$3-2-1$ foliations for Reeb flows on the tight 3-sphere

TL;DR

This paper investigates special foliations called $3-2-1$ foliations in Reeb flows on the tight 3-sphere, providing conditions for their existence and an example Hamiltonian supporting such structures.

Contribution

It introduces the concept of $3-2-1$ foliations for Reeb flows, establishes criteria for their existence, and constructs a specific Hamiltonian example on $R^4$.

Findings

Sufficient conditions for $3-2-1$ foliation existence.

Existence of a Hamiltonian on $R^4$ with $3-2-1$ foliations.

Description of the foliation structure with three binding orbits.

Abstract

We study the existence of $3-2-1$ foliations adapted to Reeb flows on the tight $3$-sphere. These foliations admit precisely three binding orbits whose Conley-Zehnder indices are $3$, $2$, and $1$, respectively. All regular leaves are disks and annuli asymptotic to the binding orbits. Our main results provide sufficient conditions for the existence of $3-2-1$ foliations with prescribed binding orbits. We also exhibit a concrete Hamiltonian on $\mathbb{R}^4$ admitting $3-2-1$ foliations when restricted to suitable energy levels.