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Dmitrii Korikov

Publications and source records attributed to Dmitrii Korikov.

10 recordsLinked to original sources
arXiv2026

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

Using an analogue of the 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 diagrams carries natural coordinates. We derive variational formulas for the determinant of the Dirichlet Laplacian 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, we compute the Dirichlet-to-Neumann (DN) conformal invariant $\frac{{\rm det}'Λ}{|Γ|}$ (here $Λ$ is the DN operator on the boundary, $Γ$, of a multiply 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) provides an 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 but one boundary components shrink; we have used this result to fix the undetermined constant of integration in our formula. As a corollary, we derive an explicit formula for the determinant of the Dirichlet Laplacian in a multiply connected domain.

math-ph↗
arXiv2026

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 $Γ$. Let $Λ$ be the Dirichlet-to-Neumann map on $Γ$ and let ${\rm det}_ζ(Λ)$ be its (modified, i. e. with zero mode excluded) $ζ$-regularized determinant. It is well-known that the quantity ${\rm det}_ζ(Λ)/|Γ|$ (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 $\mathfrak{g}=0$; in the case $\mathfrak{g}>0$ 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.

math-ph↗
arXiv2026

On a polygon version of Wiegmann-Zabrodin formula

Let $P$ be a convex polygon in ${\mathbb C}$ and let $Δ_{D, P}$ be the operator of the Dirichlet boundary value problem for the Lapalcian $Δ=-4\partial_z\partial_{\bar z}$ in $P$. We derive a variational formula for the logarithm of the $ζ$-regularized determinant of $Δ_{D, P}$ for arbitrary infinitesimal deformations of the polygon $P$ in the class of polygons (with the same number of vertices). For a simply connected domain with smooth boundary such a formula was recently discovered by Wiegmann and Zabrodin as a non obvious corollary of the Alvarez variational formula, for domains with corners this approach is unavailable (at least for those deformations that do not preserve the corner angles) and we have to develop another one.

math.SP↗
arXiv2025

On an infinitesimal Polyakov formula for genus zero polyhedra

Let $X$ be a genus zero compact polyhedral surface (the Riemann sphere equipped with a flat conical metric $m$). We derive the variational formulas for the determinant of the Laplacian, ${\rm det}\,Δ^m$, on $X$ under infinitesimal variations of the positions of the conical points and the conical angles (i. e. infinitesimal variations of $X$ in the class of polyhedra with the same number of vertices). Besides having an independent interest, this derivation may serve as a somewhat belated mathematical counterpart of the well-known heuristic calculation of ${\rm det}\,Δ^m$ performed by Aurell and Salomonson in the 90-s.

math.SP↗
arXiv2025

Laplacians in spinor bundles over translation surfaces: self-adjoint extentions and regularized determinants

We study the regularized determinants ${\rm det}\, Δ$ of various self-adjoint extensions of symmetric Laplacians acting in spinor bundles over compact Riemann surfaces with flat singular metrics $|ω|^2$, where $ω$ is a holomorphic one form on the Riemann surface. We find an explicit expression for ${\rm det}\, Δ$ for the so-called self-adjoint Szegö extension through the Bergman tau-function on the moduli space of Abelian differentials and the theta-constants (corresponding to the spinor bundle). This expression can be considered as a version of the well-known spin-$1/2$ bosonization formula of Bost-Nelson for the case of flat conformal metrics with conical singularities and a higher genus generalization of the Ray-Singer formula for flat elliptic curves. We establish comparison formulas for the determinants of two different extensions (e. g., the Szegö extension and the Friedrichs one). The paper answers a question raised by D'Hoker and Phong \cite{DH-P} more than thirty years ago. We also reconsider the results from \cite{DH-P} on the regularization of diverging determinant ratio for Mandelstam metrics (for any spin) proposing (and computing) a new regularization of this ratio.

math.DG↗
arXiv2025

Variational formulas for determinant of Laplacian on higher genus polyhedral surface

Let $X$ be a Riemann surface of genus $g\ge 1$ endowed with a flat conical metric $m$ and let ${\rm det}\,Δ$ be the $ζ$-regularized determinant of the Friedrichs Laplacian on $(X,m)$. We derive variational formulas for ${\rm det}\,Δ$ with respect to conical points and conical angles within a given conformal class. Integration of them leads to an explicit expression for ${\rm det}\,Δ$ up to moduli dependent factor. The latter, in principle, can be calculated via comparison of the above result with the well-known formulas for the case of flat conical metrics with trivial holonomy.

math.DG↗
arXiv2025

Determination of period matrix of double of surface with boundary via its DN map

As is well-known, a conformal class of a surface $M$ with boundary $Γ$ is determined by its DN map $Λ$. In the paper, the algorithm for determination of the $b$-period matrix $\mathbb{B}$ of the (Schottky) double of surface with boundary via $Λ$ is presented. Due to the Torelli theorem, $\mathbb{B}$ contains all information on the conformal class of $M$ except the proper way of attaching $Γ$ to it.

math-ph↗
arXiv2024

Determinants of pseudo-laplacians and $ζ^{({\rm reg})}(1)$ for spinor bundles over Riemann surfaces

Let $P$ be a point of a compact Riemann surface $X$. We study self-adjoint extensions of the Dolbeault Laplacians in hermitian line bundles $L$ over $X$ initially defined on sections with compact supports in $X\backslash\{P\}$. We define the $ζ$-regularized determinants for these operators and derive comparison formulas for them. We introduce the notion of the Robin mass of $L$. This quantity enters the comparison formulas for determinants and is related to the regularized $ζ(1)$ for the Dolbeault Laplacian. For spinor bundles of even characteristic, we find an explicit expression for the Robin mass. In addition, we propose an explicit formula for the Robin mass in the scalar case. Using this formula, we describe the evolution of the regularized $ζ(1)$ for scalar Laplacian under the Ricci flow. As a byproduct, we find an alternative proof for the Morpurgo result that the round metric minimizes the regularized $ζ(1)$ for surfaces of genus zero.

math.SP↗
arXiv2023

Stability estimates in determination of non-orientable surface from its Dirichlet-to-Neumann map

Let $(M,g)$ and $(M',g')$ be non-orientable Riemannian surfaces with fixed boundary $Γ$ and fixed Euler characterictic $m$, and $Λ$ and $Λ'$ be their Dirichlet-to-Neumann maps, respectively. We prove that the closeness of $Λ'$ to $Λ$ in the operator norm implies the existence of of the near-conformal diffeomorphism $β$ between $(M,g)$ and $(M',g')$ which does not move the points of $Γ$. Hence we establish the continuity of the determination $Λ\mapsto [(M,g)]$, where $[(M,g)]$ is the conformal class of $(M,g)$ and the set of such conformal classes is endowed with the natural Teichmüller-type metric $d_T$. In both orientable and non-orientable case we provide quantitative estimates of $d_T([(M,g)],[(M',g')])$ via the operator norm of the difference $Λ'-Λ$. We also obtain generalizations of the results above to the case in which the Dirichlet-to-Neumann map is given only on a segment of the boundary.

math.DG↗