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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 ℝ

^{3}

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.

Manuscripts in this family · 2

A counterexample to integer-degree harmonic dimension comparison

Sep 25, 2026·paper.pdfLean

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.

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

Sep 26, 2026·paper.pdf

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.

Full abstract

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