This paper gives a new, more elementary formula that links a spectral invariant of a surface with boundary to classical complex‑analytic data on a related closed surface. The spectral invariant is the zeta‑regularized determinant of the Dirichlet‑to‑Neumann map (the operator that sends a function on the boundary to the normal derivative of its harmonic extension). The authors show that this determinant can be written in terms of the periods of holomorphic differentials on the “double” of the surface — the closed surface obtained by gluing a mirror copy along the boundary.

More concretely, let M be a smooth orientable two‑dimensional surface of genus g with one connected boundary curve Γ of length |Γ|, and let Λ be its Dirichlet‑to‑Neumann map. It is known that detζ(Λ)/|Γ| (the zeta‑regularized determinant with the zero mode removed, divided by the boundary length) is a conformal invariant. Previous work showed this invariant equals 1 when g = 0, and for g > 0 it was related to dynamical zeta functions (Ruelle or Selberg zeta functions) of hyperbolic surfaces in the same conformal class. The new result replaces those zeta functions by the product of certain numbers µk that come from periods of holomorphic differentials on the doubled surface 2M.

The heart of the paper is the identity detζ(Λ) = detζ(∂γ) · DET(H). Here ∂γ is the tangential derivative along the boundary (whose determinant is just |Γ| in the genus zero reference case), H is the Hilbert transform on the boundary defined from Λ by H = ∂γ−1 Λ, and DET(H) is the product of the discrete eigenvalues of H that come in pairs ±iµk with µk in (0,1). The authors show DET(H) = (µ1 ··· µg)2 and thereby obtain (µ1 ··· µg)2 = detζ(Λ)/|Γ|. For surfaces of genus g>1 this same quantity matches the value at a special point of the Ruelle zeta function of the corresponding hyperbolic surface, reproducing and reinterpreting earlier results in terms of periods on 2M.