Example：10.1021/acsami.1c06204 or Chem. Rev., 2007, 107, 2411-2502
Unknotted Reeb orbits and nicely embedded holomorphic curves Journal of Symplectic Geometry (IF0.707), Pub Date : 2020-01-01, DOI: 10.4310/jsg.2020.v18.n1.a2 Alexandru Cioba, Chris Wendl
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.