Contact Forms With Large Systolic Ratio In Arbitrary Dimensions

If a contact form on a (2n + 1)-dimensional closed contact manifold admits closed Reeb orbits, then its systolic ratio 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-oriented 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 symplectic balls of any dimension. The plug we consider is a mapping torus and it is equipped with a special contact form, which leads to arbitrarily small contact volume, but with the smallest period of the Reeb flow bounded away from zero. Following the ideas of Abbondandolo et al. and using Giroux's theory of Liouville open books, we construct a special contact form on a given contact manifold whose Reeb flow leads to a circle bundle on a 'large' portion of the given contact manifold. Such a contact form can be modified on the region of periodic Reeb flow by inserting plugs so that the contact volume is sucked up while the minimal period remains the same.