Math Conjectures

Weinstein conjecture

OPENLandmarkConjectureProposed 1978 · Canonical special case

Canonical statement

Let M2n−1M^{2n-1} be a closed smooth manifold and let α\alpha be a contact form, so α∧(dα)n−1\alpha\wedge(d\alpha)^{n-1} is nowhere zero. The Reeb vector field RαR_\alpha, uniquely defined by α(Rα)=1\alpha(R_\alpha)=1 and dα(Rα,⋅)=0d\alpha(R_\alpha,\cdot)=0, has a nonconstant periodic orbit.
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Let \(M^{2n-1}\) be a closed smooth manifold and let \(\alpha\) be a contact form, so \(\alpha\wedge(d\alpha)^{n-1}\) is nowhere zero. The Reeb vector field \(R_\alpha\), uniquely defined by \(\alpha(R_\alpha)=1\) and \(d\alpha(R_\alpha,\cdot)=0\), has a nonconstant periodic orbit.
GEO-TOP-015:[Weinstein1979PeriodicOrbits][Taubes2007Weinstein3D][Ding2025WeinsteinSurvey]
Geometry and topologyDynamical systems

Notes

The Weinstein conjecture predicts that Reeb dynamics on a closed contact manifold always exhibits periodicity: if α\alpha is a contact form on a closed manifold M2n−1M^{2n-1}, the associated Reeb vector field RαR_\alpha must have a nonconstant periodic orbit. Weinstein raised the question in 1978, prompted by an analysis of the hypotheses in Rabinowitz's periodic-orbit theorems for Hamiltonian systems [Weinstein1979PeriodicOrbits].

In dimension three the conjecture is a theorem: Taubes proved it for every closed contact 33-manifold using the Seiberg–Witten equations [Taubes2007Weinstein3D], following a long line of special cases. In higher dimensions many classes of contact manifolds are known to satisfy the conjecture, and the surrounding techniques, from variational methods to holomorphic curves, are surveyed in [Ding2025WeinsteinSurvey].

The general case in dimension at least 55 remains open: for an arbitrary contact form on an arbitrary closed manifold of dimension 55 or more, no current argument produces a closed Reeb orbit, and a resolution would require methods that work without any restriction on the contact manifold.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.