Regularized determinants of Laplace-type operators, analytic surgery, and relative determinants

Abstract

Let $M$ be a compact Riemannian manifold in which $Y$ is an embedded hypersurface separating $M$ into two parts. Assume that the metric is a product on a tubular neighborhood $N$ of $Y$. Let $\Delta$ be a Laplace-type operator on $M$ adapted to the product structure on $N$. Under certain additional assumptions on $\Delta$, we establish an asymptotic expansion for the logarithm of the regularized determinant $\det \Delta$ of $\Delta$ if the tubular neighborhood $N$ is stretched to a cylinder of infinite length. We use the asymptotic expansions to derive adiabatic splitting formulas for regularized determinants