Effective bounds in E. Hopf rigidity for billiards and geodesic flows

In this paper we show that in some cases the E. Hopf rigidity phenomenon allows quantitative interpretation. More precisely, we estimate from above the measure of the set M swept by minimal orbits. These estimates are sharp, i.e. if M occupies the whole phase space we recover the E. Hopf rigidity. We give these estimates in two cases: the first is the case of convex billiards in the plane, sphere or hyperbolic plane. The second is the case of conformally flat Riemannian metrics on a torus. It seems to be a challenging question to understand such a quantitative bound for Burago-Ivanov theorem.