Birkhoff Sections for Three-Dimensional Reeb Flows

Definitions and mathematical statement

A smooth contact form on a smooth three-dimensional manifold $M$ is a smooth one-form $\alpha$ such that $\alpha\wedge{\,\mathrm{d}}\alpha$ is nowhere zero. Its Reeb vector field $R_\alpha$ is uniquely characterized by \[ \alpha(R_\alpha)=1,\qquad \iota_{R_\alpha}{\,\mathrm{d}}\alpha=0. \] Write $\varphi_t$ for its flow whenever defined.

For an explicit working formulation, suppose that $M$ is closed, meaning compact and without boundary. Use the following bounded-return convention for a Birkhoff section. It consists of a nonempty compact smooth orientable surface $S$, possibly disconnected, and a smooth map $f:S\to M$ such that:

1. The restriction $f|_{\operatorname{int}S}$ is an embedding, and its image is transverse to $R_\alpha$.
2. The set $\Gamma=f(\partial S)$ is a finite union of periodic flow orbits, disjoint from $f(\operatorname{int}S)$; each boundary component maps onto a periodic orbit as a finite covering.
3. There exists $T>0$ such that \[ \forall x\in M\setminus\Gamma\quad\exists t\in(0,T]\quad \varphi_t(x)\in f(\operatorname{int}S). \]

Under these explicitly stated conventions, the question is: \[ \text{For every closed smooth three-manifold }M\text{ and every smooth contact form }\alpha\text{ on }M, \quad R_\alpha\text{ admits a Birkhoff section?} \] Equivalently, does there exist such a pair $(M,\alpha)$ whose Reeb vector field has no Birkhoff section? The universal formulation is the supplied clean statement; the counterexample formulation is the supplied original question.

The supplied excerpt does not specify compactness, boundary conditions on the ambient manifold, regularity, or the precise definition of a Birkhoff section. Consequently, the displayed formulation is a conditional working interpretation requiring source confirmation. In particular, bounded return times, boundary coverings, and permitted boundary behavior must be checked against the report's actual conventions. No nondegeneracy, genericity, or entropy hypothesis is imposed in the supplied question.

Short English statement

The question asks whether every three-dimensional Reeb flow has a transverse surface, bounded by periodic orbits, that all other trajectories repeatedly meet. The closed-manifold, bounded-return, and boundary-covering conventions above make one working interpretation precise; the supplied excerpt does not establish these conventions. The source labels the problem open, but that status has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005586$, title Birkhoff Sections for Three-Dimensional Reeb Flows, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No proposer is named; the supplied proposed year is 2023.
• Supplied report citation: Dynamische Systeme (2024), Oberwolfach Reports, pp. 1671--1730, DOI 10.4171/owr/2023/30, https://doi.org/10.4171/owr/2023/30. The original question and surrounding comments are attributed to this report; the supplied excerpt does not identify their author.
• Supplied literature source labeled "Colin--Dehornoy--Hryniewicz--Rechtman": https://arxiv.org/abs/2202.01506.
• Supplied literature source labeled "generic Reeb-flow sections": https://arxiv.org/abs/2501.11725.
• Supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2023/30.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that an open dense set of three-dimensional contact forms admits Birkhoff sections, while the universal question remains unanswered and no counterexample is known. This is supplied dated curation, not an independently verified literature assessment; its topology, ambient-manifold assumptions, and genericity qualifications require checking.
• The supplied registry labels the statement corrected_verified and reports comparison with the official Oberwolfach report. This records the supplied verification claim only; no source has been independently visited or checked for this draft.