Aldana Domínguez, Clara Lucía: Inverse spectral theory and relative determinants of elliptic operators on surfaces with cusps. - Bonn, 2009. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
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author = {{Clara Lucía Aldana Domínguez}},
title = {Inverse spectral theory and relative determinants of elliptic operators on surfaces with cusps},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2009,
month = jan,

note = {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 $h=e^{2\varphi}g$.
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.
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.
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 $M$.},

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