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.

这项数学研究项目将进行两种相关的调查线。 这项工作基本上是精神上的几何形状，重点（i）研究了一个圆圈外部的一般极端问题的一般极端问题，以及（ii）（ii）分析得出复杂评估方向，以保留谐波平面映射。 首先，将继续努力描述在圆盘外部定义和无效的函数的线性极端问题解决方案中省略的集合。 已知省略的集合由有限的许多分析弧组成，位于水上差异的轨迹上。 除了事先已知极端的情况外，这些集合的结构仍然难以捉摸。 一些新开发的第二变化技术将应用于特定的极端问题。 对谐波图的研究来自有关三维空间中非参数最小表面的基本问题。 其表示中的坐标函数是谐波的，前两个这样的功能产生了域的单价图。 由于缺乏可接受的映射类别中的任何变化原则，因此在这种情况下引起的极端问题更加困难。 解决这个问题的一种方法是将映射写为涉及功能参数的复杂微分方程的解决方案。 可以预期，通过改变此功能，应该能够识别出极端地图的发展方式。 重点将放在从光盘到本身的映射上。