A local contact systolic inequality in dimension three

Abstract

Let $α$ be a contact form on a connected closed three-manifold $Σ$. The systolic ratio of $α$ is defined as $ρ_{\mathrm{sys}}(α):=\tfrac{1}{\mathrm{Vol}(α)}T_{\min}(α)^2$, where $T_{\min}(α)$ and $\mathrm{Vol}(α)$ denote the minimal period of periodic Reeb orbits and the contact volume. The form $α$ is said to be Zoll if its Reeb flow generates a free $S^1$-action on $Σ$. We prove that the set of Zoll contact forms on $Σ$ locally maximises the systolic ratio in the $C^3$-topology. More precisely, we show that every Zoll form $α_*$ admits a $C^3$-neighbourhood $\mathcal U$ in the space of contact forms such that, for every $α\in\mathcal U$, there holds $ρ_{\mathrm{sys}}(α)\leq ρ_{\mathrm{sys}}(α_*)$ with equality if and only if $α$ is Zoll.