arXiv subjects

Alexey Glutsyuk

Publications and source records attributed to Alexey Glutsyuk.

At least 19 recordsLinked to original sources

arXiv2026

On phase-lock area parquet in a special slow-fast limit of model of Josephson junction

B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by the family of differential equations on the 2-torus, $\frac{dθ}{dτ}=\frac1ω(\cosθ+B+A\cosτ)$, which is known as the RSJ model. It depends on three parameters: $B$ called the abscissa, $A$ called the ordinate, and a fixed frequency $ω$. We study its rotation number $ρ(B,A;ω)$ as a function of $(B,A)$ and the phase-lock areas: those its level subsets that have non-empty interiors. They exist only for integer values of the rotation number (Buchstaber, Karpov, Tertychnyi). In this paper we study asymptotics of the phase-lock area portrait in a special slow-fast limit, as $ω\to0$ and $(B,A)\to(0,1)$ so that $(B,A)-(0,1)=O(ω)$. We show that in the rescaled parameters $\ell:=\frac Bω$ and $u:=\frac{A-1}ω$ the phase-lock area portrait converges to a parquet with boundary lines being parallel to the lines $\{ u\pm\ell=0\}$. Namely, the limit of phase-lock area with rotation number $r$ is the union of an infinite chain of squares going up, with integer vertices and diagonals of length two lying on the line $\{\ell=r\}$, and an infinite strip going down (sector in the case, when $r=0$). We state and prove a generalization of this result to a wide class of slow-fast systems on 2-torus.

math.DS↗

arXiv2026

On exotic rationally integrable planar dual billiards I. Complex geometry and type of dynamics

A planar dual billiard is a planar curve $γ$ equipped with a family $(σ_P)|_{P\inγ}$ of projective involutions of the projective lines $L_P$ tangent to $γ$ at $P$ that fix $P$. A dual billiard is called rationally integrable, if there exists a rational function $R(x,y)$ of two variables (called first integral) whose restriction to each tangent line $L_P$ is $σ_P$-invariant. In the previous author's paper it was shown that rationally integrable dual billiards exist only on conics punctured at $k$ points, $0\leq k\leq 4$. Their classification given there includes standard examples with quadratic integrals, defined by conical pencils, and an infinite family of exotic examples with minimal degree of integrals being any even number greater than two. In the present paper we study a question stated by Dmitry Treschev on dynamics and conservation laws in the exotic examples. We study the dynamics acting on the two-dimensional phase space: the complex algebraic surface consisting of pairs $(Q,P)$, where $Q\in\mathbb{CP}^2\setminusγ$, $P\inγ$ and the line $QP$ is tangent to $γ$ at $P$. We show that the phase space is fibered by invariant algebraic curves along which $Q$ lies in a level curve of the integral of the dual billiard. For each example we find the type of a generic level curve of the integral and of the corresponding fiber. We show that a generic level curve is rational in most of examples and elliptic in two examples. We present formulas for an invariant area form on the phase space and for invariant holomorphic differentials on invariant curves. This yields a formula for holomorphic differentials on the elliplic level curves of the integral.

math.DS↗

arXiv2025

Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary $C^{\infty}$-smooth hypersurface $γ\subset\mathbb R^{n+1}$ that is either a global strictly convex closed hypersurface, or a germ of hypersurface. We deal with the pseudogroup generated by compositional ratios of reflections from $γ$ and of reflections from its small deformations. In the case, when $γ$ is a global convex hypersurface, we show that the latter pseudogroup is dense in the pseudogroup of Hamiltonian diffeomorphisms between subdomains of the phase cylinder: the space of oriented lines intersecting $γ$ transversally. We prove an analogous local result in the case, when $γ$ is a germ. The derivatives of the above compositional differences in the deformation parameter are Hamiltonian vector fields calculated by Ron Perline. To prove the main results, we find the Lie algebra generated by them and prove its $C^{\infty}$-density in the Lie algebra of Hamiltonian vector fields. We also prove analogues of the above results for hypersurfaces in Riemannian manifolds.

