We construct a determinant of the Laplacian for infinite-area surfaces that are hyperbolic near $\infty$ 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 $\infty$ 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.
"Determinants of Laplacians and isopolar metrics on surfaces of infinite area." Duke Math. J. 118 (1) 61 - 102, 15 May 2003. https://doi.org/10.1215/S0012-7094-03-11814-1