Sharp systolic inequalities for Reeb flows on the three-sphere

Abstract

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

78 pages, fully revised version, main results unchanged

References in corpus (1)

• Chern-Weil theory and the group of strict contactomorphisms

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