Math Conjectures

Yau First-Eigenvalue Conjecture

OPENLandmarkConjectureProposed 1982 · Full conjecture

Canonical statement

If Σn\Sigma^n is a closed embedded minimal hypersurface of the unit sphere Sn+1(1)S^{n+1}(1), then the first positive eigenvalue of its Laplace--Beltrami operator is λ1(Σ)=n\lambda_1(\Sigma)=n.
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If \(\Sigma^n\) is a closed embedded minimal hypersurface of the unit sphere \(S^{n+1}(1)\), then the first positive eigenvalue of its Laplace--Beltrami operator is \(\lambda_1(\Sigma)=n\).
GEO-SPECTRAL-027:[Yau1982ProblemSection][ChoiWang1983Eigenvalue][Zeng2025YauClaim][Yu2026Eigenvalue]
Geometry and topologyAnalysis and functional analysis

Notes

Yau conjectured that every closed embedded minimal hypersurface Mn⊂Sn+1M^n\subset S^{n+1} has first positive Laplace eigenvalue λ1(M)=n\lambda_1(M)=n [Yau1982ProblemSection]. The coordinate functions restricted from the sphere are eigenfunctions with eigenvalue nn, so the hard direction is the lower bound λ1≥n\lambda_1\ge n.

Choi and Wang proved the foundational estimate λ1≥n/2\lambda_1\ge n/2 [ChoiWang1983Eigenvalue]. Subsequent geometric estimates improve the picture under additional hypotheses, and Yu obtained further progress in the general embedded setting in 2026 [Yu2026Eigenvalue], but the sharp value nn is still out of reach.

Zeng's 2025 preprint announced the full conjecture [Zeng2025YauClaim], but its central Rayleigh-quotient step does not justify the claimed lower bound: testing one function supplies an upper bound for λ1\lambda_1, not the needed universal lower bound. With that claim unestablished and no replacement proof known, Yau's conjecture remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.