Contact forms with large systolic ratio in arbitrary dimensions

TL;DR

The paper proves that any co-orientable contact structure on a closed manifold can have a contact form with arbitrarily large systolic ratio by extending plug constructions to higher dimensions, generalizing previous 3D results.

Contribution

It extends the plug construction method to arbitrary dimensions and demonstrates the existence of contact forms with arbitrarily large systolic ratios in higher-dimensional contact manifolds.

Findings

Existence of contact forms with arbitrarily large systolic ratio in all dimensions.

Extension of plug construction to higher dimensions using symplectic ball maps.

Application of Giroux's Liouville open books to modify contact forms.

Abstract

If a contact form on a (2n+1)-dimensional closed contact manifold admits closed Reeb orbits, then its systolic ration is defined to be the quotient of (n+1)th power of the shortest period of Reeb orbits by the contact volume. We prove that every co-orientable contact structure on any closed contact manifold admits a contact form with arbitrarily large systolic ratio. This statement generalizes the recent result of Abbondandolo et al. in dimension three to higher dimensions. We extend the plug construction of Abbondandolo et. al. to any dimension, by means of generalizing the hamiltonian disc maps studied by the authors to the symplectic ball of any dimension. The plug is a mapping torus and it is equipped with a special contact form so that one can use it to modify a given contact form if the Reeb flow leads to a circle bundle on a "large" portion of the given contact manifold. Inserting the plug sucks up the contact volume while the minimal period remains the same. Following the ideas of Abbondandolo et al. and using Giroux's theory of Liouville open books, we show that any co-orientable contact structure is defined by a contact form, which is suitable to be modified via inserting plugs.