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Determinants of Laplacians and isopolar metrics on surfaces of infinite area. (English) Zbl 1040.58013

Let \(M\) be a hyperbolic surface that is complete, of finite topological type, infinite area, and without cusps. The authors define a function \(D(s)\) being is formally equivalent to \(\det(\Delta+s(s-1))\) is defined by the authors. Previous results enable them to evaluate \(D(s)\) in terms of the Selberg’s zeta function. To avoid certain complications, the authors assume the metric in question is hyperbolic near infinity – i.e. it has constant curvature \(-1\) outside a compact set. The authors then construct a relative determinant based on a conformal metric perturbation and show the relative determinant is a ratio of entire functions of order 2 with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. They establish a Polyakov formula expressing the relative determinant for a conformal perturbation in terms of the conformal parameter. 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.
The paper is organized as follows: 1. Introduction. 2. Definition of the determinant. 3. Properties of the determinant in the hyperbolic case. 4. Relative determinant (conformal deformation theory, zeta regularization of the relative determinant, Polyakov formula). 5. Hadamard factorization of the relative determinant (divisor of the relative determinant, growth estimates on the relative determinant, asymptotics and order). 6. Compactness for isopolar classes. A. (Finiteness and properness for hyperbolic surfaces, resolvent construction and estimates, logarithmic derivative of the relative scattering operator).

MSC:

58J52 Determinants and determinant bundles, analytic torsion
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P25 Scattering theory for PDEs
47A40 Scattering theory of linear operators
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