Quasihomographies in the theory of Teichmuller space

One of the most powerful tools, when studying Riemann surface, is the notion of Teichmuller space, i.e. a metrizable and complete quotient space of closed Riemann surfaces with genus g >= 2. While the concept was introduced by ingenious German mathematician O. Teichmuller before World War II, the name appears because of L. Bears and L. V. Ahlfors in the late fifties. The function – theoretic model of this, not easy understandable, original Teichmuller space, was built up by the use of equivalence classes of quasiconformal automorphisms of the unit disc or its boundary representation introduced by the author, called quasihomographies. Making use of the Poisson integral extension operator one may construct harmonic representation of the universal Teichmuller space in which, particular, boundary normalized harmonic automorphisms of the unit disc, represent elements of the space, in question.

The main purpose of the lecture is to present a number of theorems and constructions regarding metric and topological feature of harmonic and quasihomographical models of the universal Teichmuller space. This idea links once again extremal quasiconformal automorphisms of the unit disc with two classes of analytic functions, defined in the unit disc and called the conjugate Paprocki spaces of analytic functions. Some basic properties of functions from those spaces will also be presented during this lecture.