Orderable Contact Structures on Liouville-fillable Contact Manifolds

TL;DR

This paper investigates positive loops of contactomorphisms on Liouville-fillable contact manifolds, showing that in dimensions seven and higher, there exist fillable contact structures without positive loops, contrasting previous results.

Contribution

It demonstrates the existence of Liouville-fillable contact structures without positive loops in dimensions seven and higher, expanding understanding of contactomorphism dynamics.

Findings

Existence of Liouville-fillable contact structures with no positive loops in dimension ≥7.

Construction of such structures that agree with given structures outside a Darboux ball.

Contrasts with previous results showing positive loops in many fillable contact manifolds.

Abstract

We study the existence of positive loops of contactomorphisms on a Liouville-fillable contact manifold $(\Sigma,\xi=\ker(\alpha))$. Previous results show that a large class of Liouville-fillable contact manifolds admit contractible positive loops. In contrast, we show that for any Liouville-fillable $(\Sigma,\alpha)$ with $\dim(\Sigma) \geq 7$, there exists a Liouville-fillable contact structure $\xi'$ on $\Sigma$ which admits no positive loop at all. Further, $\xi'$ can be chosen to agree with $\xi$ on the complement of a Darboux ball.