Sharp systolic inequalities for Reeb flows on the three-sphere

The systolic ratio of a contact form on the three-sphere is the quantity rho(sys)(alpha) = T-min(alpha)(2)/vol(S-3, alpha boolean AND d alpha), where is the minimal period of closed Reeb orbits on . 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 in a neighbourhood of the space of Zoll contact forms on , 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 is unbounded from above on the space of tight contact forms on .