[Paper Review] A local systolic-diastolic inequality in contact and symplectic geometry

Gabriele Benedetti, Jungsoo Kang|arXiv (Cornell University)|Jan 2, 2018

Geometric and Algebraic Topology3 citations

TL;DR

This paper establishes a local systolic-diastolic inequality for Zoll contact and odd-symplectic forms on closed 3-manifolds, proving that in a $C^3$-neighborhood of a Zoll contact form $\alpha_*$, the contact volume is bounded between $t_\Sigma T_{\min}(\alpha)^2$ and $t_\Sigma T_{\max}(\alpha)^2$, with equality if and only if $\alpha$ is Zoll. The result generalizes to odd-symplectic forms via a volume-action polynomial relation, recovering known systolic inequalities and extending them to magnetic geodesics.

ABSTRACT

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.

Motivation & Objective

• To establish a local systolic-diastolic inequality for Zoll contact forms on closed 3-manifolds, relating contact volume to minimal and maximal Reeb orbit periods.
• To extend the systolic-diastolic framework from contact to odd-symplectic forms of arbitrary odd dimension, generalizing both contact volume and Calabi invariants.
• To define a generalized volume and action for odd-symplectic forms, and prove that Zoll odd-symplectic forms satisfy a polynomial equation linking these invariants.
• To recover known systolic inequalities—such as those for Hamiltonian isotopies near identity and local contact systolic bounds—as special cases of the proposed inequality.
• To explore applications of the framework to magnetic geodesics on closed 3-manifolds.

Proposed method

• Define the volume of an odd-symplectic form $\Omega$ as a generalization of both contact volume and the Calabi invariant of Hamiltonian functions.
• Introduce the action of closed characteristics of $\Omega$, generalizing both Reeb orbit periods and fixed point actions of Hamiltonian diffeomorphisms.
• Prove that for a Zoll odd-symplectic form $\Omega$, the volume and action satisfy a polynomial equation, with coefficients depending on the torsion order $t_\Sigma$.
• Establish a $C^3$-neighborhood $\mathcal{U}$ around a Zoll contact form $\alpha_*$ such that $t_\Sigma T_{\min}(\alpha)^2 \leq \mathrm{Volume}(\alpha) \leq t_\Sigma T_{\max}(\alpha)^2$ for all $\alpha \in \mathcal{U}$.
• Show that equality holds in the inequality if and only if $\alpha$ is Zoll, using the structure of the Reeb flow and $S^1$-action.
• Use the framework to recover the local systolic-diastolic inequality for Hamiltonian isotopies $C^1$-close to the identity on closed symplectic manifolds.

Experimental results

Research questions

• RQ1 Does a local systolic-diastolic inequality hold for Zoll contact forms in a $C^3$-neighborhood, with bounds involving the torsion order and minimal/maximal Reeb orbit periods?
• RQ2 Can the concept of volume and action be generalized from contact forms to odd-symplectic forms on odd-dimensional manifolds, unifying contact and Hamiltonian invariants?
• RQ3 Do Zoll odd-symplectic forms satisfy a polynomial relation between their volume and the actions of their closed characteristics?
• RQ4 Can the proposed inequality recover known local systolic bounds for Hamiltonian isotopies near the identity on symplectic manifolds?
• RQ5 What are the implications of this framework for the dynamics of magnetic geodesics on closed 3-manifolds?

Key findings

• For every Zoll contact form $\alpha_*$ on a closed 3-manifold $\Sigma$, there exists a $C^3$-neighborhood $\mathcal{U}$ such that $t_\Sigma T_{\min}(\alpha)^2 \leq \mathrm{Volume}(\alpha) \leq t_\Sigma T_{\max}(\alpha)^2$ holds for all $\alpha \in \mathcal{U}$.
• Equality in the volume bounds holds if and only if $\alpha$ is Zoll, characterizing Zoll forms via extremality of the volume-action ratio.
• The generalized volume and action for odd-symplectic forms satisfy a polynomial equation when the form is Zoll, extending the systolic-diastolic principle to odd dimensions.
• The framework recovers the local systolic-diastolic inequality for Hamiltonian isotopies $C^1$-close to the identity on closed symplectic manifolds as a special case.
• The results provide a new geometric invariant theory for magnetic geodesics, linking their action spectrum to the topology of the underlying 3-manifold via $t_\Sigma$.
• The paper establishes the equality case of a conjectural local systolic-diastolic inequality for odd-symplectic forms, verified in multiple cases.