The near-boundary Birkhoff conjecture: billiards integrable near the boundary are ellipses

Theorem 1.1 (phase foliations): if a $T$-invariant open annulus containing $(\mathbb R/L\mathbb Z)\times(0,\eta)$ in the billiard phase space is homeomorphically foliated by individually invariant essential curves, then $\Omega$ is an ellipse. The companion proves the same conclusion from a continuous collar of smooth closed convex caustics, via analytic regularity of the boundary and complex continuation. Only continuity across leaves is assumed. Not covered: integrability only on a Cantor set, rational integrability, or integrability away from the boundary; the full Birkhoff conjecture (global integrability as typically defined) is reduced to but not identical with this case.

Theorem 1.1 (phase foliations): if a $T$-invariant open annulus containing $(\mathbb R/L\mathbb Z)\times(0,\eta)$ in the billiard phase space is homeomorphically foliated by individually invariant essential curves, then $\Omega$ is an ellipse. The companion proves the same conclusion from a continuous collar of smooth closed convex caustics, via analytic regularity of the boundary and complex continuation. Only continuity across leaves is assumed. Not covered: integrability only on a Cantor set, rational integrability, or integrability away from the boundary; the full Birkhoff conjecture (global integrability as typically defined) is reduced to but not identical with this case.