Some Subspaces Of The Universal Teichmuller Space

In this thesis,we are mainly concerned with some subspaces of the universal Teichmuller space in the upper-half plane,including the symmetric Teichmuller space,the little Teichmuller space and the Weil-Petersson Teichmuller space.We have the following main results:1.A quasisymmetric homeomorphism on the real line is symmetric if and only if it can be extended an asymptotically conformal mapping to the upper-half plane(see[43],[60]).We introduce a complex Banach manifold structure on the space of normalized symmetric homeomorphisms on the real line via the Bers embedding.2.We discuss the pre-logarithmic derivative model of the little Teichmuller space on the half-plane.3.For a quasisymmetric homeomorphism h on the real line,Shen-Tang([77])proved that h is a Weil-Petersson homeomorphism if h is locally absolutely contin-uous with log h' ? HR1/2.We prove that the converse of this result is also true.4.We give a geometric characterization of a Weil-Petersson curve via its arc-length parametrization and showed that the corresponding Riemann mapping de-pends continuously on the curve.