Spectral Geometry of Infinite Volume Manifolds

摘要DMS - 0204985 PI：Borthwick主要研究者将在三个基本背景下研究光谱几何。 对于光滑的无限体积双曲流形（无尖点），他将研究具有相同共振集的曲面类的有限性问题，改进涉及他和他的合著者已经开发的行列式的技术，并研究作为双曲结构的deformationspace的函数的共振。 对于更一般的渐近双曲流形，主要目标是分析行列式和相对行列式，并利用它们从共振数据中导出几何约束。所开发的技术将被应用于定义行列式，并获得二维和三维外部区域的约束。最终目标是理解更广泛的一类负曲面的基本谱理论。 这里的目标是确定基本光谱和证明一个限制吸收原理，这将表征基本光谱，并导致建立散射理论。几何光谱理论位于微分几何和数学物理领域的接口。 在量子力学或波传播等物理理论中，人们可以自然地区分系统的几何性质（即底层结构）和分析性质（反映系统的物理行为）。分析性质（光谱就是一个很好的例子）通常是系统最容易通过实验确定的方面（例如，恒星发出的光的分量颜色，或雷达扫描中的频率尖峰）。 在许多物理应用中，基本目标是从光谱数据中获得关于几何结构的信息。 PI将在已经拥有良好的知识来源和数学工具的环境中追求这一目标。 了解这些情况下的谱理论将提供新的几何不变量的兴趣在微分几何，而发展的直觉问题的一个更实用的性质。

ABSTRACT DMS - 0204985 PI: BorthwickThe principal investigator will study spectral geometry in three basic contexts. For smooth infinite-volume hyperbolic manifolds (without cusps), he will study the question of finiteness of classes of surfaces with the same resonance set,refine techniques involving determinants that he and his co-authors have already developed, and study resonances as functions of the deformationspace of a hyperbolic structure. For the more general class of asymptoticallyhyperbolic manifolds, the main goal is to analyze the determinants and relativedeterminants and use them to derive geometric constraints from resonance data.The techniques developed will be applied to define determinants and obtainconstraints for exterior domains in two and three dimensions as well.A final goal is to understand the basic spectral theory of a broader class of negatively curved surfaces. Here the objectives are to determine the essential spectrum and prove a limiting absorption principle which will characterize the essential spectrum and lead towards the establishment of a scattering theory.Geometric spectral theory lies at the interface of the fields of differential geometry and mathematical physics. In physical theories such as quantum mechanics or wave propagation, one can draw a natural distinctionbetween geometric properties of a system, meaning the underlying structure,and analytic properties, which reflect how the physics of the system behaves.Analytic properties, of which the spectrum is a prime example,are generally the aspects of a system most readily determinedby experiment (for example, component colors of light emitted by stars, orfrequency spikes in a radar scan). In many physical applications, the basic goal is to derive information about the geometric structure from the spectral data. The PI will pursue this goal in settings for which one already has good sources of conjectures and mathematical tools. Understanding the spectral theory of these caseswill provide new geometric invariants of interest in differential geometry,while developing intuition for problems of a more applied nature.