Mathematical Sciences: Geometric Theory of Functions

点击翻译按钮获取中文摘要

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.