Differential Equations Seminar

Let M1 and M2 be two Riemannian manifolds each of which have the boundary N. Consider the Laplacian on M1 and M2 augmented with Dirichlet boundary conditions on N. A natural question to ask is if there is any relation between spectral properties of the Laplacian on M1, M2, and the Laplacian on the manifold M (without boundary) obtained gluing together M1 and M2, namely M = M1 ∪N M2. Using spectral zeta functions, a partial answer is given by the Burghelea-Friedlander- Kappeler-gluing formula for zeta-determinants. This formula contains an (in general) unknown polynomial which is completely determined by some data on a collar neighborhood of the hypersurface N. I will use conformal transformations to understand the geometric content of this polynomial. The understanding obtained will pave the way for a fairly straightforward computation of the polynomial (at least for low dimensions of M). Furthermore it leads to a partial understanding of the heat invariant for the Dirichlet-to-Neumann map, that is for the Steklov problem. Speaker(s): Klaus Kristen (Mathematical Reviews AMS and Baylor University)