The near-boundary Birkhoff conjecture

Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.

Manuscripts (2)

Continuous Phase Foliations Create Analytic Caustic Collars

September 24, 2026 · source · cite as \cite{OAI:Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026}

For a smooth strictly convex planar billiard of positive curvature, a continuous foliation of a full grazing annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics. The physical-collar rigidity theorem proved in the companion article then implies that the table is an ellipse.

Rigidity of Smooth Billiards with a Continuous Caustic Collar

September 24, 2026 · source · cite as \cite{OAI:Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026}

We prove that a smooth strictly convex planar billiard with positive curvature is an ellipse whenever a full neighborhood of its boundary is continuously foliated by smooth closed convex caustics. No differentiability across the leaves is assumed. This resolves the near-boundary Birkhoff conjecture in the stated smooth class.

Result 147 · Dynamical systems and ergodic theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.