The zeta-determinants of Dirichlet-to-Neumann operators and curvature tensors

초록

On a compact Riemannian manifold with or without boundary, a Laplacian, which is imposed with an elliptic boundary condition if the boundary is non-empty, has a small time heat trace asymptotic expansion, each of whose coefficients has some geometric informations, for example the volume, scalar curvature and principal curvatures, etc. Along the same line, on a compact Riemannian manifold without boundary, a Dirichlet-to-Neumann operator defined on the boundary has a small time heat trace asymptotic expansion, whose coefficients also contain some geometric informations. The Dirichlet-to-Neumann operator appears in the Steklov eigenvalue problem and the BFK-gluing formula for the zeta-determinants of Laplacians. In this talk, I’m going to review some basic facts about the heat trace asymptotic expansions of Laplacians and Dirichlet-to-Neumann operators. And then, I’m going to discuss the zeta-determinants of a one parameter family of Dirichlet-to-Neumann operators and its asymptotic expansion as the parameter goes to infinity. Finally, I’m going to discuss the relation between the coefficients of this asymptotic expansion and curvature tensors including the scalar curvature and principal curvatures, etc.