Quasiconformal Mappings And Quasisymmetric Mappings In Metric Spaces

In order to describe and prove a promotion of Picard theorem, Grotzsch was first to introduce the quasiconformal mappings of the plane in 1928. Afterwards, Ahlfors and Lavrentiev again studied quasiconformal mappings in 1935 from the points of the partial differential equation and the theory of functions, respectively, and thus the term of quasicomformal mappings begun to emerge. Lavrentiev initiated the study of the higher dimensional quasiconformal mappings in 1938. Then some important results were developed by Teichmuller and Ahlfors, and Gehring and Vaisala started to deliberate systematically the quasiconformal mappings in Euclidean spaces. Beurling and Ahlfors were first to put forward the quasi symmetric mapping in 1956, which is a promotion of the quasiconformal mappings from higher dimensional Euclidean spaces to general metric spaces. Very surprising, quasiconformal mappings turn out to be equivalent to the so-called quasi symmetric mappings under appropriate circumstances. The properties of quasiconformal mappings and quasisymmetric mappings in Euclidean spaces and Banach spaces have been widely studied and have many applications. As the research of the quasiconformal mappings theory proceeds, many scholars naturally take it into account in the more general metric spaces.This thesis includes three parts. In the first part, we study the local and global properties of quasihyperbolic mappings in metric spaces, and obtain a series of results, and clearly a partial answered an open problem proposed by Vaisala in 1999. In the second part, we study the properties of quasisymmetric mappings in metric spaces, and extend the results of Vaisala(1990). Finally, we investigate the geometric properties of ?-uniform domains in metric spaces, and depict the problem of ?-uniform domains invariance properties under quasisymmetric mappings. In particular, this thesis consists of the following seven chapters.In Chapter 1, we mainly introduce the background and development of the problems that will be studied in this thesis, and then state our main results.In Chapter 2, we introduce the main tools used to study our problems: quasihyperbolic metric. We introduce the definition and the properties of the quasihyperbolic metric, and state the relationships between the quasihyperbolic metric and norm metric on the quasiconvex metric spaces. In this chapter we will see that many problems on quasiconformal mappings and quasi symmetric mappings may become easy when handling them by quasihyperbolic metric.In Chapter 3, we mainly introduce some important lemmas, which play a key role in the proof of conclusions. First of all, we devote to study the relations between quasihyperbolic metric and distance ratio metric, and the relations between ?D' (f(x)) and ?G' (f(x)) in metric space, where D'(?)G'. Secondly, we mainly introduce the concepts of ?-John-ball domains and not-cut-point domains, and proved that if a homeomorphism mapping is quasisymmetric mapping whenever it is free quasiconformal mapping in metric spaces.In Chapter 4, our main consideration is the relationships between the local and global properties of quasihyperbolic mappings in metric spaces. In 1999, Vaisala proposed the following open problem:Suppose that f:G?G' is a homeomorphism, and there exists M?1 such that, for each point x?G, there exists a neighborhood D(x)(?)G such that the restriction map f?D(x):D (x)?f (D (x)) is M-QH. Is f globally M'-QH with M'=M'(M)? In this chapter, we clearly a partially answered this problem.In Chapter 5, using quasihyperbolic metric, we study the properties of quasiconformal mappings and quasisymmetric mappings on the metric spaces, and proved that if a homeomorphism mapping is quasisymmetric mapping in quasihyperbolic metric spaces and quasihyperbolic balls whenever it is a fully coarsely quasihyperbolic mapping between suitable metric spaces.In Chapter 6, we investigate some basic properties and invariance of ?-uniform domains in metric spaces. In particular, we study the geometric properties of ?-uniform domains in metric spaces, and the ?-uniform domains invariance properties problem under quasisymmetric mapping was depicted.In the final chapter, the main results of the thesis are summarized. Moreover, the difficulties we have not overcome and some new problems we will study in the future are presented.