Abstract
In this talk I shall discuss Birkhoff-Poritsky conjecture for centrally-symmetric C^2-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 C^0-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a C^1-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve 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 initiated by the author for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiards in the ellipse. Based on joint work with Andrey E. Mironov (Novosibirsk).