A counterexample to the singular Weinstein conjecture (Q6639731)

The authors consider Reeb vector fields and their dynamical properties on b-contact manifolds. They develop constructions in three dimensions that illustrate important properties of escape orbits as well as singular periodic orbits. These are immediately relevant to questions about the singular versions of the Weinstein conjecture and the Hamiltonian Seifert conjecture.\N\NThe constructions presented here lead to counterexamples of the conjectures stated in [\textit{E. Miranda} and \textit{C. Oms}, Adv. Math. 389, 41 p. (2021; Zbl 1479.53081)]. Indeed the construction identifies b-contact manifolds with no singular periodic orbits outside the critical set \(Z\) of the b-contact form.\N\NA new conjecture -- a generalized Weinstein conjecture -- is developed based on work following from these constructions. The generalized version says that every Reeb vector field of a contact manifold (even possibly a singular one) should have at least one periodic orbit or a generalized singular periodic orbit away from the critical set \(Z\).