Mathematical Sciences: Some Spectral and Isospectral Problems in Riemannian Geometry

数学科学：黎曼几何中的一些谱和等谱问题

• 批准号: 8802668
• 负责人: Peter Perry
• 金额: $ 3.57万
• 依托单位: University of Kentucky
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8802668&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Spectral; Isospectral

Peter Perry will work on two areas of spectral theory on Riemannian manifolds. These involve the scattering theory of manifolds of constant negative curvature and inverse spectral problems on compact Riemannian manifolds. These are geometric problems but the techniques to be employed will involve methods of mathematical physics. The first project will make use of the stationary scattering theory of elliptic operators invented for for the Schrodinger equation. The inverse spectral theory will employ techniques of partial differential equations. The first set of problems to be addressed concerns the spectral theory of the Laplace operator on a real hyperbolic manifold and its connections with the theory of discrete groups of hyperbolic isometries. Perry will concentrate on those manifolds which are quotients of hyperbolic space by a group of hyperbolic isometries. The work will involve investigations of the resolvent, eigenfunctions and scattering matrix of the Laplace operator on the hyperbolic space. It is hoped that this will be helpful in studying the Selberg trace formula and the Selberg zeta function of the isometry group. He will also continue his study of isospectral metrics on a Riemannian manifold. By restricting attention to metrics in a conformal class, Perry, in joint work with Brooks and Yang has, extended some two-dimensional results to the three- and four- dimensional case. Attempts will be made to remove the restriction to conformal classes and to find higher dimensional results.

彼得·佩里（Peter Perry）将在谱系理论的两个领域开展关于里曼尼（Riemannian）流形的领域。这些涉及恒定负曲率和逆光谱问题的散射理论在紧凑的riemannian歧管上。这些是几何问题，但是要采用的技术将涉及数学物理学方法。第一个项目将利用用于Schrodinger方程的椭圆算子的固定散射理论。反光谱理论将采用偏微分方程的技术。 要解决的第一组问题涉及Laplace操作员的光谱理论在实际双曲线歧管上及其与离散的双曲异构体组的联系。佩里将专注于一组双曲异构体的歧管，这些歧管是双曲线空间的商。这项工作将涉及对双曲线空间上拉普拉斯操作员的分解，特征函数和散射矩阵的研究。希望这将有助于研究Selberg痕量公式和等轴测组的Selberg Zeta功能。 他还将继续研究关于Riemannian歧管的同一指标。通过限制了保形类别中对指标的关注，佩里在与布鲁克斯和杨的联合工作中，将一些二维结果扩展到了三维和四维的情况。将尝试消除对形式类别的限制并找到更高维度的结果。