On polynomially integrable Birkhoff billiards on surfaces of constant curvature

Abstract

The polynomial version of the 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 the 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 an 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 ${\mathbb{R}}^3$ equipped with an 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 ${\mathbb{CP}}^2$ 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.