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佩里这个项目处理了光谱几何学上的几个问题：紧凑型表面的逆光谱问题；拉普拉斯的痕量公式和动力学Zeta功能在双曲线歧管和相关矢量束上；散射算子及其决定因素作为克莱琳组变形空间的函数。要采用的技术包括扰动理论，全球分析和Teichmuller理论。该项目的目的之一是表明紧凑型表面的等法集对接近恒定曲率的指标有限。 光谱几何形状与riemannian歧管（带有度量的弯曲空间）与与之相关的自然差分运算符光谱的差分几何特性的相互作用有关。操作员的光谱通常以离散且可计算的方式捕获原本棘手的差分运算符的各种分析性能。阐明各种光谱数据的几何含量是光谱几何形状的主要关注点。