Characterizations Of Uniform Domains, John Domains And Isometries In Hyperbolic Spaces, And The Quasiconvexity Of Quasigeodesics

In order to investigate the properties of approximation and injectivity of bi-Lipschitz mappings,in 1961,John introduced a class of domains in C which were named as John domians by Martio and Sarvas in 1978.In order to generalize Ahlfors' study about the properties of injectivity of conformal mappings in quasidisks,in 1978,Martio and Sarvas introduced a class of new domains which are uniform domains.It is well-known that quasidisks own a lot of properties which have been applied in many ways.Since John domains and uniform domains are generalizations of quasidisks,the problems of which properties of quasidisks are valid for these two classes of domains and what special properties these two classes of domains have have become very interesting.The isometires about Poincarémetric in H~n are one of the elementary elements in Kleinian groups and hyperbolic manifolds.This shows that the study of their characterizations is basic and important.We will focus our study on these problems.The research of this thesis includes two parts.In the first part,we discuss the problem which is how to characterize John domains and uniform domains by using quasihyperbolic metric,j_D,Apollonian metric and their corresponding inner metrics.Concerning this problem,we have got a series of results,and three open problems and conjectures raised by Broch,H(?)st(?),Ponnusamy,Sahoo, V(?)s(?)l(?)in Results in Mathematics etc have been solved.In the second part,we discuss the characterization of the isometries with respect to Poincarémetric in H~n.About this,there is a conjecture recently raised by Baokui Li and Guowu Yao in Mathematical Proceedings of Cambridge Philosophy Society.We mainly discuss this conjecture,to which a positive answer is obtained.This thesis consists of nine chapters.It is arranged as follows.In chapter one,we mainly introduce the background of our resaerch and state our obtained results.In chapter two,we discuss Riemann Mapping Theorem with respect to n-dimensional quasiconformal mappings.We prove that each bounded and convex domain in R~n is a quasiball.This result is a generalization of the known results in this line and shows that Riemann Mapping Theorem with respect to n-dimensional quasiconformal mappings holds in bounded and convex domains in R~n.In chapter three,by using the hyperbolic metric andλ-Apollonian metric, we obtain a sufficient condition for a Jordan domain in C to be a John disk.Also we construct two examples to show that the converse of the above mentioned result does not hold.These results show that we completely solve the conjecture raised by Broch in 2004.In chapter four,by using the max-min properties of geodesics,quasihyperbolic metric,Apollonian metric etc,we get three necessary and sufficient conditions for a domain in C to be John.These conditions are generalizations of the corresponding results obtained by Gehring,Hag,Herron and Broch.In chapter five,we mainly discuss the relations between Apollonian inner metric and uniform domains.By using Apollonian inner metric,we get a necessary and sufficient condition for a domain in R~n to be uniform.This result shows that the answer to the open problem raised by H(?)st(?),Ponnusamy and Sahoo in 2006 is negative.Also we get several results about the relations among isotropic domains,Apollonian quasiconvex domains and A-uniform domains.In chapter six,as the continuation of the discussion in Chapter five,by using the similar reasoning as that in chapter five,we discuss the relations between Apollonian inner metric and John domains.We get a sufficient condition for a bounded domain in R~n to be John.This completes the corresponding discussion in Chapter five.In chapter seven,we mainly study the decomposition and removability properties of John domanis in R~n.In this chapter,we first give the definition of the John domain decomposition property,and then prove that a domain in R~n is a John domain if and only if it has the John domain decompostion property. An application is given.Also we prove that for any John domain,after a finitely many points are removed from it,the new domain is still John.Further,we construct an example to show that the best possibilty of the condition "finitely many points".In chapter eight,we mainly study the uniform domain decomposition property and the quasiconvexity of the quasigeodesics in real vector normed spaces. We first define the notion of the uniform domain decomposition property which is a generalization of the notion of quasiconformally decomposable property. Then by constructing concrete simply connected domains,we prove that a domain in R~n is uniform if and only if it has the uniform domain decomposition property.As an application of our obtained result,by constructing concrete examples in R~n,we show that there are uniform domains which are not quasiballs. In the end,we prove that any quasigeodesic is quasiconvex in the norm metric in convex domains in real normed vector spaces.This result gives a positive answer to one of the conjectures raised by V(?)s(?)l(?)in 2005.In chapter nine,we discuss the characterization of the isometires about Poincarémetric in H~n.Our result is as follows:for any mapping which maps every r-hyperplane in H~n into an r-hyperplane in H~n,this mapping is an isometry if and only if it is surjective.This result shows that the answer to one of the conjectures recently raised by Baokui Li and Guowu Yao is affirmative.