Some Problems On Teichm(u|")ller Spaces

The main purpose of this dissertation is to deal with the angles between geodesic segments in Teichmüller spaces and the pull-back operators induced by quasisymmetric homeomorphisms. It is arranged as follows.The first two chapters are preliminary. In Chapter 1, we begin with some backgrounds on Teichmüller spaces, state some open problems on Teichmüller spaces and our main results obtained in this dissertation. In Chapter 2, we introduce some basic definitions and results on Teichmüller theory. These include quasiconformal mappings,Teichmüller spaces and modular groups, Bers embedding and complex analytic structure of Teichmüller spaces, Royden-Gardiner Theorem on the Kobayashi metric.In Chapter 3, we discuss the angles between geodesic segments in Teichmüller spaces. After introducing the notion of the angle between two curves in Teichmüller spaces, we prove the existence of the angle between two smooth geodesic segments.Then we show that in any infinite dimensional Teichmüller space, there exist infinitely many geodesic triangles each of which has the same three vertices and satisfies the property that its three sides have the same and arbitrarily given length while its three angles are equal to any given three possibly different numbers from 0 to π. This implies that the sum of three angles of a geodesic triangle may be equal to any given number from 0 to 3π in an infinite dimensional Teichmüller space, indicating that an infinite dimensional Teichmüller space presents all hyperbolic, Euclidean, and spherical phenomena in angle geometry.In Chapter 4, we deal with the problem whether there exist two distinct standard geodesic segments joining a pair of points that are tangent at both endpoints. We are able to 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.In the last Chapter 5, we are concerned with some sub-spaces in the universal Teichmüller space. We first introduce some integral operator induced by a quasisymmetric homeomorphism. After proving it to be a bounded operator, we use this operator to give some new characterizations for the little Teichmüller space and Weil-Petersson Teichmüller space.