Spectral geometry and minimal hypersurfaces

Yau’s First-Eigenvalue Conjecture

Collaboration beta

For every closed, connected, embedded minimal hypersurface in the unit sphere, determine whether the first nonzero Laplace–Beltrami eigenvalue equals the hypersurface dimension.

λ_{1} (Σ) = n

Known results and sources

The mathematical object and exact first-eigenvalue question, with a recent proof claim still under review.

Research problem

Exact mathematical statement

Let $X: \Sigma^n\hookrightarrow S^{n+1}(1)$ be a closed, connected, embedded minimal hypersurface, and let $L=-\Delta_\Sigma$ be the nonnegative Laplace–Beltrami operator. Yau’s first-eigenvalue conjecture asks whether

\lambda_1(\Sigma)=n.

The coordinate functions give $\lambda_1\le n$ ; the missing direction is $\lambda_1\ge n$ . The submitted source explicitly reports that no full proof is claimed.

Problem infographic

Problem at a glance

A unit sphere contains a closed embedded minimal hypersurface, beside the induced Laplacian, its coordinate eigenfunctions at eigenvalue n, and the exact question whether the first eigenvalue is n. — Problem-first view: coordinate functions give eigenvalue n and the upper bound λ₁≤n; the exact equality question remains subject to a recent proof claim under review.

Current mathematical picture

Where work on Yau’s First-Eigenvalue Conjecture stands

Recent proof claim under review

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

Useful failurePurely local symbol closure

The source states that the required pairing begins two orders lower and that local symbol analysis is insufficient. A global boundary-integral, Poisson-kernel, harmonic-measure, or shape-derivative representation remains viable.

Route status · Narrowed route

Main reductionDirectional NP-square certificate [D]

The source reports an exact completed-square constraint that forces a deficit-dependent lower threshold on the special compact side asymmetry for any hypothetical counterexample.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeDerive a publication-grade global formula for the special commutator pairing B₀ = ½⟨∇f, [N⁻¹D, ν]f⟩.Task status · Ready to work on

Work mapped so far

Yau’s First-Eigenvalue Conjecture in numbers

1kretained lines of mathematical investigation1,036 in the current working snapshot

Argument development: 847 · 82%
Explored or eliminated routes: 21 · 2%
Computational analysis: 17 · 2%
Open obligations: 55 · 5%
Definitions and setup: 96 · 9%

8selected mapped statements2routes investigated4open questions4contribution-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 byShing-Tung YauProposed1982Status checkedAug 29, 2026

Source recordedAug 29, 2026

Recommended next task

Derive a publication-grade global formula for the special commutator pairing B₀ = ½⟨∇f, [N⁻¹D, ν]f⟩.

Suggested move: Start from the two complementary domains’ boundary-integral or Poisson kernels, track the mean-zero projections and nonsymmetric operator, and isolate a sign-controlled or quantitatively bounded remainder.

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

14 selected steps

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

Scroll horizontally to explore the route

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

2 recorded

Narrowed routePurely local symbol closure

Route status · Narrowed route

Narrowed routeFixed norm-gap closure near the target eigenvalue

The source records that eigenvalue proximity does not imply closeness to the coordinate eigenspace, since the first eigenfunction stays orthogonal to all coordinate traces. A deficit-sensitive estimate tied to a geometric norm tending to zero or to a compact limit with a classified equality case remains open.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Derive a publication-grade global formula for the special commutator pairing B₀ = ½⟨∇f, [N⁻¹D, ν]f⟩.Suggested move: Start from the two complementary domains’ boundary-integral or Poisson kernels, track the mean-zero projections and nonsymmetric operator, and isolate a sign-controlled or quantitatively bounded remainder.

Ready to work on

Extract a coercive inequality from the full graph-compatibility condition rather than merely re-proving exactness of the tangential one-form.Suggested move: Insert the source’s projected field into the graph equation, apply Hodge projections, and seek control of the special asymmetry pairing, its norm, or the coexact remainder by already positive quantities.

Ready to work on

Directional side-asymmetry obstruction [O]

Exclude the exact threshold-sized action of the compact asymmetry operator on the special projected normal density.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Build a deficit-sensitive estimate whose strength scales correctly as ε = n − λ ↓ 0 and does not assume spectral-subspace closeness.Suggested move: Identify a geometric norm that tends to zero or a compact counterexample limit with a classified equality case, using source orthogonality, flat-bundle bounds, weighted isotropy, and the linked certificate.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 29, 2026

Current statusRecent proof claim under review

Recent proof claim under review. The general conjecture should not be described as solved: a 2025 arXiv v1 claims a complete proof, but no independent or peer-reviewed validation was located, and peer-reviewed 2026 literature continues to state the equality as a conjecture while pursuing lower-bound improvements. Established results include the Choi–Wang lower bound, later quantitative improvements, and proofs for special classes such as isoparametric hypersurfaces.

[5][6]

External progress

What the literature has established

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

2026-02
Peer reviewedLi and Yan's peer-reviewed article still states the general equality as Yau's conjecture and summarizes lower bounds and solved special cases. Its post-claim treatment supports retaining a conservative unresolved posture.[6]
2025-08-08
PreprintZeng's single-version arXiv preprint claims a proof that λ₁=n for every closed embedded minimal hypersurface in the unit sphere. No independent or peer-reviewed resolution was located, so this is recorded only as an unverified proof claim.[5]
2024-08-13
Peer reviewedDuncan, Sire, and Spruck proved the first explicitly computable general improvement on Choi–Wang: λ₁≥n/2+aₙ/(Λ⁶+bₙ), where Λ bounds the norm of the second fundamental form. The bound does not resolve the conjecture.[4]
2013
Peer reviewedTang and Yan proved λ₁=n for every closed minimal isoparametric hypersurface in the unit sphere, establishing the conjecture for this important special class rather than for all embedded minimal hypersurfaces.[3]

Mathematical neighborhood

Related results and reusable starting points

Current focusYau's conjecture on the first eigenvalue

Solved special caseminimal isoparametric hypersurfaces in unit spheres

Tang and Yan proved the exact eigenvalue equality for closed minimal isoparametric hypersurfaces. This verifies the conjecture on a structured family but does not reduce the general embedded case to that family.

