On The Properties Of Uniform Domains And Quaternionic M(?)bius Groups

Quasiconformal mappings are generalizations of conformal mappings.Because of the close relationships with Kleinian groups,dynamics of complex analytic func-tions and Riemann surfaces etc,quasiconformal mappings become one of the main research topics in complex analysis.In this thesis,we mainly discuss the remov-ability of uniform domains and composition of their boundaries, the properties of quaternionic MSbius groups and trace of (generalized) semi-definite matrices. The thesis is arranged as follows.In chapter 1,we introduce the background and significance of our research topic.A quasidisks is the image of the unit disk or half plain under the quasiconformal self mapping of the whole plain.As the generalization of quasidisks,Martio and others introduced the concept of uniform domains.A nature question is that what kind of subdomains of uniform domains are still uniform. The answer to this question will help us to construct examples of uniform domains and further studies of the properties of uniform domains.In chapter 2,we mainly study the removability of uniform domains.Given a uniform domain,we find its subdomains which are still uniform.This gives a partial answer to the above question.By using the theory of quasiconformal reflection,Gehring and Hag proved that a finitely connected planar domain is uniform if and only if its complementary components consist of quasidisks or points.In chapter 3,we give this result an elementary proof.Our proof does not depend on the quasiconformal reflection.Cnops and others studied some properties of quaternionic MSbius transfor-mations,quaternionic MSbius elementary groups and non-elementary groups.In Chapter 4,we introduce the concept of general elementary groups in quaternionic MSbius groups.Then we study it and obtain several properties. Yang Y and others studied some properties of inequalities about the trace of semi-definite matrices and generalized semi-definite matrices, and answered a conjecture raised by Bellman.In chapter 5,we study these properties further,and obtain their best possible generalizations. Meanwhile,we also generalize the results to the case of complex matrices.