Spectral Geometry of Infinite Volume Manifolds

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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.