A local contact systolic inequality in dimension three

Let α be a contact form on a connected closed three-manifold Σ . The systolic ratio of α is defined as ρsys(α) := 1/Vol(α) Tₘᵢₙ (α)^2 , 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 S^1 -action on Σ . We prove that the set of Zoll contact forms on Σ locally maximises the systolic ratio in the C^3 -topology. More precisely, we show that every Zoll form α_{} admits a C^3 -neighbourhood U in the space of contact forms such that ρsys(α) ≤ ρsys(α_{}) for every α ∈ U , with equality if and only if α is Zoll.