math.DS↗

arXiv2025

On complex algebraic caustics in planar and projective billiards

A caustic of a billiard is a curve whose tangent lines are reflected to its own tangent lines. A billiard is called Birkhoff caustic-integrable, if there exists a topological annulus adjacent to its boundary from inside that is foliated by closed caustics. The famous Birkhoff Conjecture, studied by many mathematicians, states that the only Birkhoff caustic-integrable billiards are ellipses. The conjecture is open even for billiards whose boundaries are ovals of algebraic curves. In this case the billiard is known to have a dense family of so-called rational caustics that are also ovals of algebraic curves. We introduce the notion of a complex caustic: a complex algebraic curve whose complex tangent lines are sent by complexified reflection to its own complex tangent lines. We show that the usual billiard on a real planar curve $γ$ has a complex caustic, if and only if $γ$ is a conic. We prove analogous result for billiards on all the surfaces of constant curvature. These results are corollaries of the solution of S.Bolotin's polynomial integrability conjecture: a joint result by M.Bialy, A.Mironov and the author. We extend them to the projective billiards introduced by S.Tabachnikov, which are a common generalization of billiards on surfaces of constant curvature. We also deal with a well-known class of projective billiards on conics that are defined to have caustics forming a dual conical pencil. We show that up to restriction to a finite union of arcs, each of them is equivalent to a billiard on appropriate surface of constant curvature.

math.DS↗

arXiv2025

Dynamical systems on torus related to general Heun equations: phase-lock areas and constriction breaking

The overdamped Josephson junction in superconductivity theory can be modeled by the family of dynamical systems on the torus, which is known as the RSJ model. This family admits an equivalent description by a family of second-order differential equations: special double confluent Heun equations. In the present paper, we construct two new families of dynamical systems on torus that can be equivalently described by a family of general Heun equations (GHE), with four singular points, and confluent Heun equations, with three singular points. The first family, related to GHE, is a deformation of the RSJ model, which will be denoted by dRSJ. The phase-lock areas of a family of dynamical systems on the torus are those level subsets of the rotation number function that have nonempty interiors. It is known that for the RSJ model, the rotation number quantization effect occurs: phase-lock areas exist only for integer rotation number values. Moreover, each phase-lock area is a chain of domains separated by points. Those separation points that do not lie on the abscissa axis are called constrictions. In the present paper, we study phase-lock areas in the new family dRSJ. The quantization effect remains valid in this family. On the other hand, we show that in the new family dRSJ the constrictions break down.

math.DS↗

arXiv2025

Phase-locking in dynamical systems and quantum mechanics

In this study, we discuss the Prufer transform that connects the dynamical system on the torus and the Hill equation, which is interpreted as either the equation of motion for the parametric oscillator or the Schrodinger equation with periodic potential. The structure of phase-locking domains in the dynamical system on torus is mapped into the band-gap structure of the Hill equation. For the parametric oscillator, we provide the relation between the non-adiabatic Hannay angle and the Poincare rotation number of the corresponding dynamical system. In terms of quantum mechanics, the integer rotation number is connected to the quantization number via the Milne quantization approach and exact WKB. Using recent results concerning the exact WKB approach in quantum mechanics, we discuss the possible non-perturbative effects in the dynamical systems on the torus and for parametric oscillator. The semiclassical WKB is interpreted in the framework of a slow-fast dynamical system. The link between the classification of the coadjoint Virasoro orbits and the Hill equation yields a classification of the phase-locking domains in the parameter space in terms of the classification of Virasoro orbits. Our picture is supported by numerical simulations for the model of the Josephson junction and Mathieu equation.

cond-mat.stat-mech↗

arXiv2024

On extended model of Josephson junction, linear systems with polynomial solutions, determinantal surfaces and Painlevé III equations

