The Manifold Structures Of BMO-Teichm(?)ller Spaces

The study of subspaces in the universal Teichmuller space plays an important role in the research of Teichmuller theory.One of the most important subspaces of the universal Teichmuller space is BMO-Teichmuller space.In this thesis,we will mainly study the structure of BMO-Teichmuler spaces,and discuss a class of opera-tors of BMO spaces which are induced by strongly quasisymmetric homemorphisms.The thesis contains four chapters.1.In chapter 1,we will introduce the background of Teichmuller space and state the main results of this thesis.2.In chapter 2,we will introduce some basic concepts and results of Teichmuller theory which are closely related to the thesis.3.Jones proved a sense preserving homeomorphism h of the unit circle induces a bounded BMO map Ph by pull-back if and only if it is strongly quasisymmetric.In this chapter,we will discuss the compactness of the projection operator Ph-sending a BMOA function φ to the anti-holomorphic part of Phφ.4.This chapter is a continuous work of[36],which is mainly concerned with the B-MO theory of the universal Teichmuller space.We will endow the BMO-Teichmuller curve τb,the VMO-Teichmuller curve τv and the BMO-Teichmuller space Tb with BMO manifold structures and prove that all these three spaces τb,τv and Tb are contractible.We will also introduce an extended BMO-Teichmuller space Tb and endow it with BMO manifold structures as well.