2015/04/30 by A. Abbondandolo, B. Bramham, U. L. Hryniewicz +1 · 2 citations
Mathematics · #math.SG #math.DG
paper · pdf · doi:10.1007/s00222-017-0755-z
published as Invent. Math. 211 (2018), 687-778 · 78 pages, fully revised version, main results unchanged
arxiv created 2017/05/23 · arxiv updated 2019/12/18
The systolic ratio of a contact form α on the three-sphere is the quantity ρsys(α) = \fracTmin(α)2vol(S3,α\wedge dα), where Tmin(α) is the minimal period of closed Reeb orbits on (S3,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 ρsys≤ 1 in a neighbourhood of the space of Zoll contact forms on S3, 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 ρsys is unbounded from above on the space of tight contact forms on S3.