A local contact systolic inequality in dimension three

Let α be a contact form on a connected closed three-manifold 6. The systolic ratio of α is defined as (equation presented), where Tmin(α) and Vol (α) denote the minimal period of periodic Reeb orbits and the contact volume. The form α is said to be Zoll if its Reeb flow generates a free S1-action on 6. We prove that the set of Zoll contact forms on 6 locally maximises the systolic ratio in the C3-topology. More precisely, we show that every Zoll form α∗> admits a C3-neighbourhood U in the space of contact forms such that ρ sys (α) ≤ ρ sys (α ∗) for every α ∈ U, with equality if and only if α is Zoll.