Mathematical Sciences: Geometric Methods in the Theory of Functions

数学科学：函数论中的几何方法

• 批准号: 8701594
• 负责人: James Jenkins
• 金额: $ 6.33万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701594&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Methods; Theory

In this project work will be done on self-conformal mappings of Riemann surfaces. In recent work the principal investigator has given a representation for arbitrary Riemann surfaces which provided simple proofs of old results together with new applications to self-conformal maps of surfaces which have a fixed point. In cases where the group of maps have several base points good progress has been made in describing the groups in terms of congruent polygons. Related to this work is on-going research measuring the number of regular mappings between a fixed (closed) Riemann surface to surfaces of lower genus. This finite bound depends only on the genus of the original surface. Similar questions regarding maps between multiply connected plane domains will be treated. The extremal metric method and Loewner's variational method of studying univalent functions have never been studied for connections. A careful analysis of this question will be undertaken. Aside from general interest, the outcome of this effort should provide a characterization of Loewner chains associated with quadratic differentials.

在这个项目中，我们将研究黎曼曲面的自共形映射。在最近的工作中，主要研究者给出了任意Riemann曲面的一个表示，它简单地证明了以前的结果，并给出了具有不动点的曲面的自共形映射的新的应用。在地图组有几个基点的情况下，在用全等多边形描述组方面取得了良好的进展。与这项工作相关的是正在进行的测量固定(闭)Riemann曲面到低亏格曲面之间的正则映射的数量的研究。这个有限界仅取决于原始曲面的亏格。关于多连通平面域之间的映射的类似问题将被处理。研究单叶函数的极值度规方法和Loewner变分方法从未被研究过。我们将对这个问题进行仔细的分析。除了普遍的兴趣之外，这一努力的结果应该提供与二次微分相关的Loewner链的特征。