{"id":"openai-math/A-counterexample-to-integer-degree-harmonic-dimension-comparison","pdf":"https://github.com/openai/math/blob/main/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026","tags":["geometric-analysis","harmonic-functions","ricci-curvature","asymptotic-geometry"],"score":0,"title":"A 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 some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝ n with nonnegative Ricci curvature whose space of real harmonic functions of pointwise polynomial growth at most k has dimension larger than the Euclidean harmonic-polynomial dimension. The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. This answers the integer-degree form of Yau's dimension comparison question in the negative.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":true,"family_title":"Failure of integer-degree harmonic dimension comparison"}