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 $\rho_{\mathrm {sys}}(\alpha) := \frac {1}{\mathrm {Vol}(\alpha)} T_{\mathrm {min}} (\alpha)^2$, where $T_{\mathrm {min}} (\alpha)$ 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 $\rho_{\mathrm {sys}}(\alpha) \leq \rho_{\mathrm {sys}}(\alpha_{\ast})$ for every $\alpha \in \mathcal U$, with equality if and only if $\alpha$ is Zoll.