A local systolic-diastolic inequality in contact and symplectic geometry
英文摘要：
Let $Σ$ be a connected closed three-manifold, and let $t_Σ$ be the order of the torsion subgroup of $H_1(Σ;\mathbb Z)$. For a contact form $α$ on $Σ$, we denote by $\mathrm{Volume}(α)$ the contact volume of $α$, and by $T_{\min}(α)$ and $T_{\max}(α)$ the minimal period and the maximal period of prime periodic orbits of the Reeb flow of $α$ respectively. We say that $α$ is Zoll if its Reeb flow generates a free $S^1$-action on $Σ$. We prove that every Zoll contact form $α_*$ on $Σ$ 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 $α$ is Zoll. We extend the above picture to odd-symplectic forms $Ω$ on $Σ$ of arbitrary odd dimension. We define the volume of $Ω$, which generalises both the contact volume and the Calabi invariant of Hamiltonian functions, and the action of closed characteristics of $Ω$, which generalises both the period of periodic Reeb orbits and the action of fixed points of Hamiltonian diffeomorphisms. We say that $Ω$ is Zoll if its characteristics are the orbits of a free $S^1$-action on $Σ$. 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.
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