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

TL;DR

This paper provides conditions under which a collection of periodic orbits in Reeb flows on 3-manifolds bounds a genus zero global surface of section, generalizing Birkhoff's classical result for geodesic flows.

Contribution

It establishes generic conditions involving linking and Conley-Zehnder indices for the existence of genus zero global surfaces of section in Reeb flows.

Findings

Conditions are generically necessary for genus zero global surfaces of section.

Linking and Conley-Zehnder index assumptions are crucial.

Reproves and extends Birkhoff's classical result for geodesic flows.

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 reprove and generalize a classical result of Birkhoff on the existence of annulus-like global surfaces of section for geodesic flows on positively curved two-spheres.