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Yau's dimension comparison for harmonic functions of integer polynomial growth under nonnegative Ricci curvature

For a complete Riemannian manifold (Mn,g)(M^n,g) let Hd(M)\mathcal H_d(M) be the space of harmonic functions with ∣u(x)∣≤C(1+d(o,x))d|u(x)|\le C(1+d(o,x))^d and hd(M)=dim⁡Hd(M)h_d(M)=\dim\mathcal H_d(M). On Rn\mathbb R^n, hk=(n+k−1k)+(n+k−2k−1)h_k=\binom{n+k-1}{k}+\binom{n+k-2}{k-1} for integer k≥1k\ge1. Li and Tam proved h1(M)≤n+1h_1(M)\le n+1 when Ric≥0\mathrm{Ric}\ge0, Colding and Minicozzi proved finite dimensionality and bounds of optimal order kn−1k^{n-1}, and Donnelly found counterexamples at non-integer degrees between one and two. Yau asked about sharp Euclidean comparison. Does Ricg≥0\mathrm{Ric}_g\ge0 imply hk(Mn)≤hk(Rn)h_k(M^n)\le h_k(\mathbb R^n) for every integer k≥1k\ge1?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Riemannian geometry; harmonic functions of polynomial growth
Posed by
Shing-Tung Yau, Open problems in geometry, Problem 48
Year posed
1994
Years open
32y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
25 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there are an even n≥8n\ge8 (the paper allows n=16n=16), an integer k≥2k\ge2 (k=50000k=50000) and a complete smooth metric on Rn\mathbb R^n with Ric≥0\mathrm{Ric}\ge0, Euclidean near the origin, asymptotic volume ratio strictly between 0 and 1, and hk>hk(Rn)h_k>h_k(\mathbb R^n); the metric has nonunique tangent cones at infinity. Companion: in dimension three, for 4/9<v<14/9<v<1, 1<c<9v/41<c<9v/4 and all large kk, metrics with Ric≥0\mathrm{Ric}\ge0, AVR vv and hk≥c(k+1)2h_k\ge c(k+1)^2, which can be made (1+ϵ)(1+\epsilon)-bi-Lipschitz to Euclidean. The metric depends on kk. Not shown: a single metric violating the bound for all large kk, or anything about the finite-dimensionality results, which stand.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Two manuscripts: the principal one (September 25, 2026) in some even dimension n >= 8, and a three-dimensional companion (September 26, 2026). The principal manuscript ships an optional exact integer check of its finite spectral selection for n = 16, k = 50000, which it says does not verify the metric.

Verification

No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 read against Yau's question as cited. Lean (in lean/formalization.yaml): OAI.HarmonicCounterexample.main in OAI/Geometry/HarmonicGrowth/Main.lean. Its statement MainClaim gives an even n≥8n\ge8, k≥2k\ge2 and a smooth metric on Rn\mathbb R^n that is complete for the intrinsic distance, has Ric≥0\mathrm{Ric}\ge0, is Euclidean near 0, has asymptotic volume ratio in (0,1)(0,1), and carries hk(Rn)+1h_k(\mathbb R^n)+1 linearly independent harmonic functions of growth kk. That states the headline negative answer. The challenge file states MainClaim through a placeholder axiom; the solution module proves main as a theorem, and the comparator's permitted axioms exclude custom ones. Not rebuilt here. The 3D companion is not formalized.

Sources

Changelog1 change

Discussion