On a closed contact manifold equipped with a chosen contact form, Mazzucchelli (Math. Ann. 2015) conjectured that every strict contactomorphism — a diffeomorphism preserving the contact form exactly — that is contact-isotopic to the identity admits an invariant Reeb orbit: a Reeb trajectory γ\gamma and a time shift T≠0T\ne 0 with ϕ(γ(t))=γ(t+T)\phi(\gamma(t))=\gamma(t+T). The conjecture contains the classical Weinstein conjecture as the case ϕ=id\phi=\mathrm{id}, where invariant orbits are precisely closed Reeb orbits. The geodesic antecedent of the question left open in the source paper — infinitely many invariant geodesics for isometries homotopic to the identity on closed manifolds with infinite cyclic fundamental group — was resolved affirmatively by Macarini–Mazzucchelli (2017), but the contact conjecture itself is confirmed only for special classes: the geodesic and unit-cotangent-bundle settings, fibrewise star-shaped hypersurfaces in cotangent bundles (Merry–Naef 2016), and, on hypertight contact manifolds, up to the alternative that an entire Reeb orbit is fixed pointwise (Meiwes–Naef 2018). It remains open for general closed contact manifolds; Cant's 2024 counterexample to the neighboring translated-point conjecture of Sandon uses non-form-preserving contactomorphisms and leaves this strict-form conjecture intact.

Background

A contact manifold (Y,ξ)(Y,\xi) is an odd-dimensional manifold carrying a hyperplane field ξ\xi that is maximally non-integrable; locally ξ=ker⁡α\xi=\ker\alpha for a 1-form α\alpha (a contact form). Any chosen contact form determines the Reeb vector field RαR_\alpha, defined by α(Rα)≡1\alpha(R_\alpha)\equiv 1 and dα(Rα,⋅)≡0\mathrm{d}\alpha(R_\alpha,\cdot)\equiv 0; its flow lines are Reeb trajectories, and a periodic Reeb trajectory is a closed Reeb orbit. The Weinstein conjecture predicts that every closed (compact, without boundary) contact manifold admits a closed Reeb orbit; it was proved for three-dimensional contact manifolds by Taubes (2007) via Seiberg–Witten Floer theory but remains open for general closed contact manifolds of dimension at least 5. A strict contactomorphism is a diffeomorphism ϕ\phi with ϕ∗α=α\phi^*\alpha=\alpha, i.e. one preserving the contact form exactly rather than only the hyperplane field ξ\xi; this is a genuine restriction, as a contactomorphism isotopic to the identity through contactomorphisms need not preserve α\alpha.

The conjecture grew out of the theory of isometry-invariant geodesics. Given an isometry II of a closed Riemannian manifold (M,g)(M,g), a geodesic γ:R→M\gamma:\mathbb{R}\to M is II-invariant if I(γ(t))=γ(t+1)I(\gamma(t))=\gamma(t+1) after reparametrization at constant positive speed. Grove initiated this theory in the early 1970s and proved that every isometry homotopic to the identity on a closed Riemannian manifold possesses an invariant geodesic (Grove 1973, Theorem 3.7); Bangert and Hingston (1984) showed that closed manifolds with infinite abelian fundamental group carry infinitely many closed geodesics. Mazzucchelli (2015) extended the Bangert–Hingston phenomenon to the invariant setting: if π1(M)≅Z⊕H\pi_1(M)\cong\mathbb{Z}\oplus H with HH a non-trivial abelian group, every isometry homotopic to the identity has infinitely many invariant geodesics (Theorem 1.1). The source paper conjectured the same conclusion for π1(M)≅Z\pi_1(M)\cong\mathbb{Z}; this was proved shortly afterwards by Macarini and Mazzucchelli (2017) for all closed MM other than the circle.

The passage from geodesics to contact geometry rests on the fact that the geodesic flow of (M,g)(M,g) is the Reeb flow of the restriction to the unit cotangent bundle S∗MS^*M (the bundle of unit-length cotangent vectors) of the Liouville form, the canonical 1-form p dqp\,dq on T∗MT^*M, and an isometry homotopic to the identity lifts to a strict contactomorphism of S∗MS^*M contact-isotopic to the identity. Mazzucchelli isolated the general phenomenon behind this correspondence: a ϕ\phi-invariant Reeb orbit is a Reeb trajectory γ:R→Y\gamma:\mathbb{R}\to Y such that ϕ(γ(t))=γ(t+T)\phi(\gamma(t))=\gamma(t+T) for some time shift T≠0T\ne 0. Because of the requirement T≠0T\ne 0, the id\mathrm{id}-invariant Reeb orbits are exactly the closed Reeb orbits, so the existence problem for invariant Reeb orbits contains the closed-orbit (Weinstein) problem. The source paper's Conjecture 1.2 asserts that on a closed contact manifold with a selected contact form, every strict contactomorphism that is contact-isotopic to the identity admits a ϕ\phi-invariant Reeb orbit.

