Preprint

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

Dmitrii Korikov Alexey Kokotov

Aug 2026 · 1 citation

Physics Mathematics

Abstract

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.

arXiv Semantic Scholar

View source

Similar papers

Preprint Sep 2026

Bounded cohomology and optimal separating constant for representations

Let $\Sigma$ be a closed oriented surface of genus $>1$ and $M$ a complete hyperbolic 3-manifold with a marking $i:\Sigma\longrightarrow M$. We consider the case that $M$ has no parabolic cusps and at least one of the two ends is simply degenerate. For $\varGamma=\pi_1(\Sigma)$, let $\rho_M:\varGamma\longrightarrow \ma...

Y. Nakano, Teruhiko Soma · 0 citations

Preprint Aug 2026

Explicit Estimates for the Bergman Kernel Form

Let $(L,e^{-\phi})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $\omega=i\partial\overline{\partial}\phi$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\,\omega\leq\omega$ and the shortest nonconstant closed geodesi...

Johannes Testorf · 0 citations

Preprint Sep 2026

Local rigidity of the Euclidean metric for the anisotropic Calder\'on problem

We prove local rigidity of the Euclidean metric and close conformally Euclidean ones for the anisotropic Calder\'on problem on smooth compact domains $M\subset\mathbb{R}^n$, $n\ge3$. A smooth Riemannian metric $g$ with the same induced boundary metric and Dirichlet-to-Neumann map as the background $g_0$ is isometric to...

P. Stefanov · 1 citation

Preprint Sep 2026

Localizing a Dirac operator via $J$-holomorphic curves

Let $(M,J)$ be a compact almost hermitian $4$-manifold with a smooth embedded $J$-holomorphic curve $C$ representing the canonical class. Motivated by the symplectic Bogomolov--Miyaoka--Yau conjecture, we choose a twisted spin$^{\text{c}}$ Dirac operator ${\mathcal D}$ on $M$ satisfying $$ \operatorname{ind}{\mathcal D...

Jun-Ho Lee · 0 citations

Preprint Sep 2026

Spheres under a Cauchy-Riemann inequality in $E(\kappa,\tau)$, and non-CMC Hopf tori

For a smooth, connected, oriented surface $\Sigma$, topologically a sphere, immersed in the homogeneous space $E(\kappa,\tau)$ with $\tau\ne0$, we show that a one-sided Cauchy--Riemann-type inequality on the original Abresch--Rosenberg differential $\mathcal{Q}_{AR}$ -- much weaker than requiring $\mathcal{Q}_{AR}$ to...

H. Alencar, H. Rosenberg · 0 citations

Preprint Aug 2026

Scalar Curvature Flexibility in the Riemannian Burnett Compactness Class

Let $M$ be a connected smooth $n$-manifold without boundary, where $n\geq3$, and let $\kappa\in\mathbb{R}$, with $\kappa\leq0$ if $M$ is open. We prove that every smooth Riemannian metric $g_0$ with $\mathrm{Scal}_{g_0}\geq\kappa$ is a locally uniform limit of smooth Riemannian metrics $g_i$ with $\mathrm{Scal}_{g_i}=\...

Jingbo Wan · 0 citations