Extremal Quasiconformal Mappings And Teichm¨¹ller Space

The purpose of this thesis is to study the problem of extremal quasiconformal mappings and the associated properties of Teichmüller space. The concept of quasiconformal mappings was born in the 1930s, around 1940, German mathematician Teichmüller applied extremal quasiconformal mappings theory to study the modular problem of Reimann surfaces, and gave a perfect answer to this classical geometry problem. Since then, the theory of quasiconformal mappings and Teichmüller space has been greatly concerned by mathematicians, such as Ahlfors, Bers, Gehring, Earle, Gardiner, Reich, Strebel, Li Zhong, Wu Shengjian, Shen Yuliang and Chen Jixiu etc. Today, the theory of quasiconformal mappings and Teichmüller space has been cross-infiltrated into differential geometry, partial differential equations, topology and other branches of mathematics.The main research contents in the theory of extremal quasiconformal mappings are the existence, uniqueness of extremal quasiconformal mappings for given boundary correspondence, and the properties and characteristics of extremal mappings, and so on. In chaptersⅡandⅢof this thesis, we will study these issues and obtain a series of results.Logarithmic derivative plays an important role in determining whether a con-formal mapping can be quaisconformally extended, in estimating the inner radius of univalency of some domains, and in the description of universal Teichmüller space, the study of logarithmic derivative will play an active role in the development of quasiconfoaml mappings theory. In chaptersⅣof this thesis, we will study some geometrical properties of the universal Teichmüller space by the model of logarithmic derivative.The tangent space of Teichmüller space (also called the infinitesimal Teichmüller space) plays an important role in the study of the Teichmüller space and in the description of extremal quaisiconformal mappings. Therefore, in this thesis we will also discuss some unknown problems in the infinitesimal Teichmüller space. The thesis is divided into five chapters.ChapterⅠ, Introduction. We briefly introduce historical background and significance of the theory of quasiconformal mappings and Teichmüller space, and then describe the origin and the development of the problems which are studied in this dissertation, and state the results we obtained.ChapterⅡ, On the extremality of quasiconformal mappings on quadratic parabolic domain. We already know a lot about the extremal quasiconformal mappings for given correspondence of all boundary points, but it remains unclear for the extremal quasiconformal mappings if we reduce the correspondence condition to a subset of the boundary. Strebel ([94] [95]) had studied this issue on several different domains, we will discuss this issue on quadratic parabolic domains.ChapterⅢ, Some unequivalence between Teichmüller space and its tangent space. The description of the characteristics of extremality and unique extremality of quasiconformal mappings is a hot point in the theory of extremal quasiconformal mappings. Characterizations of extremal quasiconformal mappings were proved by Hamilton([41]), Krushkal([45]), Reich and Strebel([76]). Later in 1998, Bo(z|ˇ)in, Lakic, Markovi(?) and Mateljevi(?) ([9]) obtained some important characteristics of uniquely extremal quasiconformal mappings. From all these papers, we find that there are many equivalent properties between Teichmüller space and its tangent space. In this chapter, we will study the equivalence problem on Strebel points, the existence of extremal Teichmüller Beltrami coefficients, the existence of extremal Beltrami coefficients with constant modulus between Teichmüller space and its tangent space.ChapterⅣ, Geometrical properties of the universal Teichmüller space by logrith-mic derivative. Logarithmic derivative is closely connected with quasiconformal extensions of univalent functions. The model of universal Teichmüller space by logarithmic derivative has many special geometric properties, Zhuravelv([116]) proved that the universal Teichmüller space by logarithmic derivative consists of infinitely many disjoint components. In this chapter, we study the geometric properties of geodesics and balls in each component of the universal Teichmüller space by logarithmic derivative.ChapterⅤ, Some problems on infinitesimal Teichmüller space. From Chapter III, we know that there are many similar and non-similar properties between Teichmüller space and its tangent space. In this chapter, we study the non-uniqueness of geodesic disks in infinitesimal Teichmüller space, and also the Hamilton sequences of infinites-imally extremal Beltrami coefficients.