We consider a 3-parameter family of linear special double confluent Heun equations introduced and studied by V.M.Buchstaber and S.I.Tertychnyi, which is an equivalent presentation of a model of Josephson junction in superconductivity. Buchstaber and Tertychnyi have shown that the set of those complex parameters for which the Heun equation has a polynomial solution is a union of explicit planar algebraic curves: the spectral curves indexed by $\ell\in\mathbb N$. In his paper with I.V.Netay, the author has shown that each spectral curve is irreducible in Heun equation parameters (consists of two irreducible components in parameters of Josephson junction model). Netay discovered numerically and conjectured a genus formula for spectral curves. He reduced it to the conjecture stating that each of them is regular in $\mathbb C^2$ with a coordinate axis deleted. Here we prove Netay's regularity and genus conjectures. For the proof we study a 4-parameter extension of a family of linear systems equivalent to the Heun equations. They yield an equivalent presentation of the extension of model of Josephson junction introduced by the author in his paper with Yu.P.Bibilo. We describe the so-called determinantal surfaces consisting of linear systems with polynomial solutions as explicit algebraic surfaces indexed by $\ell\in\mathbb N$. The spectral curves are their intersections with the hyperplane of the initial Heun equation family. We prove that each determinantal surface is regular outside appropriate hyperplane and consists of two rational irreducible components. The proofs use Stokes phenomena theory, holomorphic vector bundle technique, foliation of determinantal surfaces by isomonodromic families of linear systems governed by Painlevé 3 equation and transversality of the latter foliation to the initial model.

math.DS↗

arXiv2024

If a Minkowski billiard is projective, it is the standard billiard

In the recent paper arXiv:2405.13258, the first author of this note proved that if a billiard in a convex domain in $\mathbb{R}^n$ is simultaneously projective and Minkowski, then it is the standard Euclidean billiard in an appropriate Euclidean structure. The proof was quite complicated and required high smoothness. Here we present a direct simple proof of this result which works in $C^1$-smoothness. In addition we prove the semi-local and local versions of the result

math.DS↗

arXiv2024

On Hamiltonian projective billiards on boundaries of products of convex bodies

Let $K\subset\mathbb R^n_q$, $T\subset\mathbb R^n_p$ be two bounded strictly convex bodies (open subsets) with $C^6$-smooth boundaries. We consider the product $\overline K\times\overline T\subset\mathbb R^{2n}_{q,p}$ equipped with the standard symplectic form $ω=\sum_{j=1}^ndq_j\wedge dp_j$. The $(K,T)$-billiard orbits are continuous curves in the boundary $\partial(K\times T)$ whose intersections with the open dense subset $(K\times\partial T)\cup(\partial K\times T)$ are tangent to the characteristic line field given by kernels of the restrictions of the symplectic form $ω$ to the tangent spaces to the boundary. For every $(q,p)\in K\times \partial T$ the characteristic line in $T_{(q,p)}\mathbb R^{2n}$ is directed by the vector $(\vec n(p),0)$, where $\vec n(p)$ is the exterior normal to $T_p\partial T$, and similar statement holds for $(q,p)\in\partial K\times T$. The projection of each $(K,T)$-billiard orbit to $K$ is an orbit of the so-called $T$-billiard in $K$. In the case, when $T$ is centrally-symmetric, this is the billiard in $\mathbb R^n_q$ equipped with Minkowski Finsler structure "dual to $T$", with Finsler reflection law introduced in a joint paper by S.Tabachnikov and E.Gutkin in 2002. Studying $(K,T)$-billiard orbits is closely related to C.Viterbo's Symplectic Isoperimetric Conjecture (recently disproved by P.Haim-Kislev and Y.Ostrover) and the famous Mahler Conjecture in convex geometry. We study the special case, when the $T$-billiard reflection law is the projective law introduced by S.Tabachnikov, i.e., given by projective involutions of the projectivized tangent spaces $T_q\mathbb R^n$, $q\in\partial K$. We show that this happens, if and only if $T$ is an ellipsoid, or equivalently, if all the $T$-billiards are simultaneously affine equivalent to Euclidean billiards. As an application, we deduce analogous results for Finsler billiards.

