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arXiv · 2109.10704

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

Gonzalo Contreras·Marco Mazzucchelli
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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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Gonzalo Contreras, Marco Mazzucchelli. 2024-05-16. Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces. https://doi.org/10.1215/00127094-2023-0010

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