Friday May 3rd, 2024 | |
| Time: | 2:30 PM |
| Title: | C^2 STRUCTURALLY STABLE RIEMANNIAN GEODESIC FLOWS OF CLOSED SURFACES ARE ANOSOV |
| Speaker: | Marco Mazzucchelli, ENS Lyon |
| Abstract: | |
| A celebrated claim of Poincaré asserts that any positively-curved Riemannian 2-sphere has a (possibly degenerate) elliptic closed geodesic. This claim has been confirmed generically by Contreras and Oliveira, without requirements on the curvature: a C^2 generic Riemannian metric on the 2-sphere has an elliptic closed geodesic. In this talk, I will present a generalization of this result to arbitrary closed surfaces: a C^2 generic Riemannian metric on a closed surface has either an elliptic closed geodesic or an Anosov geodesic flow. A consequence of this statement is a confirmation of the C^2 stability conjecture for Riemannian geodesic flows of closed surfaces: any such geodesic flow that is C^2 structurally stable within Riemannian geodesic flows must be Anosov. The proof is based on a new characterization of Anosov Reeb flows of closed contact 3-manifolds. This is joint work with Gonzalo Contreras. | |