Unknotted Reeb orbits and nicely embedded holomorphic curves

We exhibit a distinctly low-dimensional dynamical obstruction to the existence of Liouville cobordisms: for any contact $3$-manifold admitting an exact symplectic cobordism to the tight $3$-sphere, every nondegenerate contact form admits an embedded Reeb orbit that is unknotted and has self-linking number $-1$. The same is true moreover for any contact structure on a closed $3$-manifold that is reducible. Our results generalize an earlier theorem of Hofer–Wysocki–Zehnder for the $3$-sphere, but use somewhat newer techniques: the main idea is to exploit the intersection theory of punctured holomorphic curves in order to understand the compactification of the space of so-called “nicely embedded” curves in symplectic cobordisms. In the process, we prove a local adjunction formula for holomorphic annuli breaking along a Reeb orbit, which may be of independent interest.

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Received 22 September 2016

Accepted 15 November 2018

Published 25 March 2020