A local systolic-diastolic inequality in contact and symplectic geometry

Abstract

Let $\Sigma$ be a connected closed three-manifold, and let $t_\Sigma$ be the order of the torsion subgroup of $H_1(\Sigma;\mathbb Z)$. For a contact form $\alpha$ on $\Sigma$, we denote by $\mathrm{Volume}(\alpha)$ the contact volume of $\alpha$, and by $T_{\min}(\alpha)$ and $T_{\max}(\alpha)$ the minimal period and the maximal period of prime periodic orbits of the Reeb flow of $\alpha$ respectively. We say that $\alpha$ is Zoll if its Reeb flow generates a free $S^1$-action on $\Sigma$. We prove that every Zoll contact form $\alpha_*$ on $\Sigma$ admits a $C^3$-neighbourhood $\mathcal U$ in the space of contact forms such that \[ t_\Sigma T_{\min}(\alpha)^2\leq \mathrm{Volume}(\alpha)\leq t_\Sigma T_{\max}(\alpha)^2,\qquad \forall\,\alpha\in\mathcal U, \] and any of the equalities holds if and only if $\alpha$ is Zoll. We extend the above picture to odd-symplectic forms $\Omega$ on $\Sigma$ of arbitrary odd dimension. We define the volume of $\Omega$, which generalises both the contact volume and the Calabi invariant of Hamiltonian functions, and the action of closed characteristics of $\Omega$, which generalises both the period of periodic Reeb orbits and the action of fixed points of Hamiltonian diffeomorphisms. We say that $\Omega$ is Zoll if its characteristics are the orbits of a free $S^1$-action on $\Sigma$. We prove that the volume and the action of a Zoll odd-symplectic form satisfy a certain polynomial equation. This builds the equality case of a conjectural local systolic-diastolic inequality for odd-symplectic forms, which we establish in some cases. This inequality recovers the inequality between the minimal action and the Calabi invariant of Hamiltonian isotopies $C^1$-close to the identity on a closed symplectic manifold, as well as the local contact systolic-diastolic inequality above. Finally, applications to magnetic geodesics are discussed.