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

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, for every integer $...

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Published inCompositio mathematica Vol. 154; no. 12; pp. 2643 - 2680
Main Authors Abbondandolo, Alberto, Bramham, Barney, Hryniewicz, Umberto L., Salomão, Pedro A. S.
Format Journal Article
LanguageEnglish
French
Published London Cambridge University Press 01.12.2018
Subjects
Convexity
Orbits
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Summary: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, for every integer $n\geqslant 2$ , a tight contact form with systolic ratio arbitrarily close to $n$ and with suitable bounds on the mean rotation number of all the closed orbits of the induced Reeb flow.
ISSN:0010-437X
1570-5846
DOI:10.1112/S0010437X18007558