# On a local systolic inequality for odd-symplectic forms

### Gabriele Benedetti

Universität Heidelberg, Germany### Jungsoo Kang

Seoul National University, Republic of Korea

## Abstract

The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases.

Let $Ω$ be an odd-symplectic form on an oriented closed manifold $Σ$ of odd dimension. We say that $Ω$ is Zoll if the trajectories of the flow given by $Ω$ are the orbits of a free $S_{1}$-action. After defining the volume of $Ω$ and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided $Ω$ is Zoll. This builds the equality case of a conjectural systolic inequality for odd-symplectic forms close to a Zoll one. We prove the conjecture when the $S_{1}$-action yields a flat $S_{1}$-bundle or when $Ω$ is quasi-autonomous. Together with previous work [BK19a], this establishes the conjecture in dimension three.

This new inequality recovers the local contact systolic inequality (recently proved in [AB19]) as well as the inequality between the minimal action and the Calabi invariant for Hamiltonian isotopies $C_{1}$-close to the identity on a closed symplectic manifold. Applications to the study of periodic magnetic geodesics on closed orientable surfaces is given in the companion paper [BK19b].

## Cite this article

Gabriele Benedetti, Jungsoo Kang, On a local systolic inequality for odd-symplectic forms. Port. Math. 76 (2019), no. 3/4, pp. 327–394

DOI 10.4171/PM/2039