The determinant of the Dirichlet-to-Neumann map for a surface with boundary and periods of holomorphic differentials on its double
Abstract
Let (M,g) be a smooth orientable 2d Riemannian manifold of genus g with Riemannian metric g and connected boundary Γ. Let Λ be the Dirichlet-to-Neumann map on Γ and let detζ(Λ) be its (modified, i. e. with zero mode excluded) ζ-regularized determinant. It is well-known that the quantity detζ(Λ)/|Γ| (where |Γ| is the length of Γ) is a conformal invariant. It was shown by Edward and Wu (\cite{EV}) that this invariant equals one for g=0; in the case g>0 Guillarmou and Guillopé \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 Guillopé 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.
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BibTeX
@misc{korikov2026determinant,
title = {{The determinant of the Dirichlet-to-Neumann map for a surface with boundary and periods of holomorphic differentials on its double}},
author = {Dmitrii Korikov and Alexey Kokotov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.09737},
eprint = {2608.09737},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09737},
}
RIS
TY - UNPB
TI - The determinant of the Dirichlet-to-Neumann map for a surface with boundary and periods of holomorphic differentials on its double
AU - Dmitrii Korikov
AU - Alexey Kokotov
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2608.09737
UR - https://arxiv.org/abs/2608.09737
AB - Let (M,g) be a smooth orientable 2d Riemannian manifold of genus g with Riemannian metric g and connected boundary Γ. Let Λ be the Dirichlet-to-Neumann map on Γ and let det_ζ(Λ) be its (modified, i. e. with zero mode excluded) ζ-regularized determinant. It is well-known that the quantity det_ζ(Λ)/|Γ| (where |Γ| is the length of Γ) is a conformal invariant. It was shown by Edward and Wu (\cite{EV}) that this invariant equals one for g=0; in the case g>0 Guillarmou and Guillopé \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 Guillopé 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.
ER -