A local contact systolic inequality in dimension three

Abstract

Let $\alpha$ be a contact form on a connected closed three-manifold $\Sigma$. The systolic ratio of $\alpha$ is defined as ${\rho }_{\mathrm{sys}}(\alpha ):=\frac{1}{\mathrm{Vol}(\alpha )}{T}_{\min }(\alpha {)}^2$, 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 ${\alpha }_{\ast }$ admits a $C^3$-neighbourhood $\mathcal{U}$ in the space of contact forms such that ${\rho }_{\mathrm{sys}}(\alpha )\le {\rho }_{\mathrm{sys}}({\alpha }_{\ast })$ for every $\alpha \in \mathcal{U}$, with equality if and only if $\alpha$ is Zoll.