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arXiv · 1710.06193

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

Alberto Abbondandolo·Barney Bramham·Umberto Hryniewicz·Pedro Salomão
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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 $\alpha_n$, $n\geq 2$, 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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Alberto Abbondandolo, Barney Bramham, Umberto Hryniewicz, Pedro Salomão. 2017-10-17. Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere. https://doi.org/10.1112/s0010437x18007558

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