The singular Weinstein conjecture

Abstract

In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in [MiO], 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 ${\mathbb{R}}^{+}$-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.

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References in corpus (15)

• Symplectic and Poisson geometry on b-manifolds
• The Seiberg-Witten equations and the Weinstein conjecture
• The plastikstufe - a generalization of the overtwisted disk to higher dimensions
• Toric actions on b-symplectic manifolds
• Contact Topology and Hydrodynamics
• The contact geometry of the restricted 3-body problem
• Oscillatory orbits in the restricted elliptic planar three body problem
• Examples of integrable and non-integrable systems on singular symplectic manifolds
• Action-angle variables and a KAM theorem for b-Poisson manifolds
• Universality of Euler flows and flexibility of Reeb embeddings
• Desingularizing $b^m$-symplectic structures
• Trapped Reeb orbits do not imply periodic ones
• Euler flows and singular geometric structures
• Periodic orbits in virtually contact structures
• Rabinowitz Floer homology for tentacular Hamiltonians