## 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 W be an odd-symplectic form on an oriented closed manifold S of odd dimension. We say that W is Zoll if the trajectories of the flow given by W are the orbits of a free S1-action. After defining the volume of W and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided W 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 S1-action yields a flat S1-bundle or when

W 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 C1-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].

Original language | English |
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Pages (from-to) | 327-394 |

Journal | Portugaliae Mathematica |

Volume | 76 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2019 |

Externally published | Yes |

### Funding

Therefore, WHdf and WH′df have the same systole and diastole. The local systolic-diastolic inequality holds for WH′df due to Proposition 7.16, and hence also for WHdf and W. r Acknowledgments. This work is part of a project in the Collaborative Research Center TRR 191 – Symplectic Structures in Geometry, Algebra and Dynamics funded by the DFG. It was initiated when the authors worked together at the University of Münster and partially carried out while J. K. was a‰liated with the Ruhr-University Bochum. We thank Peter Albers, Kai Zehmisch, and the University of Münster for having provided an inspiring academic environment. We are also grateful to Michael Kapovich for the characterisation contained in Lemma 4.6. We are indebted to the referees for the valuable suggestions on a previous version of the manuscript. G. B. would like to express his gratitude to Hans-Bert Rademacher and the whole Di¤erential Geometry group at the University of Leipzig. J. K. was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-01. This work is part of a project in the Collaborative Research Center TRR 191 – Symplectic Structures in Geometry, Algebra and Dynamics funded by the DFG. It was initiated when the authors worked together at the University of Münster and partially carried out while J. K. was a‰liated with the Ruhr-University Bochum. We thank Peter Albers, Kai Zehmisch, and the University of Munster for having provided an inspiring academic environment. We are also grateful to Michael Kapovich for the characterisation contained in Lemma 4.6. We are indebted to the referees for the valuable suggestions on a previous version of the manuscript. G. B. would like to express his gratitude to Hans-Bert Rademacher and the whole Di¤erential Geometry group at the University of Leipzig. J. K. was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-01.

Funders | Funder number |
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University of Munster | |

Deutsche Forschungsgemeinschaft | |

Samsung Science and Technology Foundation | SSTF-BA1801-01 |