A polynomial associated to the BFK-gluing formula for zeta-determinants on a compact warped product manifold

초록

The gluing formula for zeta-determinants of Laplacians was proved by Burghelea, Friedlander and Kappeler (BFK) in 1990’s by using the Dirichlet-to-Neumann operator. In the proof of this gluing formula there appears a real polynomial of degree less than half of an underlying manifold, whose coefficients are closely related to the BFK-gluing formula when a cutting hypersurface is a Riemannian product. The constant term of this polynomial is one ingredient of the BFK-gluing formula. This polynomial is completely determined by some data on an arbitrary small collar neighborhood of a cutting hypersurface. In this talk I’m going to explain briefly the BFK-gluing formula of zeta-determinants of Laplacians and the relation between this polynomial and the BFK-gluing formula. Especially, when a collar neighborhood of a cutting hypersurface is isometric to a warped product manifold of the form [a,b] ×_f Y, I’m going to discuss the computation of this polynomial in terms of the warping function f and also discuss the value of the zeta function associated to the Dirichlet-to-Neumann operator at zero.