Determinants of Laplacians as functions on spaces of metrics
Young-Heon Kim
University of Toronto
November 20, 2006
Abstract: The determinant of
the Laplacian is a global Riemannian invariant which is defined
formally as the product of all the infinitely many nonzero eigenvalues
of the Laplacian of the given Riemannian
metric. This determinant 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. We will also discuss its 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?
This talk has the same theme as the talk I have given in the
Analysis/Applied Math/PDE seminar on Oct. 13., but this time we will
focus more on the ideas of the proofs and the methods.