Fast finite-energy planes in symplectizations and applications

TL;DR

This paper introduces fast finite-energy planes in symplectizations to construct open book decompositions with disk-like pages, providing new tools for analyzing Hamiltonian dynamics on convex energy levels.

Contribution

It defines fast finite-energy planes in symplectizations and applies them to create open book decompositions with disk-like pages for tight contact structures.

Findings

Open books with disk-like pages serve as global surfaces of section.

A Hamiltonian orbit bounds a disk-like global surface iff it is unknotted with self-linking -1.

Characterization of boundary orbits for global surfaces in convex domains.

Abstract

We define the notion of fast finite-energy planes in the symplectization of a closed 3-dimensional energy level $M$ of contact type. We use them to construct special open book decompositions of $M$ when the contact structure is tight and induced by a (non-degenerate) dynamically convex contact form. The obtained open books have disk-like pages that are global surfaces of section for the Hamiltonian dynamics. Let $S \subset \R^4$ be the boundary of a smooth, strictly convex, non-degenerate and bounded domain. We show that a necessary and sufficient condition for a closed Hamiltonian orbit $P\subset S$ to be the boundary of a disk-like global surface of section for the Hamiltonian dynamics is that $P$ is unknotted and has self-linking number -1.