The BFK-gluing formula for zeta-determinants and curvature tensors on a 2-dimensional hypersurface

초록

The zeta-determinant is a global spectral invariant and its gluing formula was proved by Burghelea, Friedlander and Kappeler (BFK) in 1990’s by using one parameter family of the Dirichlet-to-Neumann operators. The asymptotic expansion of log of zeta-determinant of this family of operators plays an important role in proving the BFK-gluing formula. In this talk we are going to discuss how the coefficients of this asymptotic expansion are expressed by the scalar curvatures and principal curvatures of the cutting hypersurface in the underlying manifold. Using the similar method, we are also going to discuss how the coefficients of the heat trace asymptotic expansion of the Dirichlet-to-Neumann operator are expressed by the scalar and principal curvatures of the cutting hypersurface. Our method works for any dimension. But for computational reason, we restrict our discussion to the case that the cutting hypersurface is a 2-dimensional compact Riemannian manifold.