Mathematical Sciences: Geometric Theory of Functions

数学科学：函数的几何理论

• 批准号: 8903031
• 负责人: Robert Glassey
• 金额: $ 5.68万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8903031&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Theory; Functions

Two related lines of investigation will be carried out in this project of mathematical research. The work is basically geometric in spirit, focusing (i) on a study of general extremal problems for normalized univalent holomorphic functions in the exterior of a circle and (ii) on the analysis of complex-valued orientation preserving harmonic planar mappings. In the first instance, work will continue on efforts to describe the omitted set in the range of a solution of a linear extremal problem for the functions defined and univalent in the exterior of a disc. The omitted set is known to consist of finitely many analytic arcs lying on the trajectories of aquadratic differential. The structure of these sets remains elusive except in cases where the extremals were known beforehand. Some newly developed second variational techniques will be applied to specific extremal questions. The study of harmonic maps derives from fundamental questions about nonparametric minimal surfaces in three-dimensional space. The coordinate functions in their representation are harmonic and the first two such functions give rise to univalent maps of the domain onto itself. The extremal problems arising in this context have been much harder to work with because of the lack of any variational principles within the admissible class of mappings. One approach to this question is to write the mappings as solutions of complex differential equations involving a functional parameter. It is expected that by varying this function, one should be able to recognize how extremal maps develop. Emphasis will be placed on mappings from a disc to itself.

在这个数学研究项目中将进行两个相关的研究方向。 这项工作本质上是几何的，重点是（i）圆外部归一化单价全纯函数的一般极值问题的研究，以及（ii）复值方向保持调和平面映射的分析。 在第一种情况下，将继续努力描述在盘外部定义且单价的函数的线性极值问题的解范围内的省略集。 已知省略的集合由位于水微分轨迹上的有限多个解析弧组成。 这些集合的结构仍然难以捉摸，除非事先已知极值的情况。 一些新开发的二阶变分技术将应用于特定的极值问题。 调和映射的研究源于关于三维空间中非参数最小曲面的基本问题。 它们的表示中的坐标函数是调和的，前两个这样的函数产生域到其自身的单价映射。 由于在可接受的映射类别中缺乏任何变分原理，因此在这种情况下出现的极端问题变得更加难以处理。 解决这个问题的一种方法是将映射写成涉及函数参数的复杂微分方程的解。 预计通过改变该函数，人们应该能够识别极值图如何发展。 重点将放在从光盘到其自身的映射上。