Surfaces of section for geodesic flows of closed surfaces

TL;DR

This work addresses the problem of constructing surfaces of section for geodesic flows on closed orientable surfaces by leveraging the curve shortening flow. The authors establish existence results for Birkhoff sections under various geometric hypotheses, including absence or nondegeneracy of contractible simple closed geodesics without conjugate points, and they extend the construction to more general settings using Fried surgery on Birkhoff annuli to produce complete systems of closed geodesics and controlled trapping regions. The paper delivers topologically explicit sections: for genus $G eq0$, a genus-one surface of section with $8G-4$ boundary components; for $S^2$, Birkhoff annuli suffice; in Kupka–Smale-type generic situations, existence of Birkhoff sections is ensured, while non-generic cases admit a broken-book decomposition. The results sharpen previous approaches by avoiding holomorphic-curve techniques and by providing quantitative control over the topology of the sections and their boundary dynamics, with potential implications for the qualitative study of geodesic flows and Reeb dynamics on 3-manifolds.

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.

Figures (12)

• Figure 1: An open convex geodesic polygon $B$ that is the complement of two simple closed geodesics of a 2-torus of revolution.
• Figure 2: The geodesic triangle $T(v,r)$.
• Figure 3: The geodesic triangle $T=T(\dot\gamma_1(t_0),r)$ contained in $A$.
• Figure 4: The geodesic triangle $T'$.
• Figure 5: The point $\gamma_2(0)$ intersecting the geodesic arc $\zeta_{\ell_1}|_{[0,\rho(\ell_1))}$.

Theorems & Definitions (52)

• Theorem A
• Corollary B
• proof
• Theorem C: Bangert
• Remark 1.1
• Theorem D
• Theorem E
• Lemma 2.1
• Remark 2.2
• proof : Proof of Lemma \ref{['l:convergence_scg']}