math.DS↗

arXiv2023

On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

The billiard flow in a planar domain acts on its tangent bundle as geodesic flow with reflections from the boundary. Its trivial first integral is the squared velocity. Bolotin's Conjecture, now a joint theorem of Bialy, Mironov and the author, deals with those planar billiards whose flow admits an integral polynomial in the velocity whose restriction to the unit tangent bundle is non-constant. It states that 1) if the boundary of such a billiard is $C^2$-smooth, nonlinear and connected, then it is a conic; 2) if it is piecewise $C^2$-smooth and contains a nonlinear arc, then it consists of arcs of conics from a confocal pencil and segments of "admissible lines" for the pencil; 3) the minimal degree of the integral is either 2, or 4. In 1997 Sergei Tabachnikov introduced projective billiards: planar curves equipped with a transversal line field, defining reflection of oriented lines and the projective billiard flow. They are common generalization of billiards on constant curvature surfaces, but in general may have no canonical integral. In a previous paper the author classified those $C^4$-smooth connected nonlinear planar projective billiards whose flow admits a non-constant integral that is a rational $0$-homogeneous function of the velocity (with coefficients depending on the position): they are called rationally $0$-homogeneously integrable. It was shown that: 1) the underlying curve is a conic; 2) the minimal degree of integral is equal to two, if the billiard is defined by a dual pencil of conics; 3) otherwise it can be arbitrary even number. In the present paper we classify piecewise $C^4$-smooth rationally $0$-homogeneously integrable projective billiards. Unexpectedly, we show that such a billiard associated to a dual pencil of conics may have integral of minimal degree 2, 4, or 12. For the proof of main results we prove dual results for the so-called dual multibilliards.

math.DS↗

arXiv2023

On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlevé 3 equation

B.Josephson (Nobel Prize, 1973) predicted tunnelling effect for a system (called Josephson junction) of two superconductors separated by a narrow dielectric: existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$, $A$, $ω$. We study its rotation number $ρ(B,A;ω)$ as a function of parameters. The three-dimensional phase-lock areas are the level sets $L_r:=\{ρ=r\}$ with non-empty interiors; they exist for $r\in\mathbb Z$ (Buchstaber, Karpov, Tertychnyi). For every fixed $ω>0$ and $r\in\mathbb Z$ the planar slice $L_r\cap(\mathbb R^2_{B,A}\times\{ω\})$ is a garland of domains going vertically to infinity and separated by points; those separating points for which $A\neq0$ are called constrictions. In a joint paper by Yu.Bibilo and the author, it was shown that 1) at each constriction the rescaled abscissa $\ell:=\frac Bω$ is equal to $ρ$; 2) the family of constrictions with given $\ell\in\mathbb Z$ is an analytic submanifold $Constr_\ell$ in $(\mathbb R^2_+)_{a,s}$, $a=ω^{-1}$, $s=\frac Aω$. Here we show that the limit points of $Constr_\ell$ are $β_{\ell,k}=(0,s_{\ell,k})$, where $s_{\ell,k}>0$ are zeros of the Bessel function $J_\ell(s)$, and it lands at them regularly. Known numerical pictures show that high components of $Int(L_r)$ look similar. In his paper with Bibilo, the author introduced a candidate to the self-similarity map between neighbor components: the Poincaré map of the dynamical isomonodromic foliation governed by Painlevé 3 equation. Whenever well-defined, it preserves $ρ$. We show that the Poincaré map is well-defined on a neighborhood of the plane $\{ a=0\}\subset\mathbb R^2_{\ell,a}\times(\mathbb R_+)_s$, and it sends $β_{\ell,k}$ to $β_{\ell,k+1}$ for integer $\ell$.

