极值拟共形映射与Teichmüller空间

【作者】 范金华；

【导师】 陈纪修；

【作者基本信息】 复旦大学 ， 基础数学， 2008， 博士

【摘要】 本文主要目的在于研究拟共形映射极值问题以及与之相关的Teichmüller空间性质.拟共形映射的概念诞生于上世纪30年代,1940年左右,德国数学家Teichmüller利用极值拟共形映射理论来研究Riemann模问题,对这一经典的几何问题给予了完美的解答.此后,对拟共形映射与Teichmüller空间理论的研究就一直倍受数学家们所关注,Ahlfors,Bets,Gehring,Earle,Gardiner,Reich,Strebel,李忠,伍胜健,沈玉良和陈纪修等数学家都对该理论进行了深刻的研究.如今,拟共形映射与Teichmüller空间理论已交叉渗透到微分几何、偏微分方程、拓扑学等其它数学分支.拟共形映射的极值理论主要研究给定边界对应的拟共形映射族中极值映射的存在性、唯一性、以及极值映射的性质、特征的刻画等问题.本文的第二章和第三章将研究这些问题并得到一系列结果.对数导数在判定共形映射能否拟共形映射扩张、估计区域的单叶性内径以及描述万有Teichmüller空间的性质方面都起到非常重要的作用,对数导数的研究将对拟共形映射理论的发展起到积极的作用.本文的第四章我们将研究万有Teichmüller空间对数导数模型的一些几何性质.Teichmüiller空间的切空间(也称无限小Teichmüller空间)对研究Teichmüller空间的性质以及刻画极值拟共形映射的特征都有重要的意义.因此本文中我们也将讨论无限小Teichmüller空间中的一些未知的问题.全文共分为五章.第一章,绪论.我们简要的介绍拟共形映射与Teichmüller空间理论的历史背景和研究意义,并阐述本文所研究问题的由来和现状以及主要结果.第二章,二次抛物区域上拟共形映射的极值性.在给定所有边界点对应的前提下,我们已经知道了很多类区域上极值拟共形映射的刻画和性质,但是若降低边界对应要求,同样区域上极值拟共形映射的情况还不清楚.Strebel([94][95])曾经对几种不同的区域研究过这个问题,我们将在二次抛物区域上研究这个问题.第三章,Teichmüller空间与其切空间的一些非等价性.刻画一个极值或者唯一极值拟共形映射的特征一直是拟共形极映射值理论研究的热点.Hamilton([41]),Krushkal([45]),Reich和Strebel([76])共同研究得到了极值拟共形映射的特征刻画.1998年,Bo(?)in,Lakic,Markovi(?)和Mateljevi(?)([9])研究得到了唯一极值拟共形映射的特征刻画.从这些论文中我们发现Teichmüller空间与其切空间具有许多等价的极值性质.在本章中,我们将研究Teichmüller空间与其切空间在Strebel点、极值Teichmüller Beltrami系数的存在性、常数模极值Beltrami系数存在性之间的等价性等问题.第四章,万有Teichmüller空间对数导数模型的几何性质.对数导数和单叶函数的拟共形扩张具有深刻的联系.万有Teichmüller空间对数导数模型也具有许多奇特的几何性质,Zhuravlev([116])证明了万有Teichmüller空间对数导数模型是由无穷多个互不相交的分支组成的.在这一章,我们研究万有Teichmüller空间对数导数模型每个分支内测地线和球的几何性质.第五章,无限小Teichmüller空间中的问题.从本文第三章,我们知道Teichmüller空间与无限小Teichmüller空间具有许多相似和不相似的性质.在这一章,我们主要研究无限小Teichmüller空间中测地圆盘的个数以及无限小极值Beltrami系数的Hamilton序列问题.更多还原

【Abstract】 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.更多还原