Ranks of Reeb Action Spectra

Definitions and mathematical statement

Let $M$ be a closed smooth three-dimensional manifold, and let $\alpha$ be a contact form on $M$, meaning that $\alpha\wedge{\,\mathrm{d}}\alpha$ is nowhere zero. Its Reeb vector field $R_\alpha$ is determined by \[ \alpha(R_\alpha)=1,\qquad {\,\mathrm{d}}\alpha(R_\alpha,\cdot)=0. \] Define the Reeb action spectrum by \[ \mathcal A(\alpha)=\left\{\int_0^T\alpha_{\gamma(t)}(\dot\gamma(t))\,{\,\mathrm{d}} t:\ T>0,\ \gamma:{\mathbb{R}}/T{\mathbb{Z}}\longrightarrow M,\ \dot\gamma=R_\alpha\circ\gamma\right\}. \] Each such action equals the period $T$. This definition includes multiple covers. Interpret the additive rank of the spectrum as the rank of the additive subgroup it generates: \[ G_\alpha=\left\{\sum_{j=1}^k n_j a_j:\ k\geq 0,\ n_j\in{\mathbb{Z}},\ a_j\in\mathcal A(\alpha)\right\}\subseteq{\mathbb{R}}, \qquad r(\alpha)=\dim_{{\mathbb{Q}}}\left(G_\alpha\otimes_{{\mathbb{Z}}}{\mathbb{Q}}\right) =\dim_{{\mathbb{Q}}}\operatorname{span}_{{\mathbb{Q}}}\mathcal A(\alpha). \] The dimension is allowed to be infinite. Two periodic Reeb orbits are geometrically distinct when their images in $M$ differ; multiple covers of one orbit therefore count only once.

Questions. Which values of $r(\alpha)$ occur for contact forms on closed three-manifolds? Does every such pair $(M,\alpha)$ satisfy \[ r(\alpha)=2\quad\Longrightarrow\quad \#\{\operatorname{im}(\gamma):\gamma\text{ is a periodic Reeb orbit of }R_\alpha\}=2? \] Unresolved qualifications. The supplied record does not explicitly define additive rank; the definition above is the usual rational-rank interpretation. It does not specify connectedness of $M$, nor impose nondegeneracy or any other dynamical hypothesis on $\alpha$. Whether connectedness was intended must be checked against the original report. The first question is phrased as a classification over pairs $(M,\alpha)$; the record does not clarify whether a classification for each fixed $M$ is also intended.

Short English statement

The problem asks which rational ranks the group generated by periodic Reeb actions can have, and whether rank two forces exactly two distinct periodic-orbit images. The intended connectedness assumption and the precise source definition of rank require review. The supplied open label is a dated curation claim, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30004182$, title Ranks of Reeb Action Spectra, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year $2019$ and names no individual proposer.
• Supplied report citation: Dynamische Systeme (2020), Oberwolfach Reports, pp. 1855--1917, DOI 10.4171/owr/2019/31. Supplied literature-source label: "original OWR report." https://doi.org/10.4171/owr/2019/31.
• Supplied literature reference: Cristofaro-Gardiner--Hutchings--Pomerleano, DOI 10.2140/gt.2019.23.3601. https://doi.org/10.2140/gt.2019.23.3601. No title or precise supporting theorem is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes rank-one contact forms as understood, cites two-or-infinity results in important three-dimensional cases, and describes the possible ranks and rank-two orbit classification as unresolved. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that notation, typography, and sentence structure were repaired while preserving mathematical content. This supplied editorial label does not establish mathematical correctness.