The zeta-determinant of the Dirichlet-to-Neumann operator on forms

Abstract

On a compact Riemannian manifold M with boundary Y, we express the log of the zeta-determinant of the Dirichlet-to-Neumann operator acting on q-forms on Y as the difference of the log of the zeta-determinant of the Laplacian on q-forms on M with the absolute boundary condition and that of the Laplacian with the Dirichlet boundary condition with an additional term which is expressed by curvature tensors. When the dimension of M is 2 and 3, we compute these terms explicitly. We also discuss the value of the zeta function at zero associated to the Dirichlet-to-Neumann operator by using a metric rescaling method. As an application, we recover the result of the conformal invariance obtained in Guillarmou and Guillope (Int Math Res Not IMRN 2007(22):rnm099, 2007) when \({\text {dim}}M = 2\).