Failure of integer-degree harmonic dimension comparison

What OpenAI claims

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. (Lean)

Papers (2)

1. A counterexample to integer-degree harmonic dimension comparison

   25 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract For some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝn 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.

2. A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison

   26 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract For every 4/9 < v < 1 and 1 < c < 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 < c < 9/4, these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.