Sharp systolic inequalities for Reeb flows on the three-sphere

By Alberto Abbondandolo, Barney Bramham, Umberto L. Hryniewicz, Pedro A. S. Salomão. 2015

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The systolic ratio of a contact form $$\alpha $$α on the three-sphere is the quantity $$\begin{aligned} \rho _{\mathrm {sys}}(\alpha ) = \frac{T_{\min }(\alpha )^2}{\mathrm {vol}(S^3,\alpha \wedge d\alpha )}, \end{aligned}$$ρsys(α)=Tmin(α)2vol(S3,α∧dα),where $$T_{\min }(\alpha )$$Tmin(α) is the minimal period of closed Reeb orbits on $$(S^3,\alpha )$$(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 $$\rho _{\mathrm {sys}}\le 1$$ρsys≤1 in a neighbourhood of the space of Zoll contact forms on $$S^3$$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 $$\rho _{\mathrm {sys}}$$ρsys is unbounded from above on the space of tight contact forms on $$S^3$$S3.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3661b7e52d7bd7da532514fedcd9a5290fc9d469