Mathematical Sciences: Geometric Theory of Functions

在这个数学研究项目中，将进行两个相关的调查。该工作基本上是几何性质的，重点是(i)研究了圆外归一化一元全纯函数的一般极值问题，（ii）分析了复值取向保持调和平面映射。在第一种情况下，工作将继续努力描述一个线性极值问题的解范围内被省略的集合，该问题是在圆盘的外部定义的和一元的函数。已知省略的集合由有限多个位于二次微分轨迹上的解析弧组成。这些集合的结构仍然是难以捉摸的，除非在事先知道极值的情况下。一些新发展的二次变分技术将应用于具体的极值问题。调和映射的研究源于三维空间中非参数极小曲面的基本问题。它们表示的坐标函数是调和的，前两个这样的函数产生了域到自身的一元映射。在这种情况下出现的极端问题很难处理，因为在可接受的映射类中缺乏任何变分原则。解决这个问题的一种方法是将映射写成包含一个泛函参数的复杂微分方程的解。期望通过改变这个函数，人们能够认识到极值图是如何发展的。重点将放在从光盘到自身的映射上。

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.