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