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Failure of integer-degree harmonic dimension comparison

Proves Yau's Euclidean bound on polynomial-growth harmonic functions fails under nonnegative Ricci curvature.

2 papers September 25, 2026 Explained by AI

A harmonic function is a smooth function that equals its own average around every small sphere. On flat three-dimensional space, there are exactly (k + 1)² independent ones that grow no faster than distance to the power k. Yau asked if nonnegative Ricci curvature always keeps this count at or below the flat one. The first paper builds a counterexample in dimension 16, and Lean, a computer proof checker, has a formal statement of it. A companion paper claims counterexamples on three-dimensional space for every large k.

First step
1989
Peter Li and Luen-Fai Tam, the sharp bound for linear-growth harmonic functions
Significance
Major
It answers the integer-degree form of Yau's comparison question negatively, with a Lean statement in one dimension.
Trust
Formal proof listed
Field
Differential geometry
2 papers

The explanation

On flat three-dimensional space, list the harmonic polynomials of degree at most 2. A harmonic function equals its own average around every small sphere. There is 1 constant. There are 3 linear ones: x, y and z. There are 5 independent quadratic ones, such as xy and x² − y². That makes 1 + 3 + 5 = 9. In general, degree at most k gives 1 + 3 + … + (2k + 1) = (k + 1)². On flat space, a harmonic function that grows no faster than distance to the power k is such a polynomial.

Ricci curvature is a kind of average curvature. Nonnegative Ricci curvature means space does not spread out faster than flat space. The comparison problem asks whether nonnegative Ricci curvature always keeps the count at or below the flat count, for every whole-number k. Li and Tam proved the sharp bound for linear growth, k = 1. Colding and Minicozzi proved the count is finite for every degree. Donnelly found failures at fractional degrees between one and two. Those do not decide the whole-number case.

The first paper answers no. It builds a complete metric with nonnegative Ricci curvature in some even dimension n ≥ 8. This metric has more harmonic functions of growth at most k than flat space. It allows the choices n = 16 and k = 50000. The flat count there is about 4.68 × 10⁵⁸. The key idea is a cone whose cross-section slowly changes shape. On a cone, each angular direction carries one growth rate d, set by d(d + m − 1) = λ, where λ is an eigenvalue. On a round two-sphere, λ = l(l + 1) gives d = l. A moving cross-section lets one function use several rates in turn. The obstacle is that a tiny part in a faster direction can take over after a long stretch.

The second paper claims the same failure on three-dimensional space. For large k it builds metrics with at least c(k + 1)² harmonic functions, for any c below 9v/4. Here v, between 4/9 and 1, measures volume growth relative to flat space. With c = 2 and k = 10, that is at least 242 functions instead of 121. The metric is chosen after k and may depend on it.

How sure is this? Lean, a computer proof checker, has a formal statement of the first paper's result in dimension 16 at degree 50000. So that counterexample is proved, as far as the Lean statement matches the paper. The three-dimensional companion and the tangent-cone claims are not in the Lean scope. No specialist review is published.

What it doesn't show. This does not give one fixed metric that beats the count for every k; the metric depends on k. It does not overturn the known finite-dimensionality bounds, and the three-dimensional companion has no Lean check.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major, with a Lean statement of one counterexample. The metric depends on the degree, and the three-dimensional result is a claim without a Lean check. No specialist review is published.

Words used

Harmonic function
A smooth function equal to its own average around every small sphere.
Polynomial growth
Growing no faster than a fixed power of the distance from a base point.
Ricci curvature
An average of curvature over directions; nonnegative means volume grows no faster than in flat space.
Asymptotic volume ratio
How the volume of huge balls compares with flat balls of the same radius.
Tangent cone at infinity
The cone a space looks like when viewed from very far away.
Eigenvalue
A number λ for which some nonzero function is turned into λ times itself.

The original papers (2)

Lean notes

All quoted lines (10)
  1. For an integer k≥0k\geq0, Hk(R3,gE)\mathcal H_k(\mathbb R^3,g_{\mathrm E}) consists of harmonic polynomials of degree at most kk; summing their homogeneous dimensions gives hk(R3,gE)=∑l=0k(2l+1)=(k+1)2.h_k(\mathbb R^3,g_{\mathrm E})=\sum_{l=0}^k(2l+1)=(k+1)^2.
    A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison Back to text
  2. The integer-degree comparison problem asks whether Ric⁡g≥0\operatorname{Ric}_g \geq0 forces hk(M,g)≤hk(Rn,gE)h_k(M,g)\leq h_k(\mathbb R^n,g_{\mathrm E}) for every integer k≥1k\geq1.
    A counterexample to integer-degree harmonic dimension comparison Back to text
  3. Colding and Minicozzi proved finite dimensionality at every fixed degree and later obtained bounds of the optimal order dn−1d^{n-1} as dd grows.
    A counterexample to integer-degree harmonic dimension comparison Back to text
  4. This subquadratic excess does not by itself establish an integer-degree violation.
    A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison Back to text
  5. There are an even integer n≥8n\geq8, an integer k≥2k\geq2, and a complete smooth metric gg on Rn\mathbb R^n such that Ric⁡g≥0\operatorname{Ric}_g\geq0 and hk(Rn,g)>hk(Rn,gE).h_k(\mathbb R^n,g)>h_k(\mathbb R^n,g_{\mathrm E}).
    A counterexample to integer-degree harmonic dimension comparison Back to text
  6. The exact finite spectral selection in Appendix … permits the choices n=16n=16 and k=50000k=50000.
    A counterexample to integer-degree harmonic dimension comparison Back to text
  7. On the metric cone dr2+r2hdr^2+r^2h over an mm-dimensional link, an angular eigenfunction of the nonnegative Laplacian with eigenvalue λ\lambda has a growing homogeneous harmonic mode rdϕr^d\phi, where d(d+m−1)=λ,d≥0.d(d+m-1)=\lambda,\qquad d\geq0.
    A counterexample to integer-degree harmonic dimension comparison Back to text
  8. The challenge is to arrange that succession exactly: an arbitrarily small component in an unwanted faster direction can dominate after a sufficiently long interval.
    A counterexample to integer-degree harmonic dimension comparison Back to text
  9. For every v∈(4/9,1)v\in(4/9,1) and c∈(1,9v/4)c\in(1,9v/4), there is an integer k0(v,c)k_0(v,c) such that for every integer k≥k0(v,c)k\geq k_0(v,c) there is a complete smooth metric gkg_k on R3\mathbb R^3, Euclidean near the origin, with Ric⁡gk≥0\operatorname{Ric}_{g_k}\geq0, AVR⁡(gk)=v\operatorname{AVR}(g_k)=v, and hk(R3,gk)≥c(k+1)2>(k+1)2=hk(R3,gE).h_k(\mathbb R^3,g_k)\geq c(k+1)^2>(k+1)^2 =h_k(\mathbb R^3,g_{\mathrm E}).
    A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison Back to text
  10. The metric is chosen after kk and may depend on it.
    A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison Back to text

Other sources

  1. openai/math, Lean scope note for family 361 The construction uses dimension 1616 and degree 5000050000.
  2. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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