Systems of global surfaces of section for dynamically convex Reeb flows on the 3-sphere

TL;DR

This paper characterizes the closed Reeb orbits on the 3-sphere that bound disk-like global surfaces of section, revealing their organization into open book decompositions and applying these results to Hamiltonian dynamics.

Contribution

It provides a characterization of Reeb orbits bounding global surfaces of section without genericity assumptions, and demonstrates their organization into open book decompositions.

Findings

Reeb orbits bounding disk-like global surfaces of section are characterized.

Global surfaces of section form open book decompositions.

New global surfaces of section are constructed for convex Hamiltonian energy levels.

Abstract

We characterize which closed Reeb orbits of a dynamically convex contact form on the 3-sphere bound disk-like global surfaces of section for the Reeb flow, without any genericity assumptions. We show that these global surfaces of section come in families, organized as open book decompositions. As an application we obtain new global surfaces of section for the Hamiltonian dynamics on strictly convex 3-dimensional energy levels.