Yau's harmonic dimension comparison problem
From OpenAI Aftermath
Yau's harmonic dimension comparison problem asks whether, on a complete manifold of dimension n with nonnegative Ricci curvature, the space of harmonic functions of polynomial growth at most k has dimension at most that of the corresponding space on Euclidean Rn, for every integer k. Two manuscripts dated 25 and 26 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim counterexamples, in high even dimensions and in dimension three. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; this page does not certify them.
Background
- As recounted in the introduction of the three-dimensional manuscript:
- Yau posed the comparison together with the finite-dimensionality problem for harmonic functions of polynomial growth. Li and Tam proved the sharp bound at linear growth, and Colding and Minicozzi proved finite dimensionality at every fixed degree under nonnegative Ricci curvature, with dimension bounds of the optimal order in the degree.
- Donnelly's examples in dimensions at least five violate the Euclidean comparison at a suitable real degree between one and two, which does not by itself give an integer-degree violation.
- Comparison results hold under additional hypotheses: Cai and Lai in the locally conformally flat case, Lin, Wang and Xu in dimension three under nonnegative sectional curvature and positive asymptotic volume ratio, and Huang through the spectrum of the cone link when the tangent cone at infinity is unique.
The original results
- Claim (higher dimensions, 25 September 2026). A counterexample to integer-degree harmonic dimension comparison states that for some even n ≥ 8 and integer k ≥ 2 there is a complete smooth metric on Rn with nonnegative Ricci curvature, Euclidean near the origin, with asymptotic volume ratio strictly between zero and one and nonunique tangent cones at infinity, whose harmonic functions of growth at most k exceed the Euclidean harmonic-polynomial count. The construction uses odd-dimensional Berger sphere links.
- Claim (dimension three, 26 September 2026). Theorem 1.1 of A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison states that for every v in (4/9, 1) and c in (1, 9v/4), every sufficiently large integer k admits a complete smooth metric on R3, Euclidean near the origin, with nonnegative Ricci curvature and asymptotic volume ratio v, having at least c(k + 1)2 linearly independent harmonic functions of growth at most k, against the Euclidean count (k + 1)2. The metric depends on k. A corollary gives such metrics within any prescribed bi-Lipschitz distance of the Euclidean metric, for every c below 9/4.
- Method, as described: the three-dimensional proof does not use the Berger-link construction; its angular eigenspaces move, and it controls the exchange of selected low modes together with leakage into infinitely many higher modes.
- Sources. At commit
1581d4c: A counterexample to integer-degree harmonic dimension comparison and A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison. - Formalization, as described by the repository. The scope note lean/docs/361.md concerns the higher-dimensional manuscript: it states that the formalized result constructs one complete metric with the stated properties on an even-dimensional Euclidean space, using dimension 16 and degree 50000, and that the tangent-cone and cone-spectrum conclusions are not included. The Comparator challenge is
HarmonicGrowth.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 23 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | mxym, Simultaneous integer-degree failures of harmonic dimension comparison (mxym/math, notes/simultaneous-harmonic-degree-blocks) | Extension (quantitative strengthening) | Using that the spaces of polynomial-growth harmonic functions increase with the degree, together with the full constant of the three-dimensional theorem, a single metric violates the Euclidean comparison at every integer degree in a block whose ratio of endpoints can be any number below 3/2: if β > 1 and A > 1 satisfy Aβ2 < 9/4, the same metric has an excess by the factor A throughout the block. | Repository note prepared with AI assistance; not peer reviewed. |
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit bb5fdb6, 18 October 2026) |
Review (fidelity of the formal statements) | Rates this family "weaker-statement" and records a sandboxed Comparator pass for HarmonicGrowth. Its note states that the formal claim is a single higher-dimensional counterexample, so the three-dimensional claim of the collection's summary is not implied, and that the challenge file states the claim through an axiom rather than leaving a placeholder proof, while no axiom occurs in the solution. |
MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | cert-machine, openai/math lane K: run 38015544207 for HarmonicGrowth | Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept the solution, in 1242 s. | Repository. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_333 of the three-dimensional manuscript in JJardine919/openai-math-audit | Review (AI-assisted audit) | Reports no gap after reading every section. As a sanity check, it notes that the threshold c < 9v/4 is exactly what an obstruction of Colding–Minicozzi type allows. | Part of an audit of the collection's manuscripts that the audit describes as complete. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
