## 2008 DG-MP-PDE Seminar - 03

- Date: 01/17/2008

University of British Columbia

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 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?

3:30pm, WMAX 110