[3]

Formalization opportunities

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

Formalization targetAn exact reviewed formal statement fixing the unit-sphere normalization, embeddedness, minimality, connectedness conventions, the induced Laplace–Beltrami operator, and the meaning of the first nonzero eigenvalue.
Formalization targetA checked proof of the general lower bound λ₁≥n, together with statement alignment showing that it combines with the coordinate-function upper bound λ₁≤n to prove the exact conjecture.

Detailed research inventory

Claims, milestones, and routes in the current map

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

6 standing statements2 proposed statements4 open questions2 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction2 of 82
lemma5 of 85

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.23 displayed rows · 2 routes included

retained route statementFor a closed embedded minimal hypersurface in the unit sphere, must its first nonzero Laplace–Beltrami eigenvalue equal its dimension?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementCoordinate upper bound [S]intermediate
retained route statementTwo-sided Reilly identity [D]intermediate
retained route statementNodal fullness under the hypothetical condition λ<n [D]intermediate
retained route statementDirectional NP-square certificate [D]intermediate
retained route statementGraph compatibility [D]intermediate
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
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 failurePurely local symbol closurereported failure
Useful failureFixed norm-gap closure near the target eigenvaluereported failure
Research targetDerive a publication-grade global formula for the special commutator pairing B₀ = ½⟨∇f, [N⁻¹D, ν]f⟩.open
Research targetExtract a coercive inequality from the full graph-compatibility condition rather than merely re-proving exactness of the tangential one-form.open
Research targetBuild a deficit-sensitive estimate whose strength scales correctly as ε = n − λ ↓ 0 and does not assume spectral-subspace closeness.open
Research targetDirectional side-asymmetry obstruction [O]open
Narrowed routePurely local symbol closureThe source states that the required pairing begins two orders lower and that local symbol analysis is insufficient. A global boundary-integral, Poisson-kernel, harmonic-measure, or shape-derivative representation remains viable.
Narrowed routeFixed norm-gap closure near the target eigenvalueThe source records that eigenvalue proximity does not imply closeness to the coordinate eigenspace, since the first eigenfunction stays orthogonal to all coordinate traces. A deficit-sensitive estimate tied to a geometric norm tending to zero or to a compact limit with a classified equality case remains open.

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 bridgeDerive a publication-grade global formula for the special commutator pairing B₀ = ½⟨∇f, [N⁻¹D, ν]f⟩.

2 approaches have 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.

Continue the mathematics

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Prepared starting pointDerive a publication-grade global formula for the special commutator pairing B₀ = ½⟨∇f, [N⁻¹D, ν]f⟩.

Yau’s First-Eigenvalue Conjecture · ready to start

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Research contextPrepared context for any AI agent

For every closed, connected, embedded minimal hypersurface in the unit sphere, determine whether the first nonzero Laplace–Beltrami eigenvalue equals the hypersurface dimension.

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references6 cited works · next context review by Nov 29, 2026

The mathematical context was checked on Aug 29, 2026. Status can be refreshed sooner after a material result or claim.

1
Problem sectionoriginal source · Shing-Tung Yau · Princeton University Press · 1982 · DOI 10.1515/9781400881918-035 · MR MR0645762 · accessed Aug 29, 2026
2
A first eigenvalue estimate for minimal hypersurfacespeer reviewed result · Hyeong In Choi, Ai-Nung Wang · Journal of Differential Geometry · 1983 · DOI 10.4310/jdg/1214437788 · accessed Aug 29, 2026
3
Isoparametric foliation and Yau conjecture on the first eigenvaluepeer reviewed result · Zizhou Tang, Wenjiao Yan · Journal of Differential Geometry · 2013 · ARXIV 1201.0666 · DOI 10.4310/jdg/1370979337 · accessed Aug 29, 2026
4
An Improved Eigenvalue Estimate for Embedded Minimal Hypersurfaces in the Spherepeer reviewed result · Jonah A. J. Duncan, Yannick Sire, Joel Spruck · International Mathematics Research Notices · 2024-08-13 · ARXIV 2308.12235 · DOI 10.1093/imrn/rnae154 · accessed Aug 29, 2026
5
The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecturepreprint · Lingzhong Zeng · arXiv · 2025-08-08 · ARXIV 2508.06123 · DOI 10.48550/arXiv.2508.06123 · accessed Aug 29, 2026
6
A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifoldspeer reviewed result · Fagui Li, Junrong Yan · Differential Geometry and its Applications · 2026-02 · ARXIV 2308.02803 · DOI 10.1016/j.difgeo.2026.102330 · accessed Aug 29, 2026

Important qualifications

The 2025 arXiv v1 claims a complete proof, but this collection located no peer-reviewed publication, independent validation, authoritative acceptance, correction, or retraction. ProofAtlas did not review the proof, so the claim remains explicitly unverified.
A February 2026 peer-reviewed paper continues to treat the general statement as a conjecture. This is evidence against upgrading the statement to solved, but it is not itself a formal refutation of the 2025 preprint.
No scoped audit for formalizations, software, datasets, or maintained-problem-list recognition was performed. Empty readiness and recognition lists therefore do not establish nonexistence.

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