A note on Reeb dynamics on the tight 3-sphere

TL;DR

This paper characterizes when a tight contact form on the 3-sphere has exactly two simple closed Reeb orbits, linking it to contact homology and comparing it to irrational ellipsoids in 4-space.

Contribution

It establishes a precise condition relating the number of Reeb orbits to the vanishing of the contact homology differential and describes their dynamical properties.

Findings

Exactly two simple closed Reeb orbits occur if and only if the contact homology differential vanishes.

The Reeb orbits' Floquet multipliers and Conley-Zehnder indices match those of an irrational ellipsoid.

The result provides a dynamical characterization of certain tight contact forms on the 3-sphere.

Abstract

We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space.