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.