Browse › Differential geometry › 361 Failure of integer-degree harmonic dimension comparison
A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison
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Abstract
For every $4/9\lt v\lt 1$ and $1\lt c\lt 9v/4$, all sufficiently large integers k admit a complete smooth metric on ℝ3 with nonnegative Ricci curvature, asymptotic volume ratio v, and at least $c(k+1)^2$ linearly independent real harmonic functions of pointwise growth at most k. This answers Yau's integer-degree dimension comparison question negatively in dimension three, with a fixed-factor excess over the Euclidean count. For each $1\lt c\lt 9/4$, these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Failure of integer-degree harmonic dimension comparison: Lean scope note (OpenAI Math Release, 2026). The documented formalization covers the even-dimensional construction at dimension 16 and degree 50000, including the Euclidean core and asymptotic volume ratio in (0,1), with separate tangent-cone and cone-spectrum conclusions excluded. [family-level]
Possibly relevant literature (found by the agent, not checked)
- X.-T. Huang, On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth (arXiv:1705.05008, May 2017). This paper estimates h_d as d grows on a single manifold with maximal volume growth and a unique tangent cone, which bears on this paper's remark that its metrics depend on k and give no asymptotic bound on one fixed manifold; it is distinct from the Huang 2023 paper the manuscript cites.
Digests and expositions
- Latent Seminar, Harmonic dimension beyond the Euclidean count: changing links and averaged growth (Latent Seminar, 2026-10-07). This exposition compares both constructions, explains averaged growth and exact harmonic continuation, and reports a rerun of the first paper's finite arithmetic check while explicitly distinguishing it from verification of the complete proofs. [family-level]
Elsewhere
- Emergent Mind: family 361
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026,
author = {{OpenAI}},
title = {{A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf}{OAI:A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/361.md. The Lean statements have not been checked by people against the paper's claims.
Family 361: Failure of integer-degree harmonic dimension comparison
Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ3 has at least $(k+2)^2$ independent harmonic functions of growth at most k, exceeding the Euclidean count $(k+1)^2$. The metric may depend on k.
Other papers in this family:
- A counterexample to integer-degree harmonic dimension comparison · September 25, 2026