Author

Alexey Kokotov

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Preprint Aug 2026

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

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 $\frac{{\rm det}\Lambda}{|\Gamma|}$ (here $\Lambda$ is the DN operator on the boundary, $\Gamma$, of a multi-connected domain and $|\Gamma|$ 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 Guillop\'e 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.

Alexey Kokotov, Dmitrii Korikov, M. Nenasheva · 0 citations

Preprint Aug 2026

The determinant of the Dirichlet-to-Neumann map for a surface with boundary and periods of holomorphic differentials on its double

Let $(M,g)$ be a smooth orientable $2d$ Riemannian manifold of genus $\mathfrak{g}$ with Riemannian metric $g$ and connected boundary $\Gamma$. Let $\Lambda$ be the Dirichlet-to-Neumann map on $\Gamma$ and let ${\rm det}_\zeta(\Lambda)$ be its (modified, i. e. with zero mode excluded) $\zeta$-regularized determinant. It is well-known that the quantity ${\rm det}_\zeta(\Lambda)/|\Gamma|$ (where $|\Gamma|$ is the length of $\Gamma$) is a conformal invariant. It was shown by Edward and Wu (\cite{EV}) that this invariant equals one for $\mathfrak{g}=0$; in the case $\mathfrak{g}>0$ Guillarmou and Guillop\'e \cite{Guillarmou} found two explicit expressions for this invariant through the Rouelle and (respectively) the Selberg zeta-functions of the two surfaces of negative constant curvature from the conformal class of $(M,g)$: one is of infinite volume and complete whereas another has geodesic boundary. We present an elementary counterpart of the formulae of Guillarmou and Guillop\'e using the periods of holomorphic differentials on the double $2M$ of $M$ only. Our approach is based on the properties of the Hilbert transform of $M$ \cite{B,HilbKor} and the Kontsevich-Vishik-Friedlander-Guillemin regularization of the determinants of pseudodifferenial operators \cite{KV,Ww,F}. In particular, a connection between the length spectra of (uniformized) $M$, $2M$ and the periods of holomorphic differentials on $2M$ is established.

Dmitrii Korikov, Alexey Kokotov · 1 citation