The singular Weinstein conjecture

TL;DR

This paper explores Reeb dynamics on $b^m$-contact manifolds, revealing new phenomena such as the existence of infinitely many periodic orbits in certain cases, and extends classical conjectures to these generalized structures.

Contribution

It introduces the study of $b^m$-contact manifolds in relation to the Weinstein conjecture, providing examples without periodic orbits outside the critical set and extending Hofer's methods.

Findings

Existence of compact $b^m$-contact manifolds without periodic Reeb orbits outside $Z$.

In dimension 3, infinitely many periodic orbits on the critical set if it is compact.

Existence of traps for the $b^m$-Reeb flow in any dimension.

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.