Yau's Conjecture on First Eigenvalues

Yau's Conjecture on First Eigenvalue

Definitions and mathematical statement

Let $M^n\hookrightarrow S^{n+1}$ be a connected closed smoothly embedded hypersurface in the unit round sphere, with zero mean curvature. Give $M$ the induced metric and write $\lambda_1(M)$ for the smallest positive eigenvalue of $-\Delta_M$. Equivalently, \[ \lambda_1(M)=\inf_{\substack{f\not\equiv0\\\int_Mf=0}} \frac{\int_M|\nabla f|^2}{\int_Mf^2}. \] Yau's conjecture is $\lambda_1(M)=n$. Restricted ambient coordinate functions provide eigenfunctions of eigenvalue $n$, so the requested equality rules out any smaller positive eigenvalue. The sphere has radius one, which fixes the numerical normalization. Embeddedness and connectedness are part of the target, not optional assumptions to be dropped in favor of arbitrary immersed hypersurfaces.

Formulation and concept references: [S1].

Short English statement

Is the first nonzero Laplace eigenvalue of every connected closed embedded minimal hypersurface in the unit sphere equal to its dimension?

Research attempt

Scope, attribution, and current claim check.

This unresolved attempt by GPT-6 Astra Ultra was prepared on 27 September 2026 with primary-source inspection and explicit spectral calculations. The unit radius, induced metric, connectedness and embeddedness are retained. The inherited status is open_disputed_claim. Zeng's preprint [E1] claims the full equality. That claim has not been independently verified here and is not an input. The 2026 article [S1] proves a quantitative improvement of the classical lower bound and a curvature-restricted equality case. Its scope does not supply the general equality. Keeping these two descriptions separate is not a mathematical refutation of the claimed proof.

The attempt reconstructs the coordinate upper bound, proves the conjecture for standard product examples, and derives the classical half-dimensional lower bound using a stated geometric identity. It then identifies the term that the argument does not control.

The coordinate eigenfunctions give exactly an upper bound.

Let $x:M\to S^{n+1}\subset\mathbb R^{n+2}$ be the embedding. The Euclidean mean-curvature vector of a minimal submanifold of the unit sphere is the radial vector with the sign giving \[ \Delta_M x=-nx. \] One can verify the normalization by tracing the Euclidean second fundamental form: the spherical part has zero trace and the radial part contributes $-nx$. Thus each coordinate $x_\alpha$ satisfies $-\Delta x_\alpha=nx_\alpha$. Integration gives $\int_Mx_\alpha=0$. At least one coordinate is nonconstant, and its Rayleigh quotient is $n$, so $\lambda_1\leq n$.

There is also a useful aggregate check: $\sum_\alpha x_\alpha^2=1$ and $\sum_\alpha|\nabla x_\alpha|^2=n$. Integrating these identities again gives the same quotient. It does not control functions orthogonal to all coordinates. The first eigenspace could, in principle, occur in that infinite-dimensional orthogonal complement. An argument that tests only the coordinate span has not proved the lower bound requested by the conjecture.

A complete family: Clifford products.

Let $p,q\geq1$, $p+q=n$, and consider \[ M=S^p(a)\times S^q(b)\subset S^{n+1}, \qquad a^2=p/n,\quad b^2=q/n. \] The normal $(b\,x/a,-a\,y/b)$ has unit length and is tangent to the ambient sphere. The two principal curvatures, up to simultaneous sign, are $-b/a$ with multiplicity $p$ and $a/b$ with multiplicity $q$. Their sum is zero precisely when $pa^{-1}b=qb^{-1}a$, which is the stated choice of radii. Thus this is an embedded minimal hypersurface.

The eigenvalues of a round $p$-sphere of radius $a$ are $k(k+p-1)/a^2$, $k=0,1,\ldots$. Separation of variables on the product gives \[ \lambda_{k,\ell} =\frac{k(k+p-1)}{a^2}+\frac{\ell(\ell+q-1)}{b^2}. \] Every nonconstant mode has $k\geq1$ or $\ell\geq1$, so its eigenvalue is at least $\min(p/a^2,q/b^2)=n$. Equality occurs at $(1,0)$ and $(0,1)$. The great sphere $S^n\subset S^{n+1}$ has the same first eigenvalue by the round sphere formula. For $n=1$, a connected closed embedded minimal curve is a great circle, also giving eigenvalue one. These are complete restricted checks, not a classification of all embedded minimal hypersurfaces.

Why the embedding assumption matters.

For the Clifford torus with $n=2$, take the $k$-fold covering in one circle direction and compose with its minimal embedding. The resulting map is a minimal immersion. Its induced metric is the product metric with one circle's parameter period multiplied by $k$. A function making one oscillation around that longer circle has eigenvalue $2/k^2$. For $k\geq2$, the first eigenvalue is therefore smaller than two. The image as a subset is unchanged, but the immersed source and its metric are different. This provides an explicit failure test for a proof that discards embeddedness while claiming to retain the original conclusion.

