On The Study Of The Invariance Of Uniform Domains Under Quasim?bius Mappings In Banach Spaces

In order to investigate the properties of mappings,in 1979,Matio and Sarvas introduced the concept of uniform domains.Since then uniform domains have become a hot topic.In recent years,people pay much attention to the equivalence of uniform domains and quasihyperbolic uniform domains,the characterization of quasiconformal mappings in uniform domains,etc.The main aim of this thesis is to study the invariance of uniform domains with respect to quasim?bius mappins in Banach spaces.This thesis consists of four chapters.The arrangement is as follows:In Chapter one,we provide the background on the studied problems and the statement of the main results in this thesis.In Chapter two,several basic concepts such as quasihyperbolic metric,quasiconvexity etc are introduced.In Chapter three,several auxiliary lemmas are proved.In Chapter four,based on the obtained auxiliary lemmas,we establish the invariance of uniform domains with respect to quasim?bius mappings.This is a partial answer to one of open problems raised by V¨ais¨al¨a.