The Weinstein conjecture for closed contact manifolds of dimension at least five

The Weinstein conjecture predicts that every Reeb vector field on a closed contact manifold has at least one closed periodic orbit. It is a theorem in dimension three (Taubes, 2007) but remains open in full generality in dimensions $2n-1\ge 5$. The source, Acu and Moreno's 'Planarity in higher-dimensional contact manifolds', proved the conjecture for iterated-planar (IP) contact manifolds — those supported by open books whose pages admit iterated planar Lefschetz fibrations — and states explicitly that in higher dimensions the conjecture 'is still open' beyond the known partial results, motivating the program of extending the open-book/holomorphic-curve framework to broader classes of higher-dimensional contact manifolds. The conjecture is now additionally known for manifolds of finite planarity in the rational-symplectic-field-theory hierarchy of Moreno–Zhou (which recovers the IP theorem), for all non-orderable contact manifolds together with several newly discovered non-orderable families (via Albers–Fuchs–Merry, Hedicke–Shelukhin, Arlove), for quasiconformal Reeb fields (Finamore), and sufficient conditions follow from higher-dimensional Heegaard Floer homology (Colin–Honda–Tian); Moreno–Zhou moreover prove that non-iterated-planar contact structures exist in every fillable almost-contact homotopy class in dimensions at least five, so IP-type constructions cannot settle the general case. The only claimed general proof (Oh, 2024, via contact instantons) was withdrawn in September 2025. The general higher-dimensional Weinstein conjecture remains open.

Background

A contact manifold is a closed (compact, without boundary) odd-dimensional manifold $M^{2n-1}$ together with a hyperplane field $\xi=\ker\alpha$ defined by a 1-form $\alpha$ satisfying the maximal non-integrability condition $\alpha\wedge(d\alpha)^{n-1}\neq 0$; $\alpha$ is called a contact form. Each contact form determines a unique Reeb vector field $R_\alpha$ through the equations $\iota_{R_\alpha}d\alpha=0$ and $\alpha(R_\alpha)=1$, and the Weinstein conjecture (Weinstein, 1979) asserts that on a compact contact manifold the flow of $R_\alpha$ has at least one closed periodic orbit, for every contact form defining the given contact structure. The conjecture grew out of Rabinowitz's work on periodic orbits of Hamiltonian systems: energy hypersurfaces of contact type in symplectic manifolds carry natural Reeb flows, so the conjecture predicts periodic dynamics on all such hypersurfaces.

In dimension three the conjecture is a theorem: Taubes (2007) proved it for all closed contact 3-manifolds using the Seiberg–Witten equations. In higher dimensions only partial results were classically known: Viterbo (1987) settled compact contact-type hypersurfaces of Euclidean space $\mathbb{R}^{2n}$; Floer–Hofer–Viterbo (1990) settled certain products $P\times\mathbb{C}^l$; Hofer–Viterbo (1992) settled cases where the symplectization contains enough holomorphic spheres; and Albers–Hofer (2009) settled contact structures containing a plastikstufe, a higher-dimensional analogue of an overtwisted disk (an explicit family of Legendrian disks through a common singular point whose presence forces flexible, non-rigid behavior). Relatedly, Dörner–Geiges–Zehmisch (2014) obtained contractible periodic Reeb orbits for contact structures supported by open books whose binding is of restricted contact type in a subcritical Stein manifold, and Suhr–Zehmisch (2017) established the strong Weinstein conjecture — existence of a null-homologous link of finitely many closed orbits — for concave boundaries of cobordisms carrying an essential foliation by holomorphic spheres.