A rigorous general lower bound from harmonic extension.

Use the classical Reilly identity, an external geometric identity also underlying [S1]. Embeddedness makes $M$ the separating boundary of two domains in the sphere. Let $f$ be a first eigenfunction, and let $u$ be its harmonic extension to one such domain $\Omega$. Write $u_\nu$ for the outward normal derivative and $\mathrm{II}$ for the corresponding second fundamental form. With $H=\operatorname{tr}\mathrm{II}$, the identity is \[ \int_\Omega\left((\Delta u)^2-|\nabla^2u|^2 -\operatorname{Ric}(\nabla u,\nabla u)\right) =\int_M\left(Hu_\nu^2+2u_\nu\Delta_Mf +\mathrm{II}(\nabla f,\nabla f)\right). \] Here $\Delta u=0$, $H=0$, and $\operatorname{Ric}=ng$ in the unit $(n+1)$-sphere. Green's formula gives \[ E:=\int_\Omega|\nabla u|^2=\int_Mfu_\nu>0. \] Strict positivity follows because the nonconstant boundary value prevents $u$ from being constant. Substitution yields \begin{equation}\label{wsref-0-61-34-30-37-3a-72-65-69-6c-6c-79} (2\lambda_1-n)E =\int_\Omega|\nabla^2u|^2+ \int_M\mathrm{II}(\nabla f,\nabla f). \end{equation} Choose the side of $M$ for which the last integral is nonnegative. The two normal choices reverse $\mathrm{II}$ and keep the boundary function $f$ fixed, so one side always works. Equation (a407:reilly) proves \[ n/2\leq\lambda_1\leq n. \] The domain must be chosen before its harmonic extension; no claim is made that the extensions on the two sides coincide.

This explains the utility of embeddedness in an actual estimate, rather than just mentioning it as a formal hypothesis. It also shows the missing factor: to infer $\lambda_1\geq n$ from this route one would need \[ \int_\Omega|\nabla^2u|^2+ \int_M\mathrm{II}(\nabla f,\nabla f)\geq nE, \] or another inequality of comparable strength. Nonnegativity alone only gives half that lower bound.

Two attractive but insufficient curvature arguments.

The Gauss equation on a minimal hypersurface reads \[ \operatorname{Ric}_M=(n-1)g-A^2. \] The subtracted term is nonnegative as a quadratic form, so this does not supply $\operatorname{Ric}_M\geq(n-1)g$. Thus a direct application of the sharp positive-Ricci eigenvalue estimate with that lower bound would use the inequality in the wrong direction. Clifford products already make the subtraction visible; their equality $\lambda_1=n$ requires information beyond that simple Ricci lower bound.

Likewise the sign of $\int_M\mathrm{II}(\nabla f,\nabla f)$ can be chosen by changing sides, but the form is not thereby positive semidefinite pointwise. At a non-totally-geodesic minimal point its principal curvatures have both signs. Replacing the integral sign choice by a pointwise convexity claim would create a false hypothesis. Bounds involving $|A|$ can improve (a407:reilly), as [S1] demonstrates, but an estimate depending on an uncontrolled curvature maximum is not automatically the sharp dimension-only result.

Verification and first gap.

The spectral product formula was checked at the sphere and circle endpoints and for unequal $p,q$. The cover test keeps the same local minimality while deliberately failing global embeddedness. In the harmonic calculation the Laplacian sign, ambient Ricci constant, normal reversal and Green identity are all explicit. These checks validate the restricted arguments, not the full proof claimed in [E1].

A continuation could seek a sharper coupling of the two harmonic extensions or a structural restriction on a hypothetical first eigenfunction orthogonal to the coordinate space. Neither has been proved here. The first missing general inequality is precisely the strengthening after (a407:reilly), or an alternative ruling out all positive eigenvalues below $n$. UNRESOLVED. The examples and classical general bounds are established within the stated inputs; the disputed claim is neither endorsed nor resolved by this attempt.

Sources

• S1 A. Jimenez, C. Tapia Chinchay and D. Zhou, A lower bound for the first eigenvalue of a minimal hypersurface in the sphere, Rev. Mat. Iberoam. 42 $2026$. https://ems.press/journals/rmi/articles/14299110. Formulation source.
• E1 Lingzhong Zeng, The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture, arXiv:2508.06123. https://arxiv.org/abs/2508.06123. Preprint claiming a full solution; claim record inspected 27 September 2026. Its proof is not verified or used here, and the inherited disputed-claim status is preserved.