A Meyer–Vietoris Formula for the Determinant of the Dirichlet-to-Neumann Operator on Riemann Surfaces

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