Applications Of Hyperbolic Type Metrics And Inversions In The Theory Of Freely Quasiconformal Mappings

Univalent and analytic functions defined in the complex plane C are called conformal mappings.This class of mappings possesses many nice geometric and analytic properties.Conformal mappings have many applications not only in mathematics itself,but also in physics,engineering etc.Quasiconformal map-pings are generalizations of conformal mappings.They are homeomorphisms which have bounded perturbations with respect to a certain conformal invari-ant.Quasisymmetric mappings and quasimobius mappings are quasiconformal.In the 1990s,by using the hyperbolic type metrics,Vaisala built up the theory of quasiconformal mappings in Banach spaces,that is,the theory of freely quasi-conformal mappings.Now,this theory attracts much attention.In this thesis,we mainly study the applications of hyperbolic type metrics and inversions in theory of freely quasiconformal mappings.First,by exploiting hyperbolic type metrics,we study the properties of several classes of freely quasiconformal mapppings.Also,we get the coarsely bilipschitz continuity of inversions with respect to the distance ratio metrics,and the invariance of ?-uniform domains and ?-natural domains under inversions.Then,we obtain the invariance of?-uniform domains under quasimobius mappings.Finally,we discuss the ex-tension property of quasimobius mappings.This thesis consists of five chapters,whose arrangement is as follows.In the first chapter of this thesis,we introduce the research background of the problems,and state the main results obtained.Recently,based on the distance ratio metrics,Vuorinen et al.introduced two classes of mappings,which are quasi-isometric mappings and fully quasi-isometric mappings with respect to the distance ratio metrics,and studied the properties of these mappings.In the second chapter of this thesis,also based on the distance ratio metrics,we introduce a new class of mappings,i.e.,point quasi-isometric mappings with respect to the distance ratio metrics,and then,we prove that this class of mappings is equivalent to the one of the fully quasi-isometric mappings with respect to the distance ratio metrics.Also,we discuss the related full property of the quasi-isometric mappings with respect to the distance ratio metrics provided that their control functions are linear.The inversions in metric spaces were introduced by Buckley,Herron,and Xie in 2008.Also,they got the bilipschitz continuity of the inversions with respect to the quasihyperbolic metrics.In the third chapter of this thesis,as a supplement,we prove that the inversions are coarsely bilipschitz continuous with respect to the distance ratio metrics.Buckley,Herron,and Xie investigated the invariance of uniform domains with respect to inversions in metric spaces in 2008.As the first purpose of the fourth chapter of this thesis,we show that both ?-uniform domains and natural domains are invariant under inversions.In 2009,Xie obtained the in-variance of uniform domains with respect to quasimobius mappings.As the other purpose of the fourth chapter of this thesis,we obtain the invariance of?-uniform domains and natural domains under quasimobius mappings.Because uniform domains are both ?-uniform and natural,we see that our results are generalizations of the corresponding results obtained by Buckley et al..Haissinsky investigated the extension property of quasisymmetric mappings in metric spaces in 2009.In the last chapter of this thesis,we consider the extension property of quasimobius mappings in metric spaces.We prove that if both the restrictions of a homeomorphism on two sets are quasimobius and some other natural conditions are satisfied,then the homeomorphism is quasimobius in the union of these two sets.Several examples are constructed to show that each assumption in the obtained theorem cannot be removed.