Mathematical Sciences: Spectral Geometry of Compact Riemannian Manifolds and Kleinian Groups

数学科学：紧致黎曼流形和克莱因群的谱几何

• 批准号: 9707051
• 负责人: Peter Perry
• 金额: $ 9万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-15 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9707051&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Spectral; Geometry; Compact

9707051 Perry This project deals with several problems in spectral geometry: the inverse spectral problem for compact surfaces; trace formulas and dynamical zeta functions for the Laplacian on hyperbolic manifolds and associated vector bundles; the scattering operator and its determinant as a function on the deformation space of a Kleinian group. Techniques to be employed include perturbation theory, global analysis, and Teichmuller theory. One of the aims of this project is to show that the isopectral set of a compact surface is finite for metrics close to constant curvature. Spectral geometry is concerned with the interaction of differential-geometric properties of a Riemannian manifold (a curved space with a metric) with the spectra of natural differential operators associated with it. The spectrum of an operator often captures various analytic properties of an otherwise intractable differential operator in a discrete and computable manner. Elucidating the geometric content of various spectral data then is the main concern of spectral geometry.

9707051 Perry这个项目涉及光谱几何中的几个问题：紧曲面的逆光谱问题；双曲流形及其相关向量束上拉普拉斯算子的迹公式和动态zeta函数散射算子及其行列式作为Kleinian群变形空间的函数。所采用的技术包括摄动理论、全局分析和泰希穆勒理论。这个项目的目的之一是证明紧曲面的等谱集对于接近常数曲率的度量是有限的。谱几何是研究黎曼流形（带有度规的弯曲空间）的微分几何性质与与之相关的自然微分算子的谱之间的相互作用。一个算符的谱通常以离散和可计算的方式捕捉到另一个难以处理的微分算符的各种解析性质。阐明各种光谱数据的几何内容是光谱几何研究的主要内容。