Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces

Abstract

We prove that a $C^2$-generic Riemannian metric on a closed surface has either an elliptic closed geodesic or an Anosov geodesic flow. As a consequence, we prove the $C^2$-stability conjecture for Riemannian geodesic flows of closed surfaces: a $C^2$-structurally stable Riemannian geodesic flow of a closed surface is Anosov. In order to prove these statements, we establish a general result that may be of independent interest and provides sufficient conditions for a Reeb flow of a closed 3-manifold to be Anosov.

34 pages, 6 figures; final version, as published

References in corpus (2)

• Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds
• Contact three-manifolds with exactly two simple Reeb orbits

Cited by in corpus (1)

• Surfaces of section for geodesic flows of closed surfaces