Transversal Foliations and Global Sections for Reeb Flows

Definitions and mathematical statement

Let $M$ be a smooth closed $3$-manifold, where closed means compact without boundary, and let $\alpha$ be a contact form, so that $\alpha\wedge{\,\mathrm{d}}\alpha$ is nowhere zero. Its Reeb vector field $R_\alpha$ is characterized by \[ \alpha(R_\alpha)=1,\qquad {\,\mathrm{d}}\alpha(R_\alpha,\cdot)=0. \] Write $\varphi^t$ for its flow and $h_{\mathrm{top}}(\varphi^1)$ for the topological entropy of its time-one map. An invariant torus means a smoothly embedded torus $T\subset M$ satisfying $\varphi^t(T)=T$ for every $t\in{\mathbb{R}}$.

Under the usual compact-surface convention, a global surface of section for a nonsingular flow on a $3$-manifold $N$ is a compact embedded surface $S\subset N$, possibly with boundary, whose boundary consists of periodic trajectories, whose interior is transverse to the flow, and such that every trajectory in $N\setminus\partial S$ meets $\operatorname{int}S$ in both positive and negative time. The supplied record does not specify whether this convention, or a convention allowing noncompact surfaces or boundary on a compactification, is intended after invariant tori are removed.

Source question, with unresolved terminology. For every Reeb flow on a closed $3$-manifold that admits a transverse foliation and satisfies \[ h_{\mathrm{top}}(\varphi^1)=0, \] must there exist finitely many invariant tori $T_1,\ldots,T_k$ such that the restricted flow on \[ M\setminus\bigcup_{i=1}^{k}T_i \] admits a global surface of section? Here the interpretation of "transverse foliation" and the appropriate meaning of a global surface of section on this complement remain unspecified by the record. Consequently, this is a qualified formulation of the supplied question, not a fully determined assertion under the compact-surface convention above. The source also asks more broadly what additional dynamical structure follows from admitting a transversal foliation.

No non-degeneracy hypothesis occurs in the supplied question. The supplied literature assessment mentions results under non-degeneracy and zero-entropy assumptions; those qualifications must not be silently added to the question or treated as a complete statement of the cited results.

Short English statement

Does every zero-topological-entropy Reeb flow on a closed $3$-manifold admitting a transverse foliation have a global surface of section after finitely many invariant tori are removed? The intended foliation and the meaning of a global section on the resulting complement require clarification. The supplied record labels the problem partially solved; that assessment is not independently verified here.

Sources

• Supplied local UnsolvedMath problem registry, permanent ID $30004184$, source problem code catalogue entry, title: Transversal Foliations and Global Sections for Reeb Flows. The record lists proposed year 2019 and identifies no individual proposer.
• Supplied report citation: Dynamische Systeme (2020), Oberwolfach Reports, pp. 1855--1917, DOI 10.4171/owr/2019/31. Original OWR report: https://doi.org/10.4171/owr/2019/31.
• Supplied literature reference: Hryniewicz--Salomao--Wysocki, "global sections": https://doi.org/10.4171/JEMS/1220. The record supplies no complete article title or theorem statement.
• Literature assessment checked at 2026-08-22: supplied dated curation, not independently verified. It labels the problem partially solved and states that transversal-foliation methods yield symbolic dynamics and a global surface of section under non-degeneracy and zero-entropy assumptions, while the broad question about all resulting Reeb flows remains unsettled. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified; its accompanying explanation describes repairs to notation, typography, and sentence structure while preserving mathematical content. This supplied label is not independent mathematical verification of the formulation or its literature status.