Weinstein conjecture for general contact manifolds

Establish the Weinstein conjecture in full generality by proving that every Reeb vector field on any closed contact manifold, in any dimension, admits at least one periodic orbit.

Background

The Weinstein conjecture is a central problem in contact dynamics asserting the existence of a periodic orbit for every Reeb flow on a closed contact manifold. It is proved in various cases (e.g., all closed 3-manifolds via Taubes’s work), but remains unresolved in full generality for higher dimensions. The survey discusses how contact homology and related tools provide proofs in many settings and give quantitative refinements, while noting that the general case is not yet settled.