The Weinstein conjecture predicts that the Reeb vector field of any contact form on a closed contact manifold has at least one closed (periodic) orbit. It is proved in real dimension three (Taubes, via Seiberg-Witten Floer theory) and in many special geometric classes in all dimensions, but in full generality it remains open for closed contact manifolds of dimension 2n+12n+1 with n≥2n\geq 2. Ornea-Verbitsky (2021) settle the Sasakian case while noting that the general conjecture is still open; the problem in its open regime is therefore: prove, or refute by an explicit counterexample, that every contact form on every closed contact manifold of dimension at least five admits a closed Reeb orbit.

Background

A contact form on a (2n+1)(2n+1)-dimensional manifold MM is a 1-form λ\lambda such that λ∧(dλ)n\lambda\wedge(d\lambda)^n is a volume form; the Reeb vector field RλR_\lambda is the unique vector field satisfying λ(Rλ)=1\lambda(R_\lambda)=1 and iRλdλ=0i_{R_\lambda}d\lambda=0. Its flow is the basic dynamical system associated with the contact structure ker⁡λ\ker\lambda, and a closed Reeb orbit is a periodic trajectory of this flow. The conjecture goes back to Weinstein (1979), who asked whether every Reeb field on a closed (compact, without boundary) contact manifold has a closed orbit; it grew out of Rabinowitz's periodic-orbit theorems for hypersurfaces of contact type, the level sets of a Hamiltonian that carry a natural contact form.

The classical positive results are: Viterbo (1987), proving the conjecture for contact-type hypersurfaces in Euclidean space R2n\mathbb{R}^{2n} (in particular for compact star-shaped hypersurfaces) in all dimensions; Hofer (1993), proving it for the three-sphere and for overtwisted contact structures in dimension three (a contact structure is overtwisted if it contains a copy of a fixed explicit model disk, the overtwisted disk, and is called tight otherwise) via pseudoholomorphic curves in the symplectization (the symplectization of (M,λ)(M,\lambda) is R×M\mathbb{R}\times M with symplectic form d(etλ)d(e^t\lambda)); and Taubes (2007), proving it for all closed contact three-manifolds using Seiberg-Witten Floer theory (a gauge-theoretic invariant of three-manifolds built from solutions of the Seiberg-Witten equations). In dimensions 2n+1≥52n+1\geq 5 the conjecture is known only for restricted classes: Hofer-Viterbo (1992) proved it in the presence of holomorphic spheres (when the relevant π2\pi_2 is nonzero); Albers-Hofer (2009) proved it for PS-overtwisted manifolds, i.e. those containing a Plastikstufe (Niederkrüger's higher-dimensional analogue of an overtwisted disk); and Rukimbira (1995) proved that a compact Sasakian manifold (an odd-dimensional manifold whose cone is Kähler, equivalently a contact metric manifold with the Reeb field generating a transverse holomorphic structure) of dimension 2n+12n+1 has at least n+1n+1 closed Reeb orbits, with the sharper dichotomy that the number of closed orbits is either infinite or equal to the sum of the Betti numbers of the Kähler orbifold (a space locally modelled on quotients of Euclidean space by finite groups) obtained as an S1S^1-quotient — the result for which Ornea-Verbitsky (2021) gave a new proof via approximation by quasi-regular Sasakian structures (those whose Reeb orbits are all closed). Beyond these classes, no method is known that works for an arbitrary closed contact manifold of dimension at least five; the three-dimensional techniques (Seiberg-Witten equations, embedded contact homology — a Floer-type homology generated by the closed Reeb orbits) are dimension-specific, and symplectic-capacity arguments (capacities are symplectic-embedding-invariant size measurements of subsets of symplectic vector spaces) require ambient Euclidean or convex geometry. The multiplicity side — deciding whether a given Reeb flow has finitely or infinitely many closed orbits, and relating a finite count to topological invariants as in the Sasakian dichotomy — is likewise open for general contact manifolds.

Problem Statement

Prove or refute the Weinstein conjecture in its open regime: for every closed contact manifold (M2n+1,ξ)(M^{2n+1},\xi) with n≥2n\geq 2 (real dimension at least five) and every contact form λ\lambda with ker⁡λ=ξ\ker\lambda=\xi, does the Reeb vector field RλR_\lambda have at least one closed (periodic) orbit?

An affirmative answer must hold for all closed contact manifolds of dimension at least five without further geometric hypotheses (tight or overtwisted, fillable or non-fillable, degenerate or nondegenerate contact forms alike). A refutation must exhibit a specific closed contact manifold of dimension at least five and a specific contact form on it, together with a rigorous argument that the associated Reeb flow has no periodic orbit of any period. The central question is this existence question in full generality; the multiplicity and finiteness-of-count questions are secondary and not part of the stated problem.

References

  1. A. Weinstein, On the hypotheses of Rabinowitz' periodic orbit theorems, Journal of Differential Equations 33 (1979) 353–358, DOI 10.1016/0022-0396(79)90070-6, https://doi.org/10.1016/0022-0396(79)90070-6
  2. C. Viterbo, A proof of Weinstein's conjecture in R^{2n}, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 337–356, DOI 10.1016/S0294-1449(16)30363-8, https://doi.org/10.1016/S0294-1449(16)30363-8
  3. H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Inventiones Mathematicae 114 (1993) 515–563, DOI 10.1007/BF01232679, https://doi.org/10.1007/BF01232679
  4. C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geometry & Topology 11 (2007) 2117–2202, DOI 10.2140/gt.2007.11.2117, arXiv:math/0611007, https://arxiv.org/abs/math/0611007
  5. H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Communications on Pure and Applied Mathematics 45 (1992) 583–622, DOI 10.1002/cpa.3160450504, https://doi.org/10.1002/cpa.3160450504
  6. P. Albers and H. Hofer, On the Weinstein conjecture in higher dimensions, Commentarii Mathematici Helvetici 84 (2009) 429–436, DOI 10.4171/CMH/167, arXiv:0705.3953, https://arxiv.org/abs/0705.3953
  7. P. Rukimbira, Topology and closed characteristics of K-contact manifolds, Bulletin of the Belgian Mathematical Society - Simon Stevin 2 (1995), no. 3, 349–356, DOI 10.36045/bbms/1103408725, https://doi.org/10.36045/bbms/1103408725
  8. L. Ornea and M. Verbitsky, Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds, Mathematische Zeitschrift 299 (2021) 2287–2296, DOI 10.1007/s00209-021-02776-w, arXiv:2102.05962, https://arxiv.org/abs/2102.05962
  9. F. Pasquotto, A short history of the Weinstein conjecture, Jahresbericht der Deutschen Mathematiker-Vereinigung 114 (2012) 119–130, DOI 10.1365/s13291-012-0051-1, https://doi.org/10.1365/s13291-012-0051-1
  10. Y.-G. Oh, Contact instantons and proofs of Weinstein's conjecture and Arnold's chord conjecture, arXiv:2412.20731 (2024; withdrawn, version 2 of 17 Sep 2025), https://arxiv.org/abs/2412.20731
  11. E. Miranda and C. Oms, The singular Weinstein conjecture, Advances in Mathematics 389 (2021) 107925, DOI 10.1016/j.aim.2021.107925, arXiv:2005.09568, https://arxiv.org/abs/2005.09568
  12. J. Fontana-McNally, E. Miranda, C. Oms, and D. Peralta-Salas, A counterexample to the singular Weinstein conjecture, Advances in Mathematics 458 (2024) 109998, DOI 10.1016/j.aim.2024.109998, arXiv:2310.19918, https://arxiv.org/abs/2310.19918