# 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.
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-03357630
Contributor : Marco Mazzucchelli Connect in order to contact the contributor
Submitted on : Tuesday, September 28, 2021 - 10:49:25 PM
Last modification on : Wednesday, September 29, 2021 - 3:29:57 AM

### Identifiers

• HAL Id : ensl-03357630, version 1
• ARXIV : 2109.10704

### Citation

Marco Mazzucchelli, Gonzalo Contreras. Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces. 2021. ⟨ensl-03357630⟩

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