# Torsion contact forms in three dimensions have two or infinitely many Reeb orbits

## TL;DR

This paper proves that in three-dimensional contact geometry, nondegenerate contact forms with torsion first Chern class have either two or infinitely many Reeb orbits, with specific results depending on the manifold's properties.

## Contribution

It establishes a dichotomy for the number of Reeb orbits in three dimensions based on torsion properties of the contact structure's first Chern class.

## Key findings

- Nondegenerate contact forms with torsion first Chern class have either two or infinitely many Reeb orbits.
- If the manifold is not a three-sphere or lens space, there are infinitely many simple Reeb orbits under these conditions.
- Non-torsion contact structures guarantee at least four simple Reeb orbits for nondegenerate contact forms.

## Abstract

We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it follows that under the above assumptions, there are infinitely many simple Reeb orbits if the three-manifold is not the three-sphere or a lens space. We also show that for non-torsion contact structures, every nondegenerate contact form has at least four simple Reeb orbits.