Isoresonant conformal surfaces with cusps and boundedness of the relative determinant

We study the isoresonance problem on non-compact surfaces of finite area that are hyperbolic outside a compact set. Inverse resonance problems correspond to inverse spectral problems in the non-compact setting. We consider a conformal class of surfaces with hyperbolic cusps where the deformation takes place inside a fixed compact set. Inside this compactly supported conformal class we consider isoresonant metrics, i.e. metrics for which the set of resonances is the same, including multiplicities. We prove that sets of isoresonant metrics inside the conformal class are sequentially compact. We use relative determinants, splitting formulae for determinants and the result of B. Osgood, R. Phillips and P. Sarnak about compactness of sets of isospectral metrics on closed surfaces. In the second part, we study the relative determinant of the Laplace operator on a hyperbolic surface as function on the moduli space. We consider the moduli space of hyperbolic surfaces of fixed genus and fixed number of cusps. We consider the relative determinant of the Laplace operator and a model operator defined on the cusps. We prove that the relative determinant tends to zero as one approaches the boundary of the moduli space.