Global surfaces of section for Reeb flows in dimension three and beyond

TL;DR

This paper surveys recent advances in constructing global surfaces of section for Reeb flows in three dimensions, highlighting applications to geometry, topology, and celestial mechanics using symplectic topology methods.

Contribution

It compiles recent developments and applications of global surfaces of section for Reeb flows, emphasizing new methods and results in symplectic topology.

Findings

Existence of closed geodesics established

Sharp systolic inequalities derived

Applications to celestial mechanics demonstrated

Abstract

We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are also presented.