Determinants of Laplacians and isopolar metrics on surfaces of infinite area

• Type of Material: Article
• Authors: David Borthwick, Emory University; Chris Judge; Peter A. Perry
• Language: English
• Date: 2003
• Publisher: Duke University Press
• Publication Version: Preprint (Prior to Peer Review)
• Copyright Statement: © Duke University Press
• Final Published Version (URL): http://dx.doi.org/10.1215/S0012-7094-03-11814-1
• Title of Journal or Parent Work: Duke Mathematical Journal
• ISSN: 0012-7094
• Volume: 118
• Issue: 1
• Start Page: 61
• End Page: 102
• Research Categories: Mathematics

Abstract

We construct a determinant of the Laplacian for infinite-area surfaces that are hyperbolic near ∞ and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order 2 with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near ∞ case, the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order 2 with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.