Three Closed Reeb Orbits Beyond Lens Spaces

Definitions and mathematical statement

Let $M$ be a smooth three-dimensional manifold and let $\alpha$ be a smooth contact form, meaning that $\alpha\wedge{\,\mathrm{d}}\alpha$ is nowhere zero. Its Reeb vector field $R_\alpha$ is characterized by \[ \alpha(R_\alpha)=1,\qquad {\,\mathrm{d}}\alpha(R_\alpha,v)=0\quad\text{for every }v\in TM. \] A closed Reeb orbit is a periodic integral curve $\gamma:{\mathbb{R}}/T{\mathbb{Z}}\to M$, for some $T>0$, satisfying $\dot\gamma(t)=R_\alpha(\gamma(t))$. Count orbits by their images in $M$, so that multiple traversals of one orbit do not count as different orbits. This geometric counting convention is an interpretation requiring confirmation from the source.

For the usual finite-quotient convention, a lens space is a manifold diffeomorphic to \[ L(p,q)=S^3/\langle g\rangle,\qquad g(z_1,z_2)=\left(e^{2\pi i/p}z_1,e^{2\pi iq/p}z_2\right), \] where $S^3\subset{\mathbb{C}}^2$, $p\geq1$, $q\in{\mathbb{Z}}$, and $\gcd(p,q)=1$. This convention includes $S^3=L(1,q)$. The record does not specify its lens-space convention, including whether $S^1\times S^2$ is included as a degenerate lens space.

Source conjecture, with unresolved scope. Let $\mathcal C$ denote the intended class of smooth three-manifolds, which the supplied record does not fully specify. The proposed assertion is \[ \begin{gathered} \text{For every }M\in\mathcal C\text{ that is not a lens space, and every contact form }\alpha\text{ on }M,\\ \text{there exist }T_1,T_2,T_3>0\text{ and periodic integral curves }\gamma_i:{\mathbb{R}}/T_i{\mathbb{Z}}\to M\\ \text{of }R_\alpha\text{ such that }\operatorname{im}(\gamma_i)\neq\operatorname{im}(\gamma_j)\text{ whenever }i\neq j. \end{gathered} \] The class $\mathcal C$ remains unresolved: compactness, connectedness, absence of boundary, and any conditions at a boundary are not stated. In particular, restricting to closed connected manifolds would add hypotheses requiring source confirmation. No nondegeneracy hypothesis is supplied. The original wording says only "manifold"; the supplied cleaned statement explicitly specifies dimension three. Thus this is a parameterized statement-only draft, rather than a fully determined conjecture.

Short English statement

The supplied conjecture asks whether every Reeb vector field on a three-manifold outside the lens-space class has at least three geometrically distinct closed orbits. The intended manifold hypotheses, lens-space convention, and orbit-counting convention need confirmation. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30002923$, source problem code catalogue entry, title Three Closed Reeb Orbits Beyond Lens Spaces. Project registry locations are data/unsolvedmath/problems.json and the supplementary catalog/problems.json; this draft uses the supplied complete record. The record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2015, and names no individual proposer.
• Supplied report citation: Dynamische Systeme (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/34, pp. 1937--1988. https://doi.org/10.4171/owr/2015/34. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2016 label and its relation to the supplied 2015 provenance require verification.
• Supplied literature reference: Contact three-manifolds with exactly two simple Reeb orbits. https://arxiv.org/abs/2102.04970.
• Supplied literature reference: Oberwolfach Report on Reeb dynamics. https://doi.org/10.4171/owr/2015/34.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. The supplied status is "open." The assessment states that examples with exactly two Reeb orbits are known, describes the extracted record as broadly asking what further configurations can occur, and says that the record does not state a resolved classification theorem. Its own qualification calls this dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying explanation says mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. These are registry assertions, not independent verification performed for this draft.