We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $\Sigma $ that we construct are either Birkhoff sections, which means that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in $\partial \Sigma $ as limit sets of the orbits of the geodesic flow that do not return to $\Sigma $. 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.

We study curve-shortening flow for twisted curves in $\mathbb {R}^3$ (that is, curves with nowhere vanishing curvature $\kappa $ and torsion $\tau $) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type-II singularity which admits sequences along which ${\tau }/{\kappa ^2} \to \infty $. This contrasts strongly with Altschuler’s planarity theorem, which shows that ${\tau }/{\kappa } \to 0$ along any essential blow-up sequence.