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

Misha Bialy*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)139-153
Number of pages15
JournalCommentarii Mathematici Helvetici
Issue number1
StatePublished - 2015


  • Conjugate points
  • Convex billiards
  • Minimal geodesics
  • Minimal orbits


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