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

TL;DR

This paper uses the curve shortening flow to prove new stability and existence results for closed geodesics and Birkhoff sections on closed Riemannian surfaces, advancing understanding of geodesic flow dynamics.

Contribution

It introduces novel stability results for flat links of geodesics, a forced existence theorem for multiple geodesics, and establishes Birkhoff sections for all closed orientable surfaces.

Findings

Flat links of geodesics are stable under small metric perturbations.

Existence of infinitely many geodesics intersecting a given one on surfaces of positive genus.

Birkhoff sections exist for geodesic flows on all closed orientable surfaces.

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 $\gamma$ forces the existence of infinitely many closed geodesics intersecting $\gamma$ 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.