A Meyer-Vietoris formula for the determinant of the Dirichlet-to-Neumann operator on Riemann surfaces

TL;DR

This paper develops a Meyer-Vietoris type formula for the determinant of the Dirichlet-to-Neumann operator on Riemann surfaces, enabling analysis of its asymptotics and implications for moduli space compactness.

Contribution

It introduces a novel gluing formula for the determinant of the Dirichlet-to-Neumann operator on Riemann surfaces, extending Kirchhoff's matrix tree theorem.

Findings

Bounds the asymptotics of the invariant under degeneration

Shows properness of the height function only in genus zero

Provides asymptotics for the Laplacian determinant with boundary conditions

Abstract

This paper presents a Meyer-Vietoris type gluing formula for a conformal invariant of a Riemannian surface with boundary that is defined by the determinant of the Dirichlet-to-Neumann operator. The formula is used to bound the asymptotics of the invariant under degeneration. It is shown that the associated height function on the moduli space of hyperbolic surfaces with geodesic boundary is proper only in genus zero. Properness implies a compactness theorem for Steklov isospectral metrics in the case of genus zero. The formula also provides asymptotics for the determinant of the Laplacian with Dirichlet or Neumann boundary conditions. For the proof, we derive an extension of Kirchhoff's weighted matrix tree theorem for graph Laplacians with an external potential.