On the Dirichlet-to-Neumann conformal invariant of bounded planar domains

Abstract

Using an analogue of Mandelstam-Giddings-Wolpert diagrams, we introduce a new canonical representative of the conformal class of a bounded domain of arbitrary connectivity in $\mathbb{C}$ as a flat conical surface with geodesic boundary (a "truncated light-cone diagram", simply LC-diagram in the sequel). The space of these diagrams can be provided with natural coordinates. We derive variational formulas for the determinant of the Dirichlet boundary problem on a LC-diagram with respect to these coordinates. Then passing to the Schottky double of the LC-diagram, making use of the Burghelea-Friedlander-Kappeler formula and the known variational formulas for determinants of Laplacians on the moduli space of holomorphic differentials lead to an explicit computation of the Dirichlet-to-Neumann (DN) conformal invariant $\frac{{\rm det}Λ}{|Γ|}$ (here $Λ$ is the DN operator on the boundary, $Γ$, of a multi-connected domain and $|Γ|$ is the length of the boundary). The resulting formula (which uses the periods of the Schottky double of the domain only) presents a conspicuously elementary counterpart to the formulas of Guillarmou and Guillopé who had expressed the DN invariant through the Ruelle and Selberg zeta-functions. Our formula agrees with the recent result of Wentworth on the asymptotics of the DN invariant as all the boundary components except one shrink, we have used this result to fix the undetermined constant of integration in our formula.