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Yau's conjecture on the first eigenvalue

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Let n≥2n\ge 2 and let Sn+1={x∈Rn+2:∣x∣=1}S^{n+1}=\{x\in\mathbb{R}^{n+2}:|x|=1\} carry its standard round metric of constant sectional curvature 11. Let Mn⊂Sn+1M^{n}\subset S^{n+1} be a smooth, closed (compact and without boundary), embedded hypersurface which is minimal, i.e. whose mean curvature vanishes identically, and equip MnM^{n} with the metric induced from Sn+1S^{n+1}.

Let ΔM\Delta_{M} denote the Laplace-Beltrami operator of this induced metric, and let

0=λ0(M)<λ1(M)≤λ2(M)≤⋯0=\lambda_{0}(M)<\lambda_{1}(M)\le \lambda_{2}(M)\le\cdots

be the eigenvalues, listed with multiplicity, of the closed eigenvalue problem

ΔMf+λf=0,f∈C∞(M), f≢0,\Delta_{M} f+\lambda f=0,\qquad f\in C^{\infty}(M),\ f\not\equiv 0,

so that λ1(M)\lambda_{1}(M) is the first non-zero eigenvalue, characterized variationally by

λ1(M)=inf⁡{∫M∣∇f∣2 dv∫Mf2 dv  :  f∈C∞(M), f≢0, ∫Mf dv=0}.\lambda_{1}(M)=\inf\left\{\frac{\int_{M}|\nabla f|^{2}\,dv}{\int_{M} f^{2}\,dv}\;:\; f\in C^{\infty}(M),\ f\not\equiv 0,\ \int_{M} f\,dv=0\right\}.

Then

λ1(M)=n.\lambda_{1}(M)=n.
Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Yau's first-eigenvalue conjecture for embedded minimal hypersurfaces in spheres

    Let Σm\Sigma^m be a compact embedded minimal hypersurface in the unit sphere Sm+1S^{m+1}, and let λ1\lambda_1 denote the first eigenvalue of its Laplacian. Yau's conjecture. The first eigenvalue satisfies

    λ1=m.\lambda_1=m.

    The conjecture refines the general upper bound λ1≤m\lambda_1\leq m coming from the coordinate functions of the ambient Euclidean space. It is known in special cases, including totally geodesic spheres, while the general embedded case remains open.

    source: Yuhang Zhao, “The first eigenvalue of embedded minimal hypersurfaces in the unit sphere”, arXiv:2603.20890 (2026).

References

Primary source

Wikipedia

Additional references

  1. Wikipedia, Yau's conjecture on the first eigenvalue, the article this problem comes from.
geometryunsolved-problem
Posted

Progress summary

Refreshed
Claimed solved

A 2025 paper claimed a complete proof, but later work in 2026 continued to treat the conjecture as open.

Shing-Tung Yau posed the conjecture in 1982: every smooth, closed, embedded minimal hypersurface in the sphere should satisfy λ1=n\lambda_1=n. The full assertion remains unresolved in the available record.

Known results

  • Choi–Wang: the general lower bound λ1≥n/2\lambda_1\ge n/2.
  • Tang–Yan, 2013: the conjecture for minimal isoparametric hypersurfaces.
  • Choe–Soret: the conjecture for Lawson surfaces and Karcher–Pinkall–Sterling examples.
  • A 2024 result proves λ1=n\lambda_1=n under max⁡M∣A∣≤n\max_M|A|\le\sqrt n, covering the great sphere and Clifford torus cases.

August 2025 claim and 2026 updates

An August 2025 arXiv paper, The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau’s Conjecture, claims an affirmative resolution, λ1=n\lambda_1=n, but no verification was found. March and June 2026 papers report improved general lower bounds, not a proof of the conjecture, and describe it as unresolved.

Current status (as of September 2026): A complete proof was claimed in August 2025 but is unverified; later 2026 work reports only improved lower bounds, so the conjecture remains open.

Sources

Solutions 0

No solutions have been posted yet.