Open books and the Weinstein conjecture

TL;DR

This paper proves the existence of certain periodic orbits in contact structures supported by open books, using advanced symplectic topology techniques involving Stein manifolds and cobordisms.

Contribution

It introduces a higher-dimensional generalization of Eliashberg's theorem, linking open book decompositions with the Weinstein conjecture in symplectic topology.

Findings

Existence of contractible periodic Reeb orbits in specified contact structures

Extension of Eliashberg's theorem to higher dimensions

Connection between open book bindings and symplectic cobordisms

Abstract

We show the existence of a contractible periodic Reeb orbit for any contact structure supported by an open book whose binding can be realised as a hypersurface of restricted contact type in a subcritical Stein manifold. A key ingredient in the proof is a higher-dimensional version of Eliashberg's theorem about symplectic cobordisms from a contact manifold to a symplectic fibration.