In this talk, based on joint work with Gonzalo Contreras, I will briefly sketch the proof of the existence of global surfaces of section for the Reeb flows of closed 3-manifolds satisfying a condition à la Kupka-Smale: non-degeneracy of the closed Reeb orbits, and transversality of the stable and unstable manifolds of the hyperbolic closed Reeb orbits. I will then present an application of this theorem to hyperbolic Reeb dynamics: a Reeb flow on a closed 3-manifold is Anosov if and only if the closure of the subspace of closed Reeb orbits is hyperbolic and the Kupka-Smale transversality condition holds. This result implies the validity of the C2 stability conjecture for Riemannian geodesic flows of closed surfaces: any such geodesic flow that is C2 structurally stable within the class of Riemannian geodesic flows must be Anosov.
This video is part of the Institute for Advanced Study‘s Symplectic geometry seminar.
