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