Dynamics of modular groups on infinite dimensional Teichmuller spaces

无限维 Teichmuller 空间上的模群动力学

基本信息

批准号：
14540156
负责人：
MATSUZAKI Katsuhiko
金额：
$ 2.18万
依托单位：
Ochanomizu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540156/
关键词：
Teichmuller space modular group Riemann surface quasiconformal map hyperbolic geometry Schwarzian derivative Bers embedding univalent function モジュライ空間双曲幾何

项目摘要

Teichmueller spaces are not homogeneous spaces and their mudular groups do not act transitively. For compact Riemann surfaces, modular groups act discontinuously, but this is not the case for infinite dimensional Teichmueller spaces. We study the moduli spaces of Riemann surafces of infinite type by considering the chaotic behavior of the action of modular groups. For a viewpoint of general topology, the moduli space is either metrizable or not of the first separation axiom. However, except for a singular part, it can possess a certain geometric structure. In this research, we characterize this stable region by hyperbolic geometric structure of a Riemann surface and construct a contracted moduli space by the completion of the stable region. Consequently, we can describe the closure of a point set in terms of the geomery of Riemann surfaces, which is a point of teh contracted module space.We considered the space of pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian model of the universal Teichmueller space and classical families of univalent functions. We also investigated geometric properties of univalent functions with a prescribed growth of the Schwarzian derivative and found that they are starlike or convex according to the distance to the origin in the Bers embedding of the universal Teichmueller space.

Teichmueller空间不是均匀的空间，它们的泥泞群体不会进行传输。对于紧凑的Riemann表面，模块化组不连续起作用，但对于无限的尺寸teichmueller空间并非如此。我们通过考虑模块化基团的作用的混乱行为来研究无限类型的Riemann Surafces的模量空间。对于一般拓扑的观点，模量空间是可以超级分离公理的，或者是不可分割的。但是，除了一个奇异部分，它还可以具有一定的几何结构。在这项研究中，我们通过riemann表面的双曲几何结构来表征这个稳定区域，并通过稳定区域的完成来构建收缩的模量空间。因此，我们可以描述riemann表面的地貌的闭合，这是TEH合同模块空间的点。我们考虑了单位磁盘上无效函数的Schwarzian前衍生物的空间，该函数在单位磁盘上延伸到延伸到列出的相关词的相关效果，以调查相关的相关效果，以调查相关的相关性，并在相关的范围内调查了相关的范围，从而在相关的范围内进行了相关的效果。为了研究通用Teichmueller空间的Schwarzian模型的连接组件与单价函数的经典家族的相关组件之间的关系，该单位磁盘扩展到扩展平面的准表面映射。我们还研究了单价函数的几何特性，其规定的Schwarzian衍生物的规定生长，发现它们是根据嵌入通用Teichmueller空间的BERS的距离的距离或凸的。