The exponential neck has Q≡1, zero projective area and zero local period, while sin(kx)sin(ky) on the flat torus has 4k² critical nodal points; the model Q=m/r also gives infinitely many adaptive radius steps to the core. Threshold-frozen cells, positive divisor screening and an orthogonal Laurent corona remain viable on surfaces; projective-area-paid coherent forests remain the higher-dimensional proposal.
Route status · Narrowed routeGeometric analysis · spectral geometry · nodal sets
Yau’s Nodal-Set Upper Bound
Collaboration betaFor a Laplace eigenfunction on any smooth compact Riemannian manifold, is the hypersurface measure of its zero set bounded above by a constant times its frequency √λ?
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Listed inYau's 1982 differential-geometry problem list
Known results and sources 
Research problem
Exact mathematical statement
Let be a smooth compact -dimensional Riemannian manifold and let be a Laplace eigenfunction satisfying
For its nodal set , prove that there is a constant depending only on such that
This is the sharp upper-bound half of Yau's nodal-set conjecture for arbitrary smooth metrics. The submitted source explicitly says that it contains no complete proof.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Yau’s Nodal-Set Upper Bound stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
The harmonic extension and normalized first jet reduce the sharp upper bound to a global L1 bound for the derivative of ν, together with the retained Crofton–Prüfer module.
Evidence posture · Source-reported route statement · dependencies incompleteWe 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 We removed a duplicate or outdated task or route step.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.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Yau’s Nodal-Set Upper Bound in numbers
3kretained lines of mathematical investigation3,047 in the current working snapshot
- Argument development
- 2,662 · 87%
- Explored or eliminated routes
- 40 · 1%
- Computational analysis
- 2 · 0%
- Open obligations
- 139 · 5%
- Definitions and setup
- 204 · 7%
7selected 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.
Recommended next task
Prove bounded-overlap Laurent-corona selection and gauge-coherent amplitude-energy telescoping on surfaces.
Suggested move: Audit annular screening in one fixed isothermal disk, then build divisor-weighted ball growth and prove the parent/child amplitude inequality before summing the surface decomposition.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
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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
The exponential neck has Q≡1, zero projective area and zero local period, while sin(kx)sin(ky) on the flat torus has 4k² critical nodal points; the model Q=m/r also gives infinitely many adaptive radius steps to the core. Threshold-frozen cells, positive divisor screening and an orthogonal Laurent corona remain viable on surfaces; projective-area-paid coherent forests remain the higher-dimensional proposal.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Prove bounded-overlap Laurent-corona selection and gauge-coherent amplitude-energy telescoping on surfaces.Suggested move: Audit annular screening in one fixed isothermal disk, then build divisor-weighted ball growth and prove the parent/child amplitude inequality before summing the surface decomposition.
Ready to work on
Prove the smooth higher-dimensional projective two-area estimate with one canonical complexity N_B.Suggested move: Choose and freeze N_B, prove its comparison to frequency and doubling, then attack the projective-area inequality by a single auditable method.
Ready to work on
Close the higher-dimensional coherent neck-forest packing law.Suggested move: Use invariant first-jet tensors to glue threshold-frozen necks and quantify vector-charge transport, leakage and width-times-length cost.
Ready to work on
Sourced mathematical context
The known mathematical landscape
External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Peer reviewedLogunov proved a polynomial upper bound in smooth dimensions at least three, with exponent larger than the conjectured one-half.[3] Peer reviewedLogunov proved the sharp lower bound for smooth compact manifolds.[4] Peer reviewedDonnelly and Fefferman proved the sharp estimate for real-analytic manifolds.[2] Historical sourceYau posed the two-sided square-root growth question for nodal hypersurface measure.[1]
Mathematical neighborhood
Related results and reusable starting points
Current focusYau's nodal-set conjecture: sharp smooth upper bound
The sharp two-sided estimate holds for real-analytic metrics.
[2]The lower-bound half is proved for arbitrary smooth metrics; this workspace asks for the upper half.
[4]A polynomial smooth upper bound is known in dimensions at least three, but its exponent is larger than the sharp exponent.
[3]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetFormalized spectral geometry, Hausdorff measure, and quantitative unique-continuation infrastructure sufficient for this target was not identified.
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 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.
Correction detailsCorrected the research recordCorrection note
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.
5 standing statements2 proposed statements3 open questions1 narrowed routes
- theorem candidate
1 of 7 1 - reduction
2 of 7 2 - lemma
4 of 7 4
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
- retained route statementThe nodal-set measure should grow at most linearly with eigenfunction frequency
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementNormalized first-jet reductionintermediate
- retained route statementSharp surface scalar defectintermediate
- retained route statementGlobal polar energy budgetintermediate
- retained route statementGeneralized-analytic surface factorizationintermediate
- 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
- 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 failureLocal-charge and adaptive-chain packingreported failure
- Research targetProve bounded-overlap Laurent-corona selection and gauge-coherent amplitude-energy telescoping on surfaces.open
- Research targetProve the smooth higher-dimensional projective two-area estimate with one canonical complexity N_B.open
- Research targetClose the higher-dimensional coherent neck-forest packing law.open
- Research targetLaurent-corona theoremsuperseded
- Research targetHigher-dimensional projective routesuperseded
- Narrowed routeLocal-charge and adaptive-chain packingThe exponential neck has Q≡1, zero projective area and zero local period, while sin(kx)sin(ky) on the flat torus has 4k² critical nodal points; the model Q=m/r also gives infinitely many adaptive radius steps to the core. Threshold-frozen cells, positive divisor screening and an orthogonal Laurent corona remain viable on surfaces; projective-area-paid coherent forests remain the higher-dimensional proposal.
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
1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.
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 pointProve bounded-overlap Laurent-corona selection and gauge-coherent amplitude-energy telescoping on surfaces.
Yau’s Nodal-Set Upper Bound · ready to start
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Research contextPrepared context for any AI agent
For a Laplace eigenfunction on any smooth compact Riemannian manifold, is the hypersurface measure of its zero set bounded above by a constant times its frequency √λ?
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- Current routes and known obstacles
- What a useful result should report
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Sources and references4 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.
- 1Problem section, Seminar on Differential Geometryoriginal source · Shing-Tung Yau · Princeton University Press · 1982 · DOI 10.1515/9781400881918-035 · accessed Aug 14, 2026
- 2Nodal sets of eigenfunctions on Riemannian manifoldspeer reviewed result · Harold Donnelly, Charles Fefferman · Inventiones Mathematicae · 1988 · DOI 10.1007/BF01393691 · accessed Aug 14, 2026
- 3Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measurepeer reviewed result · Alexander Logunov · Annals of Mathematics · 2018 · DOI 10.4007/annals.2018.187.1.4 · MR 3739231 · accessed Aug 14, 2026
- 4Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecturepeer reviewed result · Alexander Logunov · Annals of Mathematics · 2018 · DOI 10.4007/annals.2018.187.1.5 · accessed Aug 14, 2026
Important qualifications
- Scoped to the original problem volume and primary journal sources; later specialist refinements were not exhaustively inventoried.
- This metadata tracks the arbitrary-smooth sharp upper-bound target, not boundary-domain or random-wave variants.
- No formalization or maintained computation was identified in the scoped search.
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