Holomorphic families of geodesic discs in infinite dimensional Teichmüller spaces

Abstract

The theory of quasiconformal mappings plays an important role in Teichmüller theory. The Teichmüller spaces of Riemann surfaces are defined as quotient spaces of the spaces of Beltrami differentials, and the Teichmüller distances are defined to measure quasiconformal deformations between the Riemann surfaces representing points in the Teichmüller spaces. The Teichmüller spaces are complex Banach manifolds equipped with natural complex structures such that the canonical projections are holomorphic. It is known (see Gardinar [4]) that the Teichmüller distance, defined independently of the complex structures, coincides with the Kobayashi distance.In spite of the naturality of the definition of a Teichmüller space as a quotient of Beltrami differentials, for given two Beltrami differentials it is hard to determine whether they are equivalent or not. For this reason, it is not trivial to describe geodesic lines with respect to the Teichmüller-Kobayashi metric.