The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables

In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric C²-smooth convex planar billiards. We assume that the domain A between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by C⁰-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a C¹-smooth foliation by convex caustics of rotation numbers in the interval $(0; 1/4]$, then the boundary curve is an ellipse. In the language of first integrals one can assert that if the billiard inside a centrally-symmetric C²-smooth convex curve γ admits a C¹-smooth first integral with non-vanishing gradient on A, then the curve γ is an ellipse. The main ingredients of the proof are (1) the non-standard generating function for convex billiards; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach for rigidity results that was invented by the first named author for circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.