Surfaces of section for geodesic flows of closed surfaces
Gonzalo Contreras, Gerhard Knieper, Marco Mazzucchelli, Benjamin H. Schulz · HAL (Le Centre pour la Communication Scientifique Directe) · 2022
We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $Σ$ that we construct are either Birkhoff sections, meaning that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in $\partialΣ$ as limit sets of the orbits of the geodesic flow that do not return to $Σ$. In order to prove these theorems, we provide a study of configurations of simple closed geodesics of closed orientable Riemannian surfaces, which may have independent interest. Our arguments are based on the curve shortening flow.