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

TL;DR

This paper provides quantitative bounds on the measure of minimal orbits in billiards and geodesic flows, offering sharp estimates that connect to the classical E.Hopf rigidity phenomenon.

Contribution

It introduces explicit measure estimates for minimal orbits in convex billiards and conformally flat tori, extending the understanding of E.Hopf rigidity quantitatively.

Findings

Sharp upper bounds for the measure of minimal orbits

Recovery of E.Hopf rigidity when the measure is maximal

Extension of bounds to convex billiards and conformally flat tori

Abstract

In this paper we show that in some cases the E.Hopf rigidity phenomenon admits 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 bounds for Burago-Ivanov theorem.