# On polynomially integrable Birkhoff billiards on surfaces of constant curvature

## TL;DR

This paper proves that polynomially integrable billiards on surfaces of constant curvature are ellipses or unions of confocal conical arcs, solving the algebraic Birkhoff Conjecture for these geometries.

## Contribution

It extends the algebraic Birkhoff Conjecture to surfaces of constant curvature, characterizing polynomially integrable billiards as conic-based boundaries.

## Key findings

- Polynomially integrable billiards are ellipses or confocal conical arcs.
- The results apply to plane, sphere, and hyperbolic surfaces.
- The paper completes the proof of the algebraic Birkhoff Conjecture in constant curvature.

## Abstract

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.