Mathematical Sciences: Geometric Methods in the Theory of Functions

在这个项目中，将对自共形映射进行研究 黎曼曲面 在最近的研究中， 给出了任意黎曼曲面的表示， 提供了简单的证明旧的结果与新的 应用于曲面的自共形映射， 不动点 如果该组地图具有多个基础， 在描述这些群体方面取得了良好的进展， 全等多边形的术语。 与这项工作有关的是正在进行的研究， 固定（闭）Riemann空间之间的正则映射数 表面到表面的低属。 这个有限的界限取决于 只在原始曲面的亏格上。 类似的问题 关于多连通平面域之间的映射， 治疗。 极值度量法和Loewner变分法 研究单叶函数的方法， 连接. 仔细分析这个问题， 进行。 除了一般的兴趣，这一结果 努力提供Loewner链的特征 与二次微分有关。

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.