The conjecture is related to, but distinct from, Sandon's translated-point conjecture. For a contactomorphism ψ\psi not necessarily preserving the form, a translated point is a point xx lying on the same Reeb orbit as ψ(x)\psi(x) with (ψ∗α)x=αx(\psi^*\alpha)_x=\alpha_x; Sandon conjectured that every contactomorphism contact-isotopic to the identity has translated points. When ψ\psi preserves α\alpha, the Reeb orbits of its translated points that are closed, or not contained in the fixed-point set of ψ\psi, are ψ\psi-invariant Reeb orbits. The two conjectures diverge sharply in general: Cant (2024) constructed contactomorphisms of the standard sphere S2n−1S^{2n-1} (n>1n>1) that are isotopic to the identity but have no translated points, refuting Sandon's conjecture — however, Cant's maps do not preserve the contact form, so the strict-form conjecture of Mazzucchelli is unaffected. Known confirmations and partial confirmations of Conjecture 1.2 are: the geodesic setting (Grove 1973); unit cotangent bundles of closed manifolds whose fundamental group is finite or has infinitely many conjugacy classes (as noted in the source paper, combining results of Albers–Merry and Kang); fibrewise star-shaped hypersurfaces in cotangent bundles (hypersurfaces bounding a domain that is star-shaped with respect to each cotangent fibre) of manifolds whose free-loop-space Betti numbers are asymptotically unbounded, where Merry and Naef (2016) in fact produce infinitely many invariant Reeb orbits; and hypertight contact manifolds (those admitting a supporting form without contractible Reeb orbits), where Meiwes and Naef (2018) prove that either a ϕ\phi-invariant Reeb orbit exists or an entire Reeb orbit is fixed pointwise by ϕ\phi.

Problem Statement

Let (Y,ξ)(Y,\xi) be a closed contact manifold with a selected contact form α\alpha (so ker⁡α=ξ\ker\alpha=\xi), and let RαR_\alpha be the associated Reeb vector field. Let ϕ:Y→Y\phi:Y\to Y be a strict contactomorphism, i.e. a diffeomorphism with ϕ∗α=α\phi^*\alpha=\alpha, that is contact-isotopic to the identity. Must there exist a ϕ\phi-invariant Reeb orbit: a Reeb trajectory γ:R→Y\gamma:\mathbb{R}\to Y (a solution of γ˙=Rα∘γ\dot\gamma=R_\alpha\circ\gamma) and a time shift T≠0T\ne 0 such that

ϕ(γ(t))=γ(t+T)for all t∈R?\phi(\gamma(t))=\gamma(t+T)\quad\text{for all }t\in\mathbb{R}?

This is Conjecture 1.2 of Mazzucchelli (Math. Ann. 362 (2015), no. 1–2, 265–280, DOI 10.1007/s00208-014-1113-8), a generalization of the Weinstein conjecture: for ϕ=id\phi=\mathrm{id} the condition T≠0T\ne 0 forces γ\gamma to be periodic, so the statement contains the existence of a closed Reeb orbit on an arbitrary closed contact manifold as a special case. An affirmative answer in full generality, or an explicit counterexample (Y,α,ϕ)(Y,\alpha,\phi), resolves the conjecture; partial confirmations restricted to additional geometric or dynamical hypotheses (fillability, hypertightness, cotangent-bundle structure) do not.

References

  1. Marco Mazzucchelli, "Isometry-invariant geodesics and the fundamental group", Mathematische Annalen 362 (2015), no. 1–2, 265–280. DOI: 10.1007/s00208-014-1113-8. https://link.springer.com/article/10.1007/s00208-014-1113-8 (also arXiv:1409.7249, https://arxiv.org/abs/1409.7249)
  2. Leonardo Macarini, Marco Mazzucchelli, "Isometry-invariant geodesics and the fundamental group, II", Advances in Mathematics 308 (2017), 671–698. DOI: 10.1016/j.aim.2016.12.023. https://arxiv.org/abs/1504.05685
  3. Will J. Merry, Kathrin Naef, "On the existence of infinitely many invariant Reeb orbits", Mathematische Zeitschrift 283 (2016), no. 1–2, 197–242. DOI: 10.1007/s00209-015-1595-4. https://arxiv.org/abs/1411.4847
  4. Matthias Meiwes, Kathrin Naef, "Translated points on hypertight contact manifolds", Journal of Topology and Analysis 10 (2018), no. 2, 289–322. DOI: 10.1142/S1793525318500097. https://arxiv.org/abs/1510.00515
  5. Dylan Cant, "Contactomorphisms of the sphere without translated points", Journal of Symplectic Geometry 22 (2024), no. 1, 1–10. DOI: 10.4310/jsg.2024.v22.n1.a1. https://arxiv.org/abs/2210.11002
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  9. Sheila Sandon, "A Morse estimate for translated points of contactomorphisms of spheres and projective spaces", Geometriae Dedicata 165 (2013), no. 1, 95–110. DOI: 10.1007/s10711-012-9741-1. https://doi.org/10.1007/s10711-012-9741-1
  10. Clifford Henry Taubes, "The Seiberg–Witten equations and the Weinstein conjecture", Geometry & Topology 11 (2007), 2117–2202. DOI: 10.2140/gt.2007.11.2117. https://doi.org/10.2140/gt.2007.11.2117
  11. Simon Allais, "Morse estimates for translated points on unit tangent bundles", Annales de l'Institut Fourier 76 (2026), no. 3, 1187–1206. DOI: 10.5802/aif.3732. https://arxiv.org/abs/2205.13946
  12. Brian Tervil, "Translated points for contactomorphisms of prequantization spaces over monotone symplectic toric manifolds", Journal of Symplectic Geometry 19 (2021), no. 6, 1421–1493. DOI: 10.4310/JSG.2021.v19.n6.a3. https://doi.org/10.4310/JSG.2021.v19.n6.a3