Abstract
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 $\mathcal M$ swept by minimal orbits. These estimates are sharp, i.e. if $\mathcal M$ 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.
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