From one Reeb orbit to two

TL;DR

This paper proves that every closed three-manifold with a contact form has at least two embedded Reeb orbits, and explores properties of their symplectic actions, revealing new relationships with contact volume and embedded contact homology.

Contribution

It establishes the existence of at least two embedded Reeb orbits for any contact form on a closed three-manifold and links their symplectic actions to contact volume and homology.

Findings

Every contact form has at least two embedded Reeb orbits.

If finitely many embedded Reeb orbits exist, their actions are not all multiples of a single number.

With exactly two embedded Reeb orbits, their actions' product is bounded by the contact volume.

Abstract

We show that every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits. We also show that if there are only finitely many embedded Reeb orbits, then their symplectic actions are not all integer multiples of a single real number; and if there are exactly two embedded Reeb orbits, then the product of their symplectic actions is less than or equal to the contact volume of the manifold. The proofs use a relation between the contact volume and the asymptotics of the amount of symplectic action needed to represent certain classes in embedded contact homology, recently proved by the authors and V. Ramos.