Periodic Orbits Bounding Global Surfaces of Section

Definitions and mathematical statement

Let $M$ be a closed smooth $3$-manifold, where closed means compact and without boundary, and let $\varphi:{\mathbb{R}}\times M\to M$ be a smooth flow. Write $\varphi_t(p)=\varphi(t,p)$ and let $X(p)=\left.\frac{{\,\mathrm{d}}}{{\,\mathrm{d}} t}\right|_{t=0}\varphi_t(p)$ be its generating vector field. Interpret a periodic orbit as the image of a nonconstant trajectory through a point $p$ such that $\varphi_T(p)=p$ for some $T>0$. For the supplied finite family $\gamma_1,\ldots,\gamma_N$, put \[ L=\bigcup_{i=1}^{N}\gamma_i. \] For an explicit embedded-surface interpretation, a global surface of section with boundary $L$ is a compact smooth embedded surface $S\subset M$ such that, writing $S^\circ=S\setminus\partial S$, \[ \partial S=L,\qquad X(q)\notin T_qS\quad\text{for every }q\in S^\circ, \] and \[ \text{for every }p\in M\setminus L\text{ there exist }t_-<0<t_+\text{ such that } \varphi_{t_-}(p),\varphi_{t_+}(p)\in S^\circ. \] No connectedness or orientability requirement is imposed in this explicit interpretation. The supplied statement does not specify whether such requirements, or more general notions of surface embedding, are intended; these conventions require mathematical review.

Problem. Given $(M,\varphi_t,\gamma_1,\ldots,\gamma_N)$ as above, determine when there exists a global surface of section $S$ whose entire boundary is exactly $L$. The clean statement asks for conditions for existence, while the original wording asks whether the given collection bounds such a surface. No additional dynamical hypotheses or particular form of an existence criterion are supplied.

Short English statement

Determine when a specified finite union of periodic orbits of a smooth flow on a closed smooth $3$-manifold is exactly the boundary of a compact surface transverse to the flow in its interior and met by every trajectory outside the boundary in both forward and backward time. Surface conventions remain to be confirmed from the source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003525$, title "Periodic Orbits Bounding Global Surfaces of Section," source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2017; no individual proposer is supplied.
• Supplied report citation: Dynamische Systeme (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/32, pp. 1987--2046. https://doi.org/10.4171/owr/2017/32. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. The literature-source list labels this same reference "Oberwolfach Report 32/2017."
• Hryniewicz--Salomao--Wysocki, "Global surfaces of section," as labeled in the supplied literature sources. https://doi.org/10.4171/JEMS/1220.
• Liu--Salomao, "Birkhoff conjecture in Hill's problem," as labeled in the supplied literature sources. https://arxiv.org/abs/2606.12912.
• Literature assessment checked at 2026-08-21: this is supplied dated curation derived from the supplied OWR curation, not independently verified here. The record labels the problem partially solved and reports strong sufficient criteria for general global-surface-of-section questions, a proof of Birkhoff's retrograde-orbit conjecture in Hill's lunar problem, and unresolved full and general restricted three-body formulations. The supplied background explicitly requests independent verification of time-sensitive status and cited claims; the precise scope of those contextual formulations is not supplied.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied provenance claims, not an independent verification of this draft or of the cited mathematics.