Genus zero global surfaces of section for Reeb flows and a result of Birkhoff (Q6057868)

Summary: 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.