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Alexey Kokotov

Publications and source records attributed to Alexey Kokotov.

At least 19 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↗
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↗
arXiv2020

Flat conical Laplacian in the square of the canonical bundle and its regularized determinants

Let $X$ be a compact Riemann surface of genus $g\geq 2$ equipped with flat conical metric $|Ω|$, where $Ω$ be a holomorphic quadratic differential on $X$ with $4g-4$ simple zeroes. Let $K$ be the canonical line bundle on $X$. Introduce the Cauchy-Riemann operators $\bar \partial$ and $\partial$ acting on sections of holomorphic line bundles over $X$ ($K^2$ in the definition of $Δ^{(2)}_{|Ω|}$ below) and, respectively, anti-holomorphic line bundles ($\bar { K}^{-1}$ below). Consider the Laplace operator $Δ^{(2)}_{|Ω|}:=|Ω| \partial |Ω|^{-2}\bar\partial$ acting in the Hilbert space of square integrable sections of the bundle $K^2$ equipped with inner product $ _{K^2}=\int_X\frac {Q_1\bar Q_2}{|Ω|}$. We discuss two natural definitions of the determinant of the operator $Δ^{(2)}_{|Ω|}$. The first one uses the zeta-function of some special self-adjoint extension of the operator (initially defined on smooth sections of $K^2$ vanishing near the zeroes of $Ω$), the second one is an analog of Eskin-Kontsevich-Zorich (EKZ) regularization of the determinant of the conical Laplacian acting in the trivial bundle. In contrast to the situation of operators acting in the trivial bundle, for operators acting in $K^2$ these two regularizations turn out to be essentially different. Considering the regularized determinant of $Δ^{(2)}_{|Ω|}$ as a functional on the moduli space $Q_g(1, \dots, 1)$ of quadratic differentials with simple zeroes on compact Riemann surfaces of genus $g$, we derive explicit expressions for this functional for the both regularizations. The expression for the EKZ regularization is closely related to the well-known explicit expressions for the Mumford measure on the moduli space of compact Riemann surfaces of genus $g$.

math.DG↗
arXiv2019

Green function and self-adjoint Laplacians on polyhedral surfaces

Using Roelcke formula for the Green function, we explicitly construct a basis in the kernel of the adjoint Laplacian on a compact polyhedral surface $X$ and compute the $S$-matrix of $X$ at the zero value of the spectral parameter. We apply these results to study various self-adjoint extensions of a symmetric Laplacian on a compact polyhedral surface of genus two with a single conical point. It turns out that the behaviour of the $S$-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.

math.SP↗
arXiv2016

Metrics of constant positive curvature with conical singularities, Hurwitz spaces, and ${\rm det}\, Δ$

Let $f: X\to {\Bbb C}P^1$ be a meromorphic function of degree $N$ with simple poles and simple critical points on a compact Riemann surface $X$ of genus $g$ and let $\mathsf m$ be the standard round metric of curvature $1$ on the Riemann sphere ${\Bbb C}P^1$. Then the pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ is a metric of curvature $1$ with conical singularities of conical angles $4π$ at the critical points of $f$. We study the $ζ$-regularized determinant of the Laplace operator on $X$ corresponding to the metric $f^*\mathsf m$ as a functional on the moduli space of the pairs $(X, f)$ (i.e. on the Hurwitz space $H_{g, N}(1, \dots, 1)$) and derive an explicit formula for the functional.

math.AP↗
arXiv2014

Moduli spaces of meromorphic functions and determinant of Laplacian

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $Δ^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $Δ^{|df|^2}$ and study it as a functional on the Hurwitz space. We prove that this functional is related to a system of PDE which admits explicit integration. This leads to an explicit expression for the determinant of the Laplace operator in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic differential $df$. The proof has several parts that can be of independent interest. As an important intermediate result we prove a decomposition formula of the type of Burghelea-Friedlander-Kappeler for the determinant of the Laplace operator on flat surfaces with conical singularities and Euclidean or conical ends. We introduce and study the $S$-matrix, $S(λ)$, of a surface with conical singularities as a function of the spectral parameter $λ$ and relate its behavior at $λ=0$ with the Schiffer projective connection on the Riemann surface $X$. We also prove variational formulas for eigenvalues of the Laplace operator of a compact surface with conical singularities when the latter move.

math.SP↗
arXiv2013

Spectral Determinants on Mandelstam Diagrams

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric $|ω|^2$, where $ω$ is a meromorphic one-form with simple poles such that all its periods are pure imaginary and all its residues are real. The main result is an explicit formula for the determinant of the Laplacian in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, canonical meromorphic bidifferential) and the divisor of the meromorphic form $ω$. As an important intermediate result we prove a decomposition formula of the type of Burghelea-Friedlander-Kappeler for the determinant of the Laplacian for flat surfaces with cylindrical ends and conical singularities.

math.SP↗
arXiv2012

Krein formula and S-matrix for Euclidean Surfaces with Conical Singularities

We use Krein formula and the S-matrix formalism to give formulas for the zeta-regularized determinant of non-Friedrichs extensions of the Laplacian on Euclidean surfaces with Conical Singularities. This formula involves S(0) and we show that the latter can be expressed using the Bergman projective connection on the underlying Riemann surface.

math.SP↗
arXiv2012

Determinant of pseudo-laplacians

Let X be a compact Riemannian manifold of dimension two or three and let P be a point of X. We derive comparison formulas relating the zeta-regularized determinant of an arbitrary self-adjoint extension of (symmetric) Laplace operator with domain, consisting of smooth functions with compact supports which does not contain P, to the zeta-regularized determinant of the self-adjoint Laplacian on X.

math.SP↗
arXiv2009

Compact polyhedral surfaces of an arbitrary genus and determinants of Laplacians

Compact polyhedral surfaces (or, equivalently, compact Riemann surfaces with conformal flat conical metrics) of an arbitrary genus are considered. After giving a short self-contained survey of their basic spectral properties, we study the zeta-regularized determinant of the Laplacian as a functional on the moduli space of these surfaces. An explicit formula for this determinant is obtained.

math.DG↗