math.DS↗

arXiv2022

On rationally integrable planar dual and projective billiards

A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then the billiard is an ellipse. It was studied by many mathematicians, including H.Poritsky, M.Bialy, S.Bolotin, A.Mironov, V.Kaloshin, A.Sorrentino and others. In the paper we study its following generalized dual version stated by S.Tabachnikov. Consider a closed smooth strictly convex curve $γ\subset\mathbb{RP}^2$ equipped with a dual billiard structure: a family of non-trivial projective involutions acting on its projective tangent lines and fixing the tangency points. Suppose that its outer neighborhood admits a foliation by closed curves (including $γ$) such that the involution of each tangent line permutes its intersection points with every leaf. Then $γ$ and the leaves are conics forming a pencil. We prove positive answer in the case, when the curve $γ$ is $C^4$-smooth and the foliation admits a rational first integral. To this end, we show that each $C^4$-smooth germ $γ$ of planar curve carrying a rationally integrable dual billiard structure is a conic and classify all the rationally integrable dual billiards on (punctured) conic. They include the dual billiards induced by pencils of conics, two infinite series of exotic dual billiards and five more exotic ones.

math.DS↗

arXiv2022

On infinitely many foliations by caustics in strictly convex open billiards

Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve $C$ whose tangent lines are reflected by the billiard to lines tangent to $C$. The famous Birkhoff Conjecture states that the only strictly convex billiards with a foliation by closed caustics near the boundary are ellipses. By Lazutkin's theorem, there always exists a Cantor family of closed caustics approaching the boundary. In the present paper we deal with an open billiard, whose boundary is a strictly convex embedded (non-closed) curve $γ$. We prove that there exists a domain $U$ adjacent to $γ$ from the convex side and a $C^\infty$-smooth foliation of $U\cupγ$ whose leaves are $γ$ and (non-closed) caustics of the billiard. This generalizes a previous result by R.Melrose, which yields existence of a germ of foliation as above at a boundary point. We show that there exists a continuum of above foliations by caustics whose germs at each point in $γ$ are pairwise different. We prove a more general version of this statement in the cases, when $γ$ is just an arc, and also when both $γ$ and the caustics are immersed curves. It also applies to a billiard bounded by a closed strictly convex curve $γ$ and yields infinitely many "immersed" foliations by immersed caustics. For the proof of the above results, we state and prove their analogue for a special class of area-preserving maps generalizing billiard reflections: the so-called $C^{\infty}$-lifted strongly billiard-like maps. We also prove a series of results on conjugacy of billiard maps near the boundary for open curves of the above type.

math.DS↗

arXiv2022

On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$ (abscissa), $A$ (ordinate), $ω$ (frequency). We study its rotation number $ρ(B,A;ω)$ as a function of $(B,A)$ with fixed $ω$. The phase-lock areas are the level sets $L_r:=\{ρ=r\}$ with non-empty interiors; they exist for $r\in\mathbb Z$ (Buchstaber, Karpov, Tertychnyi). Each $L_r$ is an infinite chain of domains going vertically to infinity and separated by points called constrictions (expect for those with $A=0$). We show that: 1) all the constrictions in $L_r$ lie in its axis $\{ B=ωr\}$ (confirming a conjecture of Tertychnyi, Kleptsyn, Filimonov, Schurov); 2) each constriction is positive: some its punctured neighborhood in the vertical line lies in $\operatorname{Int}(L_r)$ (confirming another conjecture). We first prove deformability of each constriction to another one, with arbitrarily small $ω$, of the same $ρ$, $\ell:=\frac Bω$ and type (positive or not), using equivalent description of model by linear systems of differential equations on $\bar{\mathbb C}$ (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlevé 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with $ρ\neq\ell$, or of non-positive type) with a given $\ell$ for small $ω$ is proved by slow-fast methods. In Section 6 we present applications of results and elaborated methods and open problems.

