Failure of integer-degree harmonic dimension comparison

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.

For every 4/9< v< 1 and 1< c< 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< 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.