Mathematical Sciences: Geometric Methods in the Theory of Functions

在这个项目中，我们将研究黎曼曲面的自保角映射。在最近的工作中，首席研究员给出了任意黎曼曲面的一个表示，它提供了旧结果的简单证明，并在具有不动点的曲面的自共形映射上提供了新的应用。在一组地图有几个基点的情况下，在用全等多边形描述组方面取得了很好的进展。与这项工作相关的是正在进行的研究，测量固定（封闭）黎曼曲面到下属曲面之间的正则映射的数量。这个有限界只依赖于原始曲面的格。关于多连通平面域之间的映射的类似问题将被处理。研究一元函数的极值度量方法和Loewner变分方法从未被研究过。将对这个问题进行仔细的分析。除了一般的兴趣之外，这项工作的结果应该提供与二次微分相关的Loewner链的表征。

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.