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Victor Kalvin

Publications and source records attributed to Victor Kalvin.

At least 19 recordsLinked to original sources
arXiv2026

Spectral determinants of flat metrics on the bielliptic genus-two locus

We obtain a closed explicit formula for the spectral determinant of flat conical metrics on the bielliptic genus-two locus. The metrics are generated by holomorphic one-forms with two simple zeros. For a symmetric reference metric, an exact Klein-four spectral identity reduces the spectral determinant to scalar determinants on spheres and tori; the singular anomaly formula then yields the general case. The resulting formula involves an elementary binary sextic in the coefficients of the one-form and two explicit hypergeometric areas of four-cone metrics on a quotient sphere. We apply this formula to the separating, one-node nonseparating, and simultaneous two-node degenerations of the curve and obtain complete asymptotics of the spectral determinant in all cases. The one-node degeneration has two distinct metric limits, according to whether the limiting one-form is holomorphic or meromorphic. Comparison of the cylindrical cases with the Bismut--Bost asymptotics determines the corresponding constants explicitly; in the separating case it also evaluates the relative determinant appearing in the M\"uller--M\"uller formula. As a by-product, the separating asymptotics evaluate the multiplicative constants left undetermined in earlier general conical and variational determinant formulas.

math.SP↗
arXiv2026

Spectral determinants of the Bolza surface and the Klein quartic

We obtain closed explicit formulas for the spectral determinants of the smooth hyperbolic Bolza surface and the Klein quartic. In each case, a multiplicative relation expresses the determinant of the surface in terms of determinants of singular quotient orbifolds of genera zero and one. The elliptic factors are evaluated by applying the singular Polyakov anomaly formula to explicit Belyi maps on CM elliptic curves of discriminants -8 and -7, while the genus-zero factors are evaluated by explicit determinant formulas for constant-curvature spheres with conical singularities. The same multiplicative relations hold fibrewise on the corresponding equisymmetric deformation strata and yield determinant and first-variation identities. We also prove that every compact quasiplatonic hyperbolic surface is a critical point of the spectral determinant on its Teichm\"uller space; in particular, this applies to the Bolza surface and the Klein quartic.

math.DG↗
arXiv2026

Corner contributions to Neumann jump determinants: three model calculations and a BFK conjecture

We formulate a local corner-factor conjecture for the determinant of the Neumann jump operator on a piecewise real-analytic cutting curve. For the mirror double of a simply connected geodesic polygon with interior angles $\pi\alpha_1,\ldots,\pi\alpha_N$, the conjectural determinant is \[ \Det_{\angle}'\cN = \frac{\length(\partial P)}2 \prod_{j=1}^N\alpha_j^{-1/2}. \] Here $\Det_{\angle}'$ denotes an intrinsic determinant, still to be constructed, that is required to satisfy a Burghelea--Friedlander--Kappeler gluing formula; only its quotient by $\length(\partial P)$ is purely angle-dependent. Three model calculations support the conjecture: a flat polygon and its double give $\frac12\prod_j\alpha_j^{-1/2}$; a spherical spindle split into congruent lunes gives $1/(2\alpha)$ for two angles $\pi\alpha$; and every spherical Coxeter triangle with angles $(\pi/p,\pi/q,\pi/r)$ gives $\frac12\sqrt{pqr}$, including $\sqrt2$ for the octant. The conjecture also reduces the Dirichlet determinant of a constant-curvature polygon to that of its closed double. Although the singular anomaly formula applies to the double in general, an explicit evaluation in nonzero curvature requires solving its uniformization problem. We discuss connections with the Neumann jump determinants in the work of Wiegmann--Zabrodin and Wang and with the recent Grunsky-operator approach to Coulomb gases on domains with corners.

math.SP↗
arXiv2023

Triangulations of singular constant curvature spheres via Belyi functions and determinants of Laplacians

We study the zeta-regularized spectral determinant of the Friedrichs Laplacians on the singular spheres obtained by cutting and glueing copies of constant curvature (hyperbolic, spherical, or flat) double triangle. The determinant is explicitly expressed in terms of the corresponding Belyi functions and the determinant of the Friedrichs Laplacian on the double triangle. The latter determinant was found in a closed explicit form in [V. Kalvin, Calc. Var. 62 (2023), Paper 59, arXiv:2112.02771]. In examples we consider the cyclic, dihedral, tetrahedral, octahedral, and icosahedral triangulations, and find the determinant for the corresponding spherical, Euclidean, and hyperbolic Platonic surfaces. These surfaces correspond to stationary points of the determinant.

math.AP↗
arXiv2022

Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere

We deduce an explicit closed formula for the zeta-regularized spectral determinant of the Friedrichs Laplacian on the Riemann sphere equipped with arbitrary constant curvature (flat, spherical, or hyperbolic) metric having three conical singularities of order $β_j\in(-1,0)$ (or, equivalently, of angle $2π(β_j+1)$). We show that among the metrics with a fixed value of the sum $β_1+β_2+β_3$ and a fixed surface area, those with $β_1=β_2=β_3$ correspond to a stationary point of the determinant. If, in addition, the surface area is sufficiently small, then the stationary point is a minimum. As a crucial step towards obtaining these results we find a relation between the determinant of Laplacian and the Liouville action introduced by A. Zamolodchikov and Al. Zamolodchikov in connection with the celebrated DOZZ formula for the three-point structure constants of the Liouville field theory.