math.DS↗

arXiv2020

On curves with Poritsky property

For a given closed convex planar curve $γ$ with smooth boundary and a given $p>0$, the string construction yields a family of nested billiards $Γ_p$ for which $γ$ is a caustic. The action of the corresponding reflections $T_p$ on the tangent lines to $γ$ induces their actions on the tangency points: a family of string diffeomorphisms $\mathcal T_p:γ\toγ$. We say that $γ$ has string Poritsky property, if it admits a parameter $t$ (called Poritsky string length) in which all the transformations $\mathcal T_p$ with small $p$ are translations $t\mapsto t+c_p$. These definitions also make sense for germs of curves $γ$. Poritsky property is closely related to the famous Birkhoff Conjecture. It is classically known that each conic has string Poritsky property. In 1950 H.Poritsky proved the converse: each germ of planar curve with Poritsky property is a conic. In the present paper we extend this Poritsky's result to germs of curves on simply connected complete surfaces with Riemannian metric of constant curvature and to outer billiards on all these surfaces. In the general case of curves with Poritsky property on any two-dimensional surface with Riemannian metric we prove the two following results: 1) the Poritsky string length coincides with Lazutkin parameter, introduced by V.F.Lazutkin in 1973, up to additive and multiplicative constants; 2) a germ of $C^5$-smooth curve with Poritsky property is uniquely determined by its 4-th jet. In the Euclidean case the latter statement follows from the above-mentioned Poritsky's result.

math.DS↗

arXiv2019

On spectral curves and complexified boundaries of the phase-lock areas in a model of Josephson junction

The paper deals with a three-parameter family of special double confluent Heun equations that was introduced and studied by V.M.Buchstaber and S.I.Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are $l,λ,μ\in\mathbb R$. Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a polynomial solution. They have shown that for $μ\neq0$ this happens exactly when $l\in\mathbb N$ and the parameters $(λ,μ)$ lie on an algebraic curve $Γ_l\subset\mathbb C^2_{(λ,μ)}$ called the $l$-th spectral curve and defined as zero locus of determinant of a remarkable three-diagonal $l\times l$-matrix. They studied the real spectral curves and obtained important results with applications to phase-lock areas in model of Josephson junction, which is a family of dynamical systems on 2-torus. In the present paper we prove irreducibility of complex spectral curves. We also calculate their genera for $l\leqslant20$ and present a conjecture on general genus formula. We apply the irreducibility result to the phase-lock areas, which are those level sets of the rotation number function $ρ$ on the parameter space of the above-mentioned family of dynamical systems that have non-empty interiors. The family of their boundaries is a countable union of analytic surfaces. We show that, unexpectedly, its complexification is a complex analytic subset consisting of just four irreducible components, and we describe them. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits and a partial positive result towards its confirmation.

math.DS↗

arXiv2019

On commuting billiards in higher-dimensional spaces of constant curvature

We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due to Sergei Tabachnikov. The main result is deduced from the classical theorem due to Marcel Berger saying that in higher dimensions only quadrics may have caustics. We also prove versions of Berger's theorem and the main result for billiards in spaces of constant curvature: space forms.

math.DS↗

arXiv2019

On polynomially integrable Birkhoff billiards on surfaces of constant curvature

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result to billiards with piecewise-smooth and not necessarily convex boundary on arbitrary two-dimensional surface of constant curvature: plane, sphere, Lobachevsky (hyperbolic) plane; each of them being modeled as a plane or a (pseudo-) sphere in $\mathbb R^3$ equipped with appropriate quadratic form. Namely, we show that a billiard is polynomially integrable, if and only if its boundary is a union of confocal conical arcs and appropriate geodesic segments. We also present a complexification of these results. These are joint results of Mikhail Bialy, Andrey Mironov and the author. The proof is split into two parts. The first part is given by Bialy and Mironov in their two joint papers. They considered the tautological projection of the boundary to $\mathbb{RP}^2$ and studied its orthogonal-polar dual curve, which is piecewise algebraic, by S.V.Bolotin's theorem. By their arguments and another Bolotin's theorem, it suffices to show that each non-linear complex irreducible component of the dual curve is a conic. They have proved that all its singularities and inflection points (if any) lie in the projectivized zero locus of the corresponding quadratic form on $\mathbb C^3$. The present paper provides the second part of the proof: we show that each above irreducible component is a conic and finish the solution of the Algebraic Birkhoff Conjecture in constant curvature.

math.DS↗