ProofAtlas

Contact and symplectic topology

Higher-Dimensional Weinstein Conjecture

Collaboration beta

Must every Reeb vector field on a closed contact manifold have a periodic orbit? Dimension three is known externally, but the unrestricted higher-dimensional problem remains open.

α(R)=1,ιRdα=0⇒Rhas a periodic orbit.
Known results and sources
A Reeb flow crosses nested contact slices while one amber candidate orbit nearly closes but retains a narrow gap.
The higher-dimensional conjecture asks for a periodic Reeb orbit on every closed contact manifold; the displayed orbit is not closed.

Research problem

Exact mathematical statement

Let (M2n-1,α)(M^{2n-1},\alpha) be a closed contact manifold with n≥3n\ge3, and let RαR_\alpha be its Reeb field. The conjecture is

α(Rα)=1,ιRαdα=0⇒Rαhas a periodic orbit.\alpha(R_\alpha)=1,\qquad \iota_{R_\alpha}d\alpha=0 \quad\Longrightarrow\quad R_\alpha\text{ has a periodic orbit}.

Problem infographic

Problem at a glance

A single closed layered contact manifold carries contact planes, Reeb trajectories, and one amber trajectory interrupted before closure.
The defining contact geometry and unresolved periodic-orbit question are shown directly, without a holonomy or flux route.

Current mathematical picture

Where work on Higher-Dimensional Weinstein Conjecture stands

Partially resolved

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureRigid affine leaf-space reduction outside its hypotheses

The current work identifies nonlinear dense holonomy as the surviving regular branch after the rigid model is excluded. A projective cocycle analysis directly in the nonlinear transverse coordinate remains viable.

Route status · Narrowed route
Main reductionCurrent reduction

Under hypothetical aperiodicity, pass from positive loop quantization to invariant measures with a symplectic zero center, then force a fixed point or globalize center flux.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeEliminate the nonlinear dense-holonomy T^3-bundle branch without assuming smooth affine linearization.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Higher-Dimensional Weinstein Conjecture in numbers

2.3kretained lines of mathematical investigation2,265 in the current working snapshot
Argument development
1,929 · 85%
Explored or eliminated routes
75 · 3%
Computational analysis
4 · 0%
Open obligations
64 · 3%
Definitions and setup
193 · 9%
6selected mapped statements1routes investigated3open questions3contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byAlan WeinsteinProposed1978Status checkedAug 14, 2026
Source recordedAug 14, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

10 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

10 selected steps

Scroll horizontally to explore the route

Working route overview for Higher-Dimensional Weinstein ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Every closed contact manifold should carry a periodic Reeb orbit. — Depends on missing premiseEvery closed contactmanifold should carry aperiodic…Current reduction — Depends on missing premiseCurrent reductionCanonical current identity — Depends on missing premiseCanonical current identityClosing target — Depends on missing premiseClosing targetPeriodic Reeb orbit — Depends on missing premisePeriodic Reeb orbitRegularity firewall — Depends on missing premiseRegularity firewallRigid affine leaf-space reduction outside its hypotheses — stoppedRigid affine leaf-spacereduction outside itshypothesesEliminate the nonlinear dense-holonomy T^3-bundle branch without assuming smooth affine linearization. — OpenEliminate the nonlineardense-holonomy T^3-bundlebranch…Prove a flux-to-time globalization lemma producing a closed one-form with nonzero Reeb average. — OpenProve a flux-to-timeglobalization lemmaproducing…Develop a reduced center fixed-point index that survives measurable or non-Hausdorff center geometry. — OpenDevelop a reduced centerfixed-point index thatsurvives…
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeRigid affine leaf-space reduction outside its hypotheses

The current work identifies nonlinear dense holonomy as the surviving regular branch after the rigid model is excluded. A projective cocycle analysis directly in the nonlinear transverse coordinate remains viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Eliminate the nonlinear dense-holonomy T^3-bundle branch without assuming smooth affine linearization.Suggested move: Construct its transverse suspension model and analyze the equivariant projective conormal cocycle under recurrence.
Ready to work on
02
Prove a flux-to-time globalization lemma producing a closed one-form with nonzero Reeb average.Suggested move: Track local Hamiltonian deck increments through the recurrent center pseudogroup and patch them globally.
Ready to work on
03
Develop a reduced center fixed-point index that survives measurable or non-Hausdorff center geometry.Suggested move: Use positivity and additivity of the return-time cocycle to obtain nonzero degree on one center ball.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusPartially resolved

The conjecture is proved for every closed three-dimensional contact manifold and for significant higher-dimensional classes, but the unrestricted assertion for all closed contact manifolds of dimension at least five remains open.

