11/17/2006

Young-Heon Kim
University of Toronto

Determinants of Laplacians as functions on spaces of metrics

The determinant of the Laplacian is a global Riemannian invariant which is defined formally as the product of all the countably many nonzero eigenvalues of the Laplacian of the given Riemannian  metric, and it gives us 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?