isometric extension to the 3-dimensional hyperbolic space, thus confirming the well known Schoen Conjecture in dimension 3.Comment: Final Versio

On the Forelli-Rudin projection theorem

Motivated by the Forelli--Rudin projection theorem we give in this paper a criterion for boundedness of an integral operator on weighted Lebesgue spaces in the interval $(0,1)$. We also calculate the precise norm of this integral operator. This is the content of the first part of the paper. In the second part, as applications, we give some results concerning the Bergman projection and the Berezin transform. We derive a generalization of the Dostani\'{c} result on the norm of the Berezin transform acting on Lebesgue spaces over the unit ball in $\mathbf{C}^n$.Comment: to appear in Integral Equations and Operator Theor

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