On the $C^2$-local systolic optimality of Zoll odd-symplectic forms

TL;DR

This paper proves that Zoll odd-symplectic forms are locally optimal for the systolic ratio, generalizing known results from Zoll contact forms and establishing local inequalities near Zoll structures.

Contribution

It introduces a normal form theorem for near-Zoll odd-symplectic forms and demonstrates their local systolic optimality, extending previous contact form results.

Findings

Zoll odd-symplectic forms are local maximizers of the systolic ratio.

Normal form theorem for forms close to Zoll forms.

Application to hypersurfaces near Zoll in symplectic manifolds.

Abstract

An odd-symplectic form is a closed and maximally non-degenerate $2$-form on a compact odd-dimensional manifold. It describes the dynamics of an autonomous Hamiltonian system on a regular energy level. It is called Zoll if the induced dynamics is a free circle action, up to a global time reparameterization. This article establishes a normal form theorem for odd-symplectic forms close to a Zoll one and cohomologous to it, which is then used to prove that Zoll odd-symplectic forms are the local maximizers of the associated systolic ratio. This generalizes the known systolic optimality of Zoll contact forms in the $C^3$-topology. As an application, local systolic inequalities are established in symplectic manifolds for hypersurfaces close to Zoll ones. In particular, this applies to certain non-exact twisted cotangent bundles of manifolds of dimension greater than two.