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 .

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.

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.

For a Riemannian metric g on the two-sphere, let ℓ min (g) be the length of the shortest closed geodesic and ℓ max (g) be the length of the longest simple closed geodesic. We prove that if the curvature of g is positive and sufficiently pinched, then the sharp systolic inequalitieshold, and each of these two inequalities is an equality if and only if the metric g is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and symplectic geometry.