Convex billiards and a theorem by E. Hopf (Q1320990)

We consider billiards in convex compact planar domains. Our main result states that if the phase space of the billiard ball map is foliated by not null-homotopic continuous invariant curves then the domain is circular. The result solves in part a Birkhoff's conjecture that only for elliptic domains billiards are integrable. We give a variational version of this result: the only billiards without conjugate points are circular billiards. The main tool of the paper is to apply the ``discrete version'' of E. Hopf's method invented for Riemannian manifolds without conjugate points.