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arXiv · 1609.01660

Unknotted Reeb orbits and nicely embedded holomorphic curves

Alexandru Cioba·Chris Wendl
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Abstract

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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Alexandru Cioba, Chris Wendl. 2016-09-06. Unknotted Reeb orbits and nicely embedded holomorphic curves. https://arxiv.org/abs/1609.01660

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