Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces - Archive ouverte HAL Access content directly
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## Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces

(1) ,
1
Marco Mazzucchelli
Gonzalo Contreras
• Function : Author

#### 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.

#### Domains

Mathematics [math]

### Dates and versions

ensl-03357630 , version 1 (28-09-2021)

### Identifiers

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

### Cite

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

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