Genus zero global surfaces of section for Reeb flows and a result of Birkhoff

Abstract

We exhibit sufficient conditions for a finite collection of periodic orbits of a Reeb flow on a closed $3$-manifold to bound a positive global surface of section with genus zero. These conditions turn out to be $C^{\infty }$-generically necessary. Moreover, they involve linking assumptions on periodic orbits with Conley–Zehnder index ranging in a finite set determined by the ambient contact geometry. As an application, we re-prove and generalize a classical result of Birkhoff on the existence of annulus-like global surfaces of section for geodesic flows on positively curved $2$-spheres.