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arXiv · 2204.11977

Surfaces of section for geodesic flows of closed surfaces

Gonzalo Contreras·Gerhard Knieper·Marco Mazzucchelli·Benjamin H. Schulz
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Abstract

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.

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Gonzalo Contreras, Gerhard Knieper, Marco Mazzucchelli, Benjamin H. Schulz. 2024-10-08. Surfaces of section for geodesic flows of closed surfaces. https://doi.org/10.1017/etds.2024.86

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