{"id":"openai-math/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison","pdf":"https://github.com/openai/math/blob/main/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026","tags":["riemannian-geometry","harmonic-functions","geometric-analysis"],"score":0,"title":"A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison","domain":"Differential geometry","family":"361","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"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 ℝ 3 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 .","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-26","reactions":{},"formalized":false,"family_title":"Failure of integer-degree harmonic dimension comparison"}