From curve shortening to flat link stability and Birkhoff sections of geodesic flows

TL;DR

The paper develops a strategy to translate curve shortening flow dynamics into global facts about geodesic flows on closed surfaces. Using Angenent’s Morse theory in flat knot types and a robust $C^0$-perturbation analysis, it establishes (i) $C^0$-stability for flat links of closed geodesics, (ii) forcing results that guarantee infinitely many geodesics intersecting a given one (and a sphere analog with two geodesics), and (iii) the unconditional existence of Birkhoff sections for geodesic flows on any closed orientable surface. The approach replaces Floer-theoretic methods with curvature-controlled flow arguments, local homology in flat knot types, and model metrics (focusing caps) to derive geometric and topological consequences for geodesic dynamics. The results yield new rigidity under perturbations, constructive forcing phenomena, and a framework to obtain global surfaces of section, with potential connections to Reeb dynamics and entropy considerations. Overall, the work broadens the toolkit for understanding geodesic flows through curve shortening flow and Morse-theoretic techniques on flat knot types.

Abstract

We employ the curve shortening flow to establish three new results on the dynamics of geodesic flows of closed Riemannian surfaces. The first one is the stability, under $C^0$-small perturbations of the Riemannian metric, of certain flat links of closed geodesics. The second one is a forced existence theorem for closed connected orientable Riemannian surfaces: for surfaces of positive genus, the existence of a contractible simple closed geodesic $γ$ forces the existence of infinitely many closed geodesics intersecting $γ$ in every primitive free homotopy class of loops; for the 2-sphere, the existence of two disjoint simple closed geodesics forces the existence of a third one intersecting both. The final result asserts the existence of Birkhoff sections for the geodesic flow of any closed connected orientable Riemannian surface.

Figures (8)

• Figure 1: A $\rho$-subloop of $\gamma$, with a filling $D$ of area less than or equal to $\rho$.
• Figure 2: A loop $\alpha\in\Delta(\zeta)$ with a tangency to $\zeta$, and a loop $\beta\in\Delta(\zeta)$ with a self-tangency. We stress that, while the tangencies depicted are without crossing, tangencies can be topologically transverse.
• Figure 3: The family of meridians (in gray) in a 2-sphere of revolution, intersecting the equator $\lambda$ orthogonally.
• Figure 5: The cone bundle $C$.
• Figure 6: A loop $\gamma\in\mathcal{U}_2$. Here, $\mathcal{U}$ is the connected component of the loop $\beta$.

Theorems & Definitions (69)

• Definition 1.1
• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Remark 1.2
• Remark 2.1
• Lemma 2.2: Angenent:2005aa, Lemmas 5.3-4
• Lemma 2.3: Angenent:2005aa, Lemmas 3.3 and 6.2
• Remark 2.4