On Two-Dimensional Polynomially Integrable Billiards on Surfaces of Constant Curvature

Abstract

The polynomial version of Birkhoff Conjecture on integrable billiards on complete simply connected surfaces of constant curvature (plane, sphere, hyperbolic plane) was first stated, studied and solved in a particular case by Sergei Bolotin in 1990-1992. Here we present a complete solution of the polynomial version of Birkhoff Conjecture. Namely we show that every polynomially integrable real bounded planar billiard with C 2 -smooth connected boundary is an ellipse. We extend this result to billiards with piecewise-smooth and not necessarily convex boundary on arbitrary two-dimensional simply connected complete surface of constant curvature: plane, sphere, Lobachevsky-Poincaré (hyperbolic) plane; each of them being modeled as a plane or a (pseudo-) sphere in 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 in two papers by Bialy and Mironov (in Euclidean and non-Euclidean cases respectively). Their geometric construction reduced the Polynomial Birkhoff Conjecture to a purely algebro-geometric problem to show that an irreducible algebraic curve in CP 2 with certain properties is a conic. They have shown that its singular and inflection points lie in the complex light conic of the above-mentioned quadratic form. In the present paper we solve the above algebro-geometric problem completely.

On polynomially integrable planar outer billiards and curves with symmetry property

“…Its complete proof for billiards on any surface of constant curvature was recently obtained as a result of the three following papers: two joint papers by M.Bialy and A.E.Mironov [6,8]; a very recent paper [14] (see also its short version [15]) of the first author of the present article. For more detailed surveys on Birkhoff Conjecture and its algebraic version see [6,14,18] and references therein.…”

“…Remark 1.8 The second part of the proof of the Algebraic Birkhoff Conjecture given in [14,15] uses results of [6,8] and techniques elaborated in the present article and in the previous paper of the first author [13].…”

“…He wishes to thank Andrey Mironov and Misha Bialy for their invitations and hospitality and both institutions for their hospitality and support. We are grateful to the referee for a very careful reading of the paper and very helpful remarks, which also led to major improvement of the first author's paper [14].…”

On the integrability of Birkhoff billiards

“…The relationship between this notion of integrability and Birkhoff conjecture (algebraic Birkhoff conjecture) has been studied and has led to interesting results [37,38]. Recently, using previous results of [37], Glutsyuk [39] proved the algebraic Birkhoff conjecture.…”

Open problems, questions and challenges in finite- dimensional integrable systems

“…The above formulated conjectures and problems have algebraic versions, and much progress has been made in this area recently. For inner billiards, the polynomial Birkhoff conjecture was proved in [18,90], including the cases of non-zero constant curvature, and for outer billiards in the affine plane -in [91]. The next problem is a generalization of the previous one.…”