The BFK type gluing formula of zeta-determinants for the Robin Boundary condition

TL;DR

This paper develops a gluing formula for zeta-determinants of Laplacians with Robin boundary conditions on compact manifolds, extending previous results and providing explicit computations for cylinders.

Contribution

It introduces a BFK type gluing formula for Robin boundary conditions and computes the zeta-determinant differences between Robin, Dirichlet, and Neumann conditions.

Findings

Derived a gluing formula for Robin boundary conditions.

Computed the zeta-determinant on a cylinder with Robin boundary conditions.

Extended previous results to more general boundary conditions.

Abstract

In this paper we discuss the BFK type gluing formula for zeta-determinants of Laplacians with respect to the Robin boundary condition on a compact Riemannian manifold. As a special case, we discuss the gluing formula with respect to the Neumann boundary condition. We also compute the difference of two zeta-determinants with respect to the Robin and Dirichlet boundary conditions. We use this result to compute the zeta-determinant of a Laplacian on a cylinder when the Robin boundary condition is imposed, which extends a result in [25]. We also discuss the gluing formula more precisely when the product structure is given near a cutting hypersurface.