Fast finite-energy planes in symplectizations and applications

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 $S\subset {\mathbb{R}}^4$ 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.

73 pages, some minor corrections made. To appear in Transactions of the American Mathematical Society

References in corpus (1)

• On the existence of disk-like global sections for Reeb flows on the tight 3-sphere

Cited by in corpus (5)

• Torsion contact forms in three dimensions have two or infinitely many Reeb orbits
• A dynamical characterization of universally tight lens spaces
• A Poincaré-Birkhoff theorem for tight Reeb flows on $S^3$
• Global properties of tight Reeb flows with applications to Finsler geodesic flows on $S^2$
• Computational symplectic topology and symmetric orbits in the restricted three-body problem