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 ${\alpha }_{\ast }$ on $\Sigma$ admits a $C^3$-neighbourhood $\mathcal{U}$ in the space of contact forms such that \[ t_ΣT_{\min}(α)^2\leq \mathrm{Volume}(α)\leq t_ΣT_{\max}(α)^2,\qquad \forall\,α\in\mathcal U, \] 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.

The contents of the article is now divided into three articles available at arXiv:1902.01249, arXiv:1902.01261, and arXiv:1902.01262