Orderable contact structures on Liouville-fillable contact manifolds

We study the existence of positive loops of contactomorphisms on a Liouville-fillable contact manifold $(\Sigma, \xi = \ker (\alpha))$. Previous results (see [6]) 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 $\xi^\prime$ on $\Sigma$ which admits no positive loop at all. Further, $\xi^\prime$ can be chosen to agree with $\xi$ on the complement of a Darboux ball.

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Published 13 April 2015