A central tool in higher dimensions is the open book decomposition: a codimension-two binding $B\subset M$ together with a fibration of $M\setminus B$ over $S^1$ whose pages are the closures of the fibers. A fundamental correspondence asserts that every closed contact manifold is supported by some open book with Liouville pages (pages carrying an exact symplectic form whose primitive restricts to a contact form on the boundary, which is the binding). Abbas–Cieliebak–Hofer (2005) proved the Weinstein conjecture for planar contact 3-manifolds, i.e. those supported by open books with genus-zero pages. The source, Acu–Moreno ('Planarity in higher-dimensional contact manifolds', arXiv 2018, published in Int. Math. Res. Not. 2022), generalized this to higher dimensions: a page is iterated planar if it admits an iterated planar Lefschetz fibration — an exact symplectic Lefschetz fibration (a map to the disk with finitely many isolated critical points modeled on complex singularities) whose generic fiber is itself the total space of a lower-dimensional such fibration, the bottom layer having genus-zero (planar) fibers. Their Theorem C establishes that every iterated-planar (IP) contact manifold satisfies the Weinstein conjecture, via a symplectic handle attachment that produces a moduli space of holomorphic spheres, regularized using polyfold technology (an analytic framework for achieving transversality in moduli spaces of holomorphic curves). Their paper states explicitly that in higher dimensions the conjecture 'is still open' beyond the known partial results.

Problem Statement

Let $(M,\xi)$ be a closed contact manifold of dimension $2n-1$ with $n\ge 3$, i.e. of dimension at least five. Does every contact form $\alpha$ with $\ker\alpha=\xi$ have at least one closed periodic Reeb orbit? Equivalently: prove, or refute by an explicit closed contact manifold of dimension at least five together with a contact form whose Reeb flow has no periodic orbit, the Weinstein conjecture in the dimensions in which it remains unresolved. This is the famous Weinstein conjecture restricted to the open dimension range (dimension three having been settled by Taubes). Results that settle further classes of manifolds — such as the iterated-planar, plastikstufe-containing, non-orderable, finite-planarity, or quasiconformal classes — constitute partial progress toward this problem, and the extension of open-book and holomorphic-curve methods to broader classes is the associated research program, but only a proof or refutation of the general statement in dimensions $2n-1\ge 5$ answers the question.

References

1. Alan Weinstein, On the hypotheses of Rabinowitz' periodic orbit theorems, Journal of Differential Equations 33 (1979), no. 3, 353–358, DOI 10.1016/0022-0396(79)90070-6, https://doi.org/10.1016/0022-0396(79)90070-6
2. Claude Viterbo, A proof of Weinstein's conjecture in R^{2n}, Annales de l'Institut Henri Poincaré C, Analyse non linéaire 4 (1987), no. 4, 337–356, DOI 10.1016/S0294-1449(16)30363-8 (Zbl 0631.58013), https://doi.org/10.1016/S0294-1449(16)30363-8
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8. Max Dörner, Hansjörg Geiges, Kai Zehmisch, Open books and the Weinstein conjecture, The Quarterly Journal of Mathematics 65 (2014), no. 3, 869–885, DOI 10.1093/qmath/hat055, https://doi.org/10.1093/qmath/hat055
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11. Bahar Acu, Agustin Moreno, Planarity in higher-dimensional contact manifolds, International Mathematics Research Notices 2022 (2022), no. 6, 4222–4258, DOI 10.1093/imrn/rnaa155 (preprint arXiv:1810.11448), https://doi.org/10.1093/imrn/rnaa155
12. Vincent Colin, Ko Honda, Yin Tian, Applications of higher-dimensional Heegaard Floer homology to contact topology, preprint arXiv:2006.05701 (2020), https://arxiv.org/abs/2006.05701
13. Agustin Moreno, Zhengyi Zhou, A landscape of contact manifolds via rational SFT, Geometry & Topology 29 (2025), no. 7, 3465–3565, DOI 10.2140/gt.2025.29.3465 (preprint arXiv:2012.04182), https://doi.org/10.2140/gt.2025.29.3465
14. Douglas Finamore, Quasiconformal contact foliations, preprint arXiv:2301.01738 (2023), https://arxiv.org/abs/2301.01738
15. Jakob Hedicke, Egor Shelukhin, Non-orderability and the contact Hofer norm, preprint arXiv:2411.19887 (2024; v2 March 2025), https://arxiv.org/abs/2411.19887
16. Pierre-Alexandre Arlove, Spectral selectors on strongly orderable contact manifolds and applications, preprint arXiv:2509.12856 (2025), https://arxiv.org/abs/2509.12856
17. Yong-Geun Oh, Contact instantons and Proofs of Weinstein's conjecture and Arnold's chord conjecture, preprint arXiv:2412.20731 (2024; withdrawn by the author, v2 of 17 September 2025), https://arxiv.org/abs/2412.20731