BFK-gluing formula for zeta-determinants of Laplacians and a warped product metric

초록

The gluing formula for zeta-determinants of Laplacians on a compact Riemannian manifold was given by Burghelea, Friedlander and Kappeler (BFK-gluing formula) in 1992 by using the Dirichlet boundary condition and the Dirichlet-to-Neumann operator. The BFK-gluing formula contains a constant which is determined by a data on a collar neighborhood of a cutting hypersurface. This constant is known explicitly when the dimension of a manifold is even or the product metric is given near the cutting hypersurface. In this talk we discuss this constant when a warped product metric is given on a collar neighborhood of the cutting hypersurface. We apply this result to express the relative zeta-determinant of a Laplacian on a manifold with cusps as the product of the zeta-determinant of a Laplacian on a compact manifold with boundary with the Dirichlet condition and the zeta-determinant of a Dirichlet-to-Neumann operator.