Let LL be a closed manifold and (T∗L,ωstd=dλstd)(T^*L,\omega_{\mathrm{std}}=d\lambda_{\mathrm{std}}) its exact cotangent bundle, the phase space of a classical mechanical system with configuration space LL. The Weinstein conjecture in cotangent bundles asks whether every compact hypersurface Σ⊂T∗L\Sigma\subset T^*L of contact type carries a closed characteristic, i.e. a periodic orbit of the Hamiltonian flow on Σ\Sigma. The enclosing case (where the bounded component of T∗L∖ΣT^*L\setminus\Sigma contains the zero section) was settled by Hofer–Viterbo (1988) and reproved by Asselle–Starostka (2020) via a mixed-regularity Palais–Smale condition; the latter explicitly remark that the full conjecture in cotangent bundles seems not to be known. The literature leaves a precise open core: existence is proven for hypersurfaces enclosing the zero section, for bases with finite fundamental group (Viterbo 1997), for bases of the form Q×S1Q\times S^1 (Geiges–Zehmisch 2012, strong version), for surface bases (Taubes 2007), and for twisted/magnetic cotangent bundles with magnetic form nonvanishing on π2\pi_2 (Groman–Merry 2023) — but remains open for compact contact-type hypersurfaces over closed bases of dimension at least 3 with infinite fundamental group admitting no S1S^1-product splitting, such as the product L=Σg×ΣhL=\Sigma_g\times\Sigma_h of closed orientable surfaces of genera g,h≥2g,h\ge 2.

Background

Let LL be a closed (compact, boundaryless, connected) smooth manifold. Its cotangent bundle T∗LT^*L carries the canonical Liouville one-form λstd\lambda_{\mathrm{std}}, defined at a point (q,p)∈T∗L(q,p)\in T^*L by evaluation of the covector pp on tangent vectors of LL at qq; the exact two-form ωstd=dλstd\omega_{\mathrm{std}}=d\lambda_{\mathrm{std}} is the standard symplectic structure, making T∗LT^*L the phase space of a classical mechanical system with configuration space LL. A smooth function H:T∗L→RH:T^*L\to\mathbb{R} (the Hamiltonian, e.g. kinetic plus potential energy) defines the Hamiltonian vector field XHX_H by ωstd(XH,⋅)=−dH\omega_{\mathrm{std}}(X_H,\cdot)=-dH. Since HH is constant along trajectories of XHX_H, motion at fixed energy κ\kappa is confined to the energy hypersurface Σ=H−1(κ)\Sigma=H^{-1}(\kappa). If Σ\Sigma is compact, connected, and regular (XHX_H nowhere zero on Σ\Sigma), the restriction of ωstd\omega_{\mathrm{std}} to Σ\Sigma has a one-dimensional kernel ℓΣ\ell_\Sigma, the characteristic line bundle, which integrates to the characteristic foliation of Σ\Sigma; its leaves are the unparametrized Hamiltonian trajectories, independent of the choice of defining Hamiltonian up to time-reparametrization. A closed characteristic is a compact leaf of this foliation, i.e. a periodic orbit of the system at energy κ\kappa; finding periodic motions at fixed energy is equivalent to finding closed characteristics.

