The singular Weinstein conjecture
In this article, we investigate Reeb dynamics on bm-contact manifolds, previously introduced in [37], which are contact away from a hypersurface Z but satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The se...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2072/531265 |
| Acceso en línea: | http://hdl.handle.net/2072/531265 |
| Access Level: | acceso abierto |
| Palabra clave: | b-Contact forms b-Symplectic geometry Overtwisted contact structures Periodic orbits Reeb dynamics Singular Weinstein conjecture 51 |
| Sumario: | In this article, we investigate Reeb dynamics on bm-contact manifolds, previously introduced in [37], which are contact away from a hypersurface Z but satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact bm-contact manifolds without periodic Reeb orbits outside Z are provided. Furthermore, we prove that in dimension 3, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the bm-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are R+-invariant in the open ends, obtaining as a corollary the existence of periodic bm-Reeb orbits away from the critical set. The study of bm-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three body problem are described by the Reeb vector field of a b3-contact form that admits an infinite number of periodic orbits at the critical set. © 2021 The Author(s) |
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