The systolic ratio of a contact form α on the three-sphere is the quantitywhere T min (α) is the minimal period of closed Reeb orbits on (S 3 , α). A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that ρ sys ≤ 1 in a neighbourhood of the space of Zoll contact forms on S 3 , with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a local middle-dimensional non-squeezing theorem, and a sharp systolic inequality for Finsler metrics on the two-sphere which are close to Zoll ones. Our second main result is that ρ sys is unbounded from above on the space of tight contact forms on S 3 .
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.Comment: 73 pages, some minor corrections made. To appear in Transactions of
the American Mathematical Societ
Abstract. We consider Reeb dynamics on the 3-sphere associated to a tight contact form. Our main result gives necessary and sufficient conditions for a periodic Reeb orbit to bound a disk-like global section for the Reeb flow, when the contact form is assumed to be non-degenerate.
Abstract. We introduce a local version of contact homology for an isolated periodic orbit of the Reeb flow and prove that its rank is uniformly bounded for isolated iterations. Several applications are obtained, including a generalization of Gromoll-Meyer's theorem on the existence of infinitely many simple periodic orbits, resonance relations and conditions for the existence of nonhyperbolic periodic orbits.
We give necessary and sufficient conditions for a closed connected co-orientable contact 3-manifold (M, ξ) to be a standard lens space based on assumptions on the Reeb flow associated to a defining contact form. Our methods also provide rational global surfaces of section for nondegenerate Reeb flows on (L(p, q), ξ std ) with prescribed binding orbits.
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