万有Teichmüller空间与拟共形扩张及区域的单叶性内径

【作者】 康悦明；

【导师】 陈纪修；

【作者基本信息】 复旦大学 ， 基础数学， 2008， 博士

【摘要】 本文围绕万有Teichmuller空间的几何性质展开,将万有Teichmuller空间与单叶函数,拟共形映射,Loewner链理论结合起来,研究了万有Teichmuller空间的不同模型下的测地线唯一性性质,解释了Pre-Schwarz导数模型下单叶性内径的几何意义,给出了在一个区域内局部单叶的全纯(亚纯)函数成为整体单叶函数的充分条件,并得到了区域的Pre-Schwarz导数单叶性内径的下界估计公式。论文共分五章,第一章是引言,我们将简要的介绍拟共形映射的发展以及应用背景,Teichmuller空间理论,Loewner理论,并叙述本文研究的主要问题及所获得的结果。在第二章中,我们将讨论万有Teichmuller空间的测地线唯一性性质,研究万有Teichmuller空间不同模型下测地线的唯一性问题是否等价。通过构造具体的反例,我们给予这个问题一个否定的回答,即万有Teichmuller空间的不同模型下测地线的唯一性不具有等价性。在第三章中,我们将研究以无穷远点为内点的区域的Pre-Schwarz导数单叶性内径,结合万有Teichmuller空间的定义与性质,我们指出以∞为内点的拟圆区域D的Pre—Schwarz导数单叶性内径σ_I~*(D),即为该模型中一点到边界的最小距离。同时,我们给出了与Ahlfors—Lehto公式相对应的关于Pre—Schwarz导数的单叶性内径的公式,并应用它得到了椭圆外区域的Pre—Schwarz导数单叶性内径的下界估计值。在第四章中,我们利用了Loewner理论,给出了对于单位圆内局部单叶的全纯(亚纯)函数成为整体单叶函数的充分条件。通过构造函数的拟共形扩张表达式,并结合万有Teichmuller空间的性质与Pre-Schwarz导数单叶性内径的几何意义,得到了拟圆区域Pre—Schwarz导数单叶性内径的下界估计公式。在第五章中,我们将研究Pre-Schwarz导数与拟共形扩张的问题.这个问题同样和万有Teichmuller空间Pre-Schwarz导数嵌入模型密切相关。如何用Pre-Schwarz导数来判定一个函数能否拟共形扩张,且其扩张后的函数的复特征与Pre-Schwarz导数有何联系,这是一个十分有趣的问题,如何写出它的具体的扩张表达式则是一个十分困难却有意义的工作。在本章中,我们将就上述两方面进行研究。更多还原

【Abstract】 This paper is concerned with the geometric property of the Universal Teichmüller space.Using the theory of Universal Teichmüller space,univalent functions,quasiconformal extension and the Loewner chain,we study the properties of different models of Universal Teichmüller space,explain the geometric meaning of the inner radius of univalency in the Pre-Schwarzian derivative model,and get some sufficient conditions for locally univalent and holomorphie(or meromorphic)functions to be wholly univalent in a given domain.There are five chapters in this thesis.The first chapter is the preface of it.We introduce the theory of quasiconformal mappings,the development of quasiconformal mapping and its application,the Universal Teichmüller space.Furthermore,the problems discussed in this thesis and our main results are introduced.In chapter two,we mainly study the geodesic property of the Universal Teichmüller Space and discuss whether the uniqueness of geodesics in different models of the Universal Teichmüller Space is equivalent.By construct a counterexample, we give a negative answer to this question,that is when there is a unique geodesic segment joining two points in one model of the Universal Teichmüller Space,while in other models,there may exist more than one geodesic segment joining the corresponding points.In chapter three,we discuss the inner radius of univalency by Pre-Schwarzian derivative of domains where the infinity is an inner point.By the definition and property of Universal Teichmüller Space,we get that the inner radius of univalency by Pre-Schwarzian derivative is the distance from a point in the Pre-Schwarzian derivative model of the Universal Teichmüller Space to its boundary.As an application, an estimation of the lower bound of inner radius of univalency by Pre-Schwarzian derivative of the infinitive domain bounded by an elliptic is obtained.In chapter four,by the theory of Loewner chain,we get some sufficient conditions for locally univalent and holomorphic(meromorphic) functions to be wholly univalent. We construct the quasiconformal extension with the Loewner chain and get the lower bound formula of the inner radius of univalency by Pre-Schwarzian derivative of the quasidisk domain. In chapter five,we discuss the problem of Pre-Schwarzian derivative and quasiconformal extension.This problem also relates to the Universal Teichmüller Space embedded by Pre-Schwarzian derivative.We find some connections between the complex dilatations of the quasiconformal extensions and the norms of the Pre-Schwarzian derivatives.Furthermore,we find another proof for the lower bound of the inner radius of univalency for angular domains by constructing an explicit quasiconformal extension of a class of holomorphic functions.更多还原