2008 DG-MP-PDE Seminar - 03

The determinant of the Laplacian is a global Riemannian invariant which is defined formally as the product of the nonzero eigenvalues of the Laplacian of a given Riemannian metric. It gives a continuous function on the space of Riemannian metrics. In this talk we are interested in the case of compact surfaces with boundary and will discuss the properness of the determinant function on the moduli space of hyperbolic surfaces with geodesic boundary and on the moduli space of flat surfaces with boundary of constant geodesic curvature. We will also discuss an application to the following isospectral compactness problem: On a given compact surface with boundary, consider the set of all smooth flat metrics having the same Dirichlet Laplacian spectrum. Is it compact in C^\infty topology?