A compact hypersurface Σ⊂(T∗L,ωstd)\Sigma\subset(T^*L,\omega_{\mathrm{std}}) is of contact type if there exists a one-form α\alpha on Σ\Sigma with dα=ωstd∣Σd\alpha=\omega_{\mathrm{std}}|_\Sigma and α\alpha nonzero on ℓΣ\ell_\Sigma; equivalently, a Liouville vector field YY (one with LYωstd=ωstd\mathcal{L}_Y\omega_{\mathrm{std}}=\omega_{\mathrm{std}}, e.g. the radial fiber-dilation field) exists near Σ\Sigma and is everywhere transverse to Σ\Sigma. Then α\alpha is a contact form and its Reeb vector field — the unique field RαR_\alpha with α(Rα)=1\alpha(R_\alpha)=1 and ιRαdα=0\iota_{R_\alpha}d\alpha=0 — spans ℓΣ\ell_\Sigma, so closed characteristics on Σ\Sigma are exactly closed Reeb orbits of α\alpha. The Weinstein conjecture (Weinstein 1979) predicts that every closed (compact, boundaryless) contact manifold carries a closed Reeb orbit, equivalently that every compact contact-type hypersurface in any symplectic manifold carries a closed characteristic. It is known for R2n\mathbb{R}^{2n} (Viterbo 1987), for all three-dimensional closed contact manifolds (Taubes 2007, via Seiberg–Witten theory, a gauge-theoretic equation imported from four-manifold topology), and in many special classes, but is open in general in contact dimension at least five.

For dim⁡L≥2\dim L\ge 2, every compact connected hypersurface of T∗LT^*L separates the total space into a bounded and an unbounded part (by Alexander duality, the standard duality between the homology of a subspace and the cohomology of its complement). Following Hofer–Viterbo, Σ\Sigma is said to enclose the zero section (or be OL\mathbb{O}_L-separating) if the bounded component contains the zero section OL⊂T∗L\mathbb{O}_L\subset T^*L. Hofer–Viterbo (1988) proved that every compact connected contact-type hypersurface enclosing the zero section carries a closed characteristic; Asselle–Starostka (2020) gave a simplified proof by showing that the Hamiltonian action functional (which integrates λstd\lambda_{\mathrm{std}} along a loop in the energy hypersurface, and whose critical points are exactly its closed characteristics) satisfies the Palais–Smale condition — the requirement that any sequence of loops with bounded action and action gradient tending to zero possess a convergent subsequence — on a mixed-regularity loop space (HsH^s in the base, H1−sH^{1-s} in the fibers, s∈(12,1)s\in(\tfrac12,1)), and proved unconditionally that every enclosing hypersurface, even without the contact-type assumption, has arbitrarily nearby hypersurfaces carrying closed characteristics. The contact-type assumption cannot simply be dropped from the conjecture itself: compact hypersurfaces without closed characteristics exist already in R4\mathbb{R}^{4} (Ginzburg–Gürel 2003, a C2C^2 counterexample to the Hamiltonian Seifert conjecture, which asks whether every compact regular energy surface of a Hamiltonian system in R2n\mathbb{R}^{2n} carries a periodic orbit). Beyond the enclosing case, Viterbo (1997) settled the conjecture for simply connected bases (and, as generally attributed, for bases with finite fundamental group) using exact Lagrangian obstruction theory and generating functions (a finite-dimensional encoding of Lagrangian submanifolds of cotangent bundles), and Geiges–Zehmisch (2012) settled it in the strong form for bases splitting as Q×S1Q\times S^1 using holomorphic curves in symplectic cobordisms; Taubes (2007) covers all surface bases, and Groman–Merry (2023) handle twisted cotangent bundles carrying a magnetic two-form nonvanishing on π2(L)\pi_2(L). What no method reaches is the remaining class of bases with infinite fundamental group and no S1S^1-factor: as Ginzburg's survey explains, the action-selector techniques — homological functionals used to force existence of periodic orbits by bounding their action — lose their a priori bounds precisely when π1(L)\pi_1(L) is infinite, because candidate periodic orbits can migrate between free homotopy classes.

Problem Statement

Let LL be a closed smooth manifold and (T∗L,ωstd=dλstd)(T^*L,\omega_{\mathrm{std}}=d\lambda_{\mathrm{std}}) its cotangent bundle with the canonical exact symplectic structure. The problem is the Weinstein conjecture in exact cotangent bundles in its remaining generality: does every compact hypersurface Σ⊂T∗L\Sigma\subset T^*L of contact type carry a closed characteristic leaf, i.e. a periodic orbit of the characteristic (Hamiltonian) flow on Σ\Sigma — equivalently, a closed Reeb orbit for the contact form induced on Σ\Sigma?

