Inverse spectral theory and relative determinants of elliptic operators on surfaces with cusps

Abstract

This thesis concerns relative determinants for Laplacians on surfaces with asymptotically cusps ends and the inverse spectral problem on surfaces with cusps. We consider $(M,g)$, a surface with cusps, and a metric on the surface that is a conformal transformation of the initial metric $h=e^{2\varphi}g$.<br /> In the first part we find conditions $\varphi$ that make it possible to define the relative determinant of the pair $(\Delta_{h},\Delta_{g})$. We prove Polyakov's formula for the relative determinant and study the extremal values of this determinant as a function of unit area metrics inside a conformal class. We prove that if the maximum exists it has to be attained at the metric of constant curvature. We discuss necessary conditions for the existence of a maximizer.<br /> In the second part we restrict our attention to hyperbolic surfaces of fixed genus and a fixed number of cusps. We study the relative determinant as a function on the moduli space for this kind of surfaces and use the results of J. Jorgenson and R Lundelius in [19] to prove that it tends to zero at the boundary of the moduli space. <br /> In the third part we return to general surfaces with cusps. We prove a splitting formula for the relative determinant and use it to prove compactness in the $C^{\infty}$-topology of sets of isospectral metrics in a given conformal class. We assume that the conformal factors $\varphi$ have support in a fixed compact set of $M$.