math.DG↗
arXiv2020

Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area $4π$ the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomonson formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.

math-ph↗
arXiv2020

Determinant of Friederichs Dirichlet Laplacians on $2$-dimensional hyperbolic cones

We explicitly express the spectral determinant of Friederichs Dirichlet Laplacians on the 2-dimensional hyperbolic (Gaussian curvature -1) cones in terms of the cone angle and the geodesic radius of the boundary. The related results in the recent paper "Riemann-Roch isometries in the non-compact orbifold setting," J. Eur. Math. Soc. 22 (2020) by G. Freixas i Montplet and A. von Pippich turn out to be incorrect.

math.SP↗
arXiv2020

Spectral determinant on Euclidean isosceles triangle envelopes of fixed area as a function of angles: absolute minimum and small-angle asymptotics

We study extremal properties of the determinant of Friederichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. Small-angle asymptotics show that the determinant grows without any bound as an angle of triangle envelope goes to zero. We prove that the equilateral triangle envelope (the most symmetrical geometry) always gives rise to a critical point of the determinant and find the critical value. Moreover, if the area of envelopes is not too large, then the determinant achieves its absolute minimum only on the equilateral triangle envelope and there are no other critical points, whereas for sufficiently large area the equilateral triangle envelope corresponds to a local maximum of the determinant.

math.AP↗
arXiv2018

On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions

Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singularity of $\mathsf m$ or $\{\infty\}$. The pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ has conical singularities of angles $4π$ at the critical points of $f$ and other conical singularities that are the preimages of those of $\mathsf m$. We study the $ζ$-regularized determinant $\operatorname{Det}' Δ_F$ of the (Friedrichs extension of) Laplace-Beltrami operator on $(X,f^*\mathsf m)$ as a functional on the moduli space of pairs $(X, f)$ and obtain an explicit formula for $\operatorname{Det}' Δ_F$.

math.AP↗
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

Analysis of Perfectly Matched Layer operators for acoustic scattering on manifolds with quasicylindrical ends

We prove stability and exponential convergence of the Perfectly Matched Layer (PML) method for acoustic scattering on manifolds with axial analytic quasicylindrical ends. These manifolds model long-range geometric perturbations (e.g. bending or stretching) of tubular waveguides filled with homogeneous or inhomogeneous media. We construct non-reflective infinite PMLs replacing the metric on a part of the manifold by a non-degenerate complex symmetric tensor field. We prove that the problem with PMLs of finite length is uniquely solvable and solutions to this problem locally approximate scattered solutions with an error that exponentially tends to zero as the length of PMLs tends to infinity.

math.AP↗
arXiv2011

Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.

math.AP↗
arXiv2011

Aguilar-Balslev-Combes theorem for the Laplacian on a manifold with an axial analytic asymptotically cylindrical end

We develop the complex scaling for a manifold with an asymptotically cylindrical end under an assumption on the analyticity of the metric with respect to the axial coordinate of the end. We allow for arbitrarily slow convergence of the metric to its limit at infinity, and prove a variant of the Aguilar-Balslev-Combes theorem for the Laplacian $Δ$ on functions. In the case of a manifold with (noncompact) boundary it is either the Dirichlet or the Neumann Laplacian. We introduce resonances as the discrete non-real eigenvalues of non-selfadjoint operators, obtained as deformations of the Laplacian by means of the complex scaling. The resonances are identified with the poles of the resolvent matrix elements $((Δ-μ)^{-1}F, G)$ meromorphic continuation in $μ$ across the essential spectrum of $Δ$, where $F$ and $G$ are elements of an explicitly given set of analytic vectors. The Laplacian has no singular continuous spectrum, the eigenvalues can accumulate only at thresholds.

math-ph↗
arXiv2011

Complex scaling for the Dirichlet Laplacian in a domain with asymptotically cylindrical end

We develop the complex scaling method for the Dirichlet Laplacian in a domain with asymptotically cylindrical end. We define resonances as discrete eigenvalues of non-selfadjoint operators, obtained as deformations of the selfadjoint Dirichlet Laplacian $Δ$ by means of the complex scaling. The resonances are identified with the poles of the resolvent matrix elements $((Δ-μ)^{-1}F, G)$ meromorphic continuation in $μ$ across the essential spectrum of $Δ$, where $F$ and $G$ are elements of an explicitly given set of partial analytic vectors. It turns out that the Dirichlet Laplacian has no singular continuous spectrum, and its eigenvalues can accumulate only at threshold values of the spectral parameter.

math.AP↗
arXiv2010

Exponential Decay of Eigenfunctions and Accumulation of Eigenvalues on Manifolds with Axial Analytic Asymptotically Cylindrical Ends

In this paper we continue our study of the Laplacian on manifolds with axial analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using the complex scaling method and the Phragmén-Lindelöf principle we prove exponential decay of the eigenfunctions corresponding to the non-threshold eigenvalues of the Laplacian on functions. In the case of a manifold with (non-compact) boundary it is either the Dirichlet Laplacian or the Neumann Laplacian. We show that the rate of exponential decay of an eigenfunction is prescribed by the distance from the corresponding eigenvalue to the next threshold. Under our assumptions on the behaviour of the metric at infinity accumulation of isolated and embedded eigenvalues occur. The results on decay of eigenfunctions combined with the compactness argument due to Perry imply that the eigenvalues can accumulate only at thresholds and only from below. The eigenvalues are of finite multiplicity.

math.SP↗