Contact forms with large systolic ratio in dimension three

TL;DR

This paper proves that any co-orientable contact structure on a closed three-manifold can be represented by a contact form with arbitrarily large systolic ratio, challenging the universality of certain geometric inequalities.

Contribution

It demonstrates that large systolic ratios are achievable for all co-orientable contact structures on closed three-manifolds, highlighting the topological nature of systolic inequalities.

Findings

Existence of contact forms with arbitrarily large systolic ratio

Systolic inequalities are not solely contact-topological phenomena

Counterexamples to universal systolic bounds in contact geometry

Abstract

The systolic ratio of a contact form on a closed three-manifold is the quotient of the square of the shortest period of closed Reeb orbits by the contact volume. We show that every co-orientable contact structure on any closed three-manifold is defined by a contact form with arbitrarily large systolic ratio. This shows that the many existing systolic inequalities in Finsler and Riemannian geometry are not purely contact-topological phenomena.