Journal of the
Korean Mathematical Society JKMS

The purpose of this paper is to discuss the constant term appearing in the BFK-gluing formula for the zeta-determinants of Laplacians on a complete Riemannian manifold when the warped product metric is given on a collar neighborhood of a cutting compact hypersurface. If the dimension of a hypersurface is odd, generally this constant is known to be zero. In this paper we describe this constant by using the heat kernel asymptotics and compute it explicitly when the dimension of a hypersurface is 2 and 4. As a byproduct we obtain some results for the value of relative zeta functions at $s=0$.

Keywords: (relative) zeta-determinant, BFK-gluing formula, Dirichlet-to-Neumann operator, Dirichlet boundary condition, warped product metric

MSC numbers: 58J52, 58J50