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. (Lean)

Manuscripts (2)

A counterexample to integer-degree harmonic dimension comparison

September 25, 2026 · source · Lean formalization available · cite as \cite{OAI:A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026}

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.

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

September 26, 2026 · source · cite as \cite{OAI:A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026}

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.

Result 361 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.