[2][3]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedAlbers and Hofer proved the conjecture for higher-dimensional contact structures containing a Plastikstufe.[3]
  2. Peer reviewedTaubes proved the Weinstein conjecture for closed oriented three-manifolds using Seiberg–Witten theory.[2]
  3. Historical sourceWeinstein established periodic-orbit results for convex Hamiltonian systems and formulated the conjectural closed-characteristic framework.[1]
3 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHigher-Dimensional Weinstein Conjecture
Solved special casecontact dimension three

Every Reeb vector field of a contact form on a closed oriented three-manifold has a closed orbit.

[2]
Solved special casehigher-dimensional contact structures with a Plastikstufe

The existence of a Plastikstufe implies the conjecture for all supporting contact forms.

[3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA formal statement needs closed contact manifolds, the Reeb vector field, and periodic integral curves in arbitrary odd dimension at least five.
  • Formalization targetA checked proof would require substantial contact/symplectic topology infrastructure; no checked formal proof of the unrestricted higher-dimensional statement was located.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

4 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction1 of 61
  • lemma3 of 63
  • negative result1 of 61
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
  • retained route statementEvery closed contact manifold should carry a periodic Reeb orbit.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementPeriodic Reeb orbitintermediate
  • retained route statementCanonical current identityintermediate
  • retained route statementRegularity firewallintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureRigid affine leaf-space reduction outside its hypothesesreported failure
  • Research targetEliminate the nonlinear dense-holonomy T^3-bundle branch without assuming smooth affine linearization.open
  • Research targetProve a flux-to-time globalization lemma producing a closed one-form with nonzero Reeb average.open
  • Research targetDevelop a reduced center fixed-point index that survives measurable or non-Hausdorff center geometry.open
  • Research targetNonlinear dense holonomysuperseded
  • Research targetFlux globalizationsuperseded
  • Research targetReduced center indexsuperseded
  • Narrowed routeRigid affine leaf-space reduction outside its hypothesesThe current work identifies nonlinear dense holonomy as the surviving regular branch after the rigid model is excluded. A projective cocycle analysis directly in the nonlinear transverse coordinate remains viable.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeEliminate the nonlinear dense-holonomy T^3-bundle branch without assuming smooth affine linearization.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

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Prepared starting pointEliminate the nonlinear dense-holonomy T^3-bundle branch without assuming smooth affine linearization.

Higher-Dimensional Weinstein Conjecture · ready to start

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Research contextPrepared context for any AI agent

Must every Reeb vector field on a closed contact manifold have a periodic orbit? Dimension three is known externally, but the unrestricted higher-dimensional problem remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Periodic orbits for convex Hamiltonian systemsoriginal source · Alan Weinstein · Annals of Mathematics 108(3), 507–518 · 1978 · DOI 10.2307/1971185 · MR 0512430 · accessed Aug 14, 2026
  2. 2
    The Seiberg-Witten equations and the Weinstein conjecturepeer reviewed result · Clifford Henry Taubes · Geometry & Topology 11, 2117–2202 · 2007 · ARXIV math/0611007 · DOI 10.2140/gt.2007.11.2117 · accessed Aug 14, 2026
  3. 3
    On the Weinstein conjecture in higher dimensionspeer reviewed result · Peter Albers, Helmut W. Hofer · Commentarii Mathematici Helvetici 84(2), 429–436 · 2009 · ARXIV 0705.3953 · DOI 10.4171/CMH/167 · accessed Aug 14, 2026

Important qualifications

  • External metadata was collected independently of the unreviewed source material; packet claims are not external evidence.
  • Scoped searches do not establish that no additional result, formalization, or dataset exists elsewhere.
  • No external computation was rerun, and no source-submitted URL or packet attachment was fetched, executed, or rendered.
  • This metadata grants no proof, acceptance, credit, publication, redistribution, deployment, or public-write authority.

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