Failure of integer-degree harmonic dimension comparison

1 · The essentials

These manuscripts ask whether a curved space can support more independent harmonic functions than ordinary flat space, when the functions obey the same limit on growth. Harmonic functions are functions satisfying a particular equation determined by the space’s geometry. The comparison concerns spaces whose curvature satisfies a specified nonnegativity condition. The main manuscript gives a counterexample in some even dimension at least eight, and its main result has a Lean formalization. A companion manuscript shows in the manuscript that counterexamples also exist in three dimensions, for every sufficiently large integer growth limit. In those three-dimensional examples, the number of independent functions exceeds the flat-space count by a fixed factor greater than one. The geometry may change with the chosen growth limit, so this is not a claim that one example works for all limits.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine comparing how many independent tunes two instruments can play while obeying the same loudness limit. The proposed comparison would make the flat-space instrument the benchmark that cannot be exceeded; these manuscripts describe geometries that exceed it. The image does not capture the equation or the curvature restrictions.

Why it matters

The results give a negative answer to the integer-degree version of Yau’s dimension comparison question under nonnegative Ricci curvature. The companion manuscript goes beyond a small excess: its three-dimensional examples have a fixed-factor excess over the Euclidean count. It also states that these examples can be chosen arbitrarily close to Euclidean geometry in global bi-Lipschitz distance, so that kind of closeness does not preserve the proposed bound.

2 · Going further

The precise question compares the dimension of the space of real harmonic functions of pointwise polynomial growth at most an integer $k$ on a complete smooth manifold with nonnegative Ricci curvature against the Euclidean harmonic-polynomial dimension. The main manuscript constructs a counterexample on $R^{n}$ for some even $n \geq 8$ and some integer $k \geq 2$ . Its metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. The main result has a Lean formalization.

The companion manuscript gives a three-dimensional statement with explicit parameters. For every $4/9 < v < 1$ and $1 < c < 9 v /4$ , every sