Mathematical Sciences: Geometric Properties of Domains, Extremal Quasiconformal Mappings and Integrability of Conformal Mappings

数学科学：域的几何性质、极值拟共形映射和共形映射的可积性

• 批准号: 9203407
• 负责人: John Garnett
• 金额: $ 1.8万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-15 至 1994-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203407&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Properties; Domains

This award provides support for postdoctoral research on problems arising in the general area of geometric function theory. The work focuses on the geometric properties of quasiextremal distance domains. These are domains characterized by the property that points and closed sets can be joined by curves within the domain whose lengths do not be arbitrarily long. They are singled out because they have the characteristics believed to be fundamental for the study of quasiconformal mappings. Work on this project will concentrate on showing how the modulus of quasiextremal distance domains carries over to the dilatation constant for quasiconformal mapping defined on them. A second line of research will consider the length of level sets of univalent mappings and the question of the largest power to which such a mapping's derivative has a finite integral. Complex function theory encompasses the study of differentiable functions of a complex variable and related classes of functions such as harmonic functions and quasiconformal mappings. The subject is highly geometric; many of the problems concern the properties of various sets under transform by functions from one of the above classes. Applications of the theory to potential theory and fluid dynamics is now standard in engineering circles

该奖项为几何函数理论一般领域中出现的问题的博士后研究提供支持。本文主要研究拟极值距离域的几何性质。这些域的特征在于，域中的点和闭集可以由长度不是任意长的曲线连接起来。它们之所以被挑出来，是因为它们具有被认为是研究拟共形映射的基础的特征。这个项目的工作将集中在展示拟极值距离域的模如何传递到定义在其上的拟共形映射的伸缩常数。第二行的研究将考虑单叶映射的水平集的长度，以及这种映射的导数对其具有有限积分的最大幂的问题。复变函数论研究复变数的可微函数和相关的函数类，如调和函数和拟共形映射。这个主题是高度几何的；许多问题涉及上述类中的一个函数变换下的各种集合的性质。该理论在位势理论和流体力学中的应用现已成为工程界的标准。