Mathematical Sciences: Differential Geometry and Function Theory

数学科学：微分几何和函数论

• 批准号: 8801439
• 负责人: C. David Minda
• 金额: --
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8801439&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Differential; Geometry; Function

This project continues mathematical studies on the role of hyperbolic geometry in the theyro of complex function theory. The goals of the work are to relate hyperbolic geometry to Euclidean quantities such as Euclidean estimates of the hyperbolic metric and descriptions of Euclidean properties of hyperbolic geodesics. Results of this type have application to questions on distortion of univalent functions and estimates for Bloch and Landau constants. Geometric notions of symmetry and reflection play a central role in this research. Typical of the type of problems to be undertaken is establishing the existence and exact values of natural constants associated with classes of functions. The Bloch constant, for instance, measures the largest schlicht disk in the Riemann image surface of a function, whereas the Marden constant measures the size of the largest disk in the domain where a function must be univalent. These fundamental values have been obtained for some classes but not for some of the more important univalent functions. Work also continues in the study of hyperbolic geometry of regions where results point to appropriate generalizations to multiply connected regions. Two main goals are pursued here, the first being one of finding lower bounds for the density of the hyperbolic metric in terms of the distance to the boundary. The second goal is to obtain estimates for the Euclidean curvature and center of curvature for a hyperbolic geodesic. Much of this work will concentrate initially on special domains such as those which are spherically k-convex or hyperbolically k-convex before moving on to multiply connected ones.

这个项目继续数学研究的作用， 复变函数论中的双曲几何。 这项工作的目标是将双曲几何与 欧几里德量，例如 双曲度量及其欧氏性质描述 双曲测地线 这种类型的结果适用于 单叶函数的畸变问题和估计 布洛赫和朗道常数。 对称的几何概念， 反思在这项研究中起着核心作用。 要解决的典型问题类型是 确定自然常数的存在和精确值 与函数类相关。 Bloch常数， 例如，测量黎曼图像中最大的施利希特盘 表面的功能，而马尔登常数测量 函数必须所在的域中最大磁盘的大小 单价体 这些基本值已经获得了一些 类，但不是一些更重要的单价 功能协调发展的 工作还继续在双曲几何的研究， 结果表明， 多个连通区域。 在这里追求两个主要目标， 第一个是找到密度的下限， 用到边界的距离表示的双曲度量。 的 第二个目标是获得欧氏曲率的估计 和双曲测地线的曲率中心。 这在很大程度 工作最初将集中在特殊领域， 是球面k凸或双曲k凸的 然后是多重连接。