Some notes on geodesic segments in infinite dimensional Teichmüller spaces

We discuss the class of quasiconformal mappings of the unit disk which keep the boundary points fixed and study the geodesic segments in infinite dimensional Teichmüller spaces. In particular, we prove that in any infinite dimensional Teichmüller space, if there exist more than one geodesic segments between two points, then there must exist infinitely many geodesic segments joining them such that each pair of these geodesic segments are tangent to each other at both endpoints.