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

TL;DR

This paper derives a formula for the zeta-determinant of the Dirichlet-to-Neumann operator on forms on a manifold with boundary, relating it to Laplacian determinants and curvature, with explicit computations in low dimensions.

Contribution

It provides a new expression for the zeta-determinant of the Dirichlet-to-Neumann operator on forms, including explicit formulas in 2D and 3D, and discusses conformal invariance properties.

Findings

Explicit formulas for zeta-determinant in 2D and 3D

Relation between Dirichlet-to-Neumann and Laplacian determinants

Conformal invariance of the zeta function at zero

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 absolute boundary conditions and that of the Laplacian with Dirichlet boundary conditions with some additional terms which are expressed by curvature tensors. When the dimension of $M$ is $2$ or $3$, 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 conformal rescaling method. As an application, we recover the result of the conformal invariance obtained in C. Guillarmou and L. Guillop\'e, The determinant of the Dirichlet-to-Neumann map for surfaces with boundary, Int. Math. Res. Not. IMRN 2007, no. 22, Art. ID rnm099, when the dimension of $M$ is $2$.