The singular Weinstein conjecture

In this article, we investigate Reeb dynamics onbm-contact manifolds, previously introduced in [MO], which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$.
In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in \cite{MO}, 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 $b^m$-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 $b^m$-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 $b^m$-Reeb orbits away from the critical set. The study of $b^m$-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 $b^3$-contact form that admits an infinite number of periodic orbits at the critical set.
Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA AcademiaPrize 2016. C ́edric Oms is supported by an AFR-Ph.D. grant of FNR - Luxembourg National Research Fund. Eva Mi-randa and C ́edric Oms are partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER)and reference number 2017SGR932 (AGAUR). Eva Miranda was supported by aChaire d’Excellenceof theFondationSciences Math ́ematiques de Pariswhen this project started and this work has been supported by a public grant overseenby the French National Research Agency (ANR) as part of the“Investissements d’Avenir”program (reference: ANR-10-LABX-0098). This material is based upon work supported by the National Science Foundation under Grant No.DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, Califor-nia, during the Fall 2018 semester.
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