Birkhoff–Poritsky conjecture (full generality)

Prove that the only integrable Birkhoff billiards in the plane are ellipses, thereby establishing the Birkhoff–Poritsky conjecture in full generality.

Background

The classical Birkhoff billiard is a planar billiard inside a strictly convex smooth closed curve. Birkhoff and Poritsky conjectured that integrability forces the boundary to be an ellipse. While many partial and local results are known, a complete proof for all smooth convex tables remains elusive.

The survey reviews several approaches (algebraic, variational/total integrability, KAM/local) and presents rigidity theorems under additional assumptions (e.g., central symmetry or total integrability on specific regions), but none settle the conjecture in full generality.

References

Birkhoff and Poritsky conjectured that the only integrable Birkhoff billiards in the plane are ellipses. It was explicitly formulated in [Poritsky] and since then it remains unsolved in full generality.

— Integrable Billiards and Related Topics  (2510.03790 - Bialy et al., 4 Oct 2025) in Introduction

Among bounded, strictly convex planar domains with $\mathcal{C}2$ boundary, the only integrable billiard tables are the ellipses.

— Analytic rigidity and symbolic dynamics for two-centre billiards  (2609.02310 - Baranzini et al., 2 Sep 2026) in Conjecture (Birkhoff–Poritsky), Introduction; Section “Around the Birkhoff Conjecture”