Abstract
Let
α be a contact form on a connected closed three-manifold
Σ. The systolic ratio of
α is defined as
ρsys(α):=Vol(α)1Tmin(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
S1-action on
Σ. We prove that the set of Zoll contact forms on
Σ 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
ρsys(α)≤ρsys(α∗) for every
α∈U, with equality if and only if
α is Zoll.