From curve shortening to flat link stability and Birkhoff sections of geodesic flows
Marcelo R. R. Alves, Marco Mazzucchelli · Journal de l’École polytechnique — Mathématiques · 2025
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.