Geometric Analysis on Moduli Spaces of Riemann Surfaces and Locally Symmetric Spaces

黎曼曲面模空间和局部对称空间的几何分析

• 批准号: 1104696
• 负责人: Lizhen Ji
• 金额: $ 18.55万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104696&HistoricalAwards=false
• 关键词: Geometric; Analysis; Moduli; Spaces; Riemann

The moduli space of compact Riemann surfaces or compact Riemann surfaces with punctures is one of the most important spaces in mathematics and has been extensively studied in algebraic geometry, complex geometry, topology and mathematics physics. A closely related space is the Teichmueller space of Riemann surfaces where the mapping class group acts, and the quotient is equal to the moduli space of Riemann surfaces.The mapping class group is a basic group intensively studied in geometric group theory. The Teichmueller space and hence the moduli space of Riemann surfaces admit several natural Riemannian metrics such as the Weil-Petersson metric, the Ricci metric and Poincare type metrics. The main goal of this proposal is to study the spectral theory of the moduli space with respect to these Riemannian metrics and to understand its relations with the geometry and topology of the moduli space. Pursuing similarities between the moduli space and locally symmetric spaces, and between mapping class groups and arithmetic groups has been fruitful, the second goal of this proposal is to study some related problems for locally symmetric spaces. Specifically, the proposal consists of the following 4 projects:(1) Spectral theory of the incomplete Weil-Petersson metric on the moduli space.(2) Spectral theory for complete Riemannian metrics on the moduli space: geometric scattering theory.(3) Simplicial volume and spines of the moduli space.(4) Equivariant spines of symmetric spaces and L^p-spectral theory of locally symmetric spaces.A drum corresponds to a domain in the plane, and its tones correspond to the eigenvalues of the Laplace operator of the domain with the Dirichlet boundary condition. A perhaps naive question is how these eigenvalues are related to and reflect the shape of the drum, i.e., the geometry of the domain. For example, a very large drum has a low pitch. A famous question raised by Marc Kac in 1966 is "Can one hear the shape of a drum?". This question has been one of the motivating forces for the subject of spectral geometry. Domains are special examples of Riemannian manifolds, and mathematicians have been trying to understand geometry and spectral theory of various Riemannian manifolds, for example, closed surfaces inside the three dimensional Euclidean space. An important class of spaces comes from collections of mathematical objects sharing similar properties, the so-called moduli spaces. The main purpose of this proposal is to understand the geometry and spectral of moduli spaces of Riemann surfaces.

紧Riemann曲面或带穿孔的紧Riemann曲面的模空间是数学中最重要的空间之一，在代数几何、复几何、拓扑学和数学物理中有着广泛的研究。一个密切相关的空间是黎曼曲面的Teichmueller空间，映射类群作用于该空间，其商等于黎曼曲面的模空间，映射类群是几何群论中研究最多的一个基本群。Teichmueller空间和黎曼曲面的模空间允许一些自然的黎曼度量，如Weil-Petersson度量、Ricci度量和Poincare型度量。这个提议的主要目标是研究模空间关于这些黎曼度量的谱理论，并理解它与模空间的几何和拓扑的关系。由于模空间与局部对称空间、映射类群与算术群之间的相似性研究已经取得了一定的成果，本文的第二个目标是研究局部对称空间的相关问题。具体地说，该计划包括以下四个方面：（1）模空间上不完全Weil-Petersson度量的谱理论。(2)模空间上完备黎曼度量的谱理论：几何散射理论。(3)模空间的单形体体积和刺。(4)对称空间的等变刺与局部对称空间的L^p-谱理论，鼓对应于平面上的一个区域，鼓的音调对应于该区域的具有Dirichlet边界条件的拉普拉斯算子的特征值.一个也许天真的问题是这些特征值如何与鼓的形状相关并反映鼓的形状，即，域的几何形状。例如，一个非常大的鼓有一个低的音调。1966年，马克·卡茨（Marc Kac）提出了一个著名的问题：“人们能听到鼓的形状吗？".这个问题一直是动力之一的主题谱几何。Domain是黎曼流形的特殊例子，数学家一直试图理解各种黎曼流形的几何和谱理论，例如三维欧氏空间中的闭曲面。一类重要的空间来自于具有相似性质的数学对象的集合，即所谓的模空间。这个建议的主要目的是了解黎曼曲面的模空间的几何和谱。