  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.

Symplectic Geometry;Differential Geometry