The Removability Property Of ψ-uniform Domains In Metric Spaces

In 1928, Grotzsch introduced the concept of quasiconformal mapping when he did some research on Riemann mapping theorem on a kind of special map-pings from squares to rectangles remaining the vertexes invariant in E2. Qua-siconformal mapping has become an important branch of complex analysis through several decades of development, and it is relevant to Klein group, com-plex analytic dynamic system and Teichmuller space. How to describe quasicon-formal mapping through special domains is worth thinking, and so the analytic properties and geometric properties of special domains become very important objects to study in complex analysis.This paper is devoted to study the removability properties of ψ-uniform domains in metric spaces. And it makes up of six chapters, which are arranged as follows.In the first chapter, we introduce the background and main result of this paper, especially the removability properties of ψ1-uniform domains.In the second chapter, some basic concepts such as the definitions of quasi-hyperbolic distance and sphere convex domains are presented.In the following chapter, to prove the main result of this paper some es-sential lemmas and their proofs are introduced.In the forth chapter, we discuss the sufficient condition of the main result by splitting up it into several cases according to the distance from the point to the boundary.In the fifth chapter, the necessary condition of the main result is derived.In the last chapter, in order to prove the main lemma of this paper, which is labeled as lemma 3.1.10, we try to find a kind of special curves, and divide them into pieces according to the increasing property of the distance from the points of the curve to the boundary of the domain.