The conjecture is already settled whenever Σ\Sigma encloses the zero section (bounded component of T∗L∖ΣT^*L\setminus\Sigma containing OL\mathbb{O}_L; Hofer–Viterbo 1988, new proof by Asselle–Starostka 2020), whenever π1(L)\pi_1(L) is finite (Viterbo 1997), whenever L=Q×S1L=Q\times S^1 with QQ closed (Geiges–Zehmisch 2012), and whenever dim⁡L≤2\dim L\le 2 (Taubes 2007 for dim⁡L=2\dim L=2; the case dim⁡L=1\dim L=1 is immediate). The question therefore targets compact contact-type hypersurfaces — in particular hypersurfaces that do not enclose the zero section — over closed bases with dim⁡L≥3\dim L\ge 3, π1(L)\pi_1(L) infinite, and no S1S^1-product splitting, such as the product L=Σg×ΣhL=\Sigma_g\times\Sigma_h of closed orientable surfaces of genera g,h≥2g,h\ge 2, or a closed hyperbolic 3-manifold. The conjecture is settled either by a proof of existence of a closed characteristic for every such LL and Σ\Sigma, or by a compact contact-type hypersurface in some exact cotangent bundle whose characteristic foliation has no closed leaf. Restricting to enclosing, fiberwise star-shaped, or otherwise special hypersurfaces does not answer the question; dropping the contact-type hypothesis is not part of the conjecture, since without it existence is known to fail.

References

  1. Luca Asselle, Maciej Starostka, The Palais–Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications, Calculus of Variations and Partial Differential Equations 59 (2020), no. 4, Art. 113. DOI: 10.1007/s00526-020-01762-0, https://arxiv.org/abs/2003.10133
  2. H. Hofer, C. Viterbo, The Weinstein conjecture in cotangent bundles and related results, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Serie IV 15 (1988), no. 3, 411–445. https://www.numdam.org/item/ASNSP_1988_4_15_3_411_0/
  3. C. Viterbo, Exact Lagrange submanifolds, periodic orbits and the cohomology of free loop spaces, Journal of Differential Geometry 47 (1997), no. 3, 420–468. DOI: 10.4310/jdg/1214460546
  4. Hansjörg Geiges, Kai Zehmisch, Symplectic cobordisms and the strong Weinstein conjecture, Mathematical Proceedings of the Cambridge Philosophical Society 153 (2012), no. 2, 261–279. DOI: 10.1017/S0305004112000163, https://arxiv.org/abs/1109.1389
  5. Viktor L. Ginzburg, The Weinstein conjecture and theorems of nearby and almost existence, in: The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics 232, Birkhäuser Boston, 2005, pp. 139–172. DOI: 10.1007/0-8176-4419-9_6, https://arxiv.org/abs/math/0310330
  6. Clifford Henry Taubes, The Seiberg–Witten equations and the Weinstein conjecture, Geometry & Topology 11 (2007), 2117–2202. DOI: 10.2140/gt.2007.11.2117
  7. Yoel Groman, Will J. Merry, The symplectic cohomology of magnetic cotangent bundles, Commentarii Mathematici Helvetici 98 (2023), no. 2, 365–424. DOI: 10.4171/cmh/555, https://arxiv.org/abs/1809.01085
  8. A. 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
  9. C. Viterbo, A proof of Weinstein's conjecture in R2n\mathbb{R}^{2n}, Annales de l'Institut Henri Poincaré C, Analyse non linéaire 4 (1987), no. 4, 337–356. https://www.numdam.org/item/AIHPC_1987__4_4_337_0/
  10. Viktor L. Ginzburg, Başak Z. Gürel, A C2C^2-smooth counterexample to the Hamiltonian Seifert conjecture in R4\mathbb{R}^4, Annals of Mathematics 158 (2003), no. 3, 953–976. DOI: 10.4007/annals.2003.158.953