Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere

2017/10/17 by Alberto Abbondandolo, Barney Bramham, Umberto Hryniewicz +1 · 1 citation
Mathematics · #math.SG

paper · pdf · doi:10.1112/s0010437x18007558

published as Compositio Math. 154 (2018) 2643-2680 · 44 pages

arxiv created 2017/10/17 · arxiv updated 2019/12/11

Abstract

We construct a dynamically convex contact form on the three-sphere whose systolic ratio is arbitrarily close to 2. This example is related to a conjecture of Viterbo, whose validity would imply that the systolic ratio of a convex contact form does not exceed 1. We also construct a sequence of tight contact forms α_n, n≥ 2, with systolic ratio arbitrarily close to n and suitable bounds on the mean rotation number of all the closed orbits of the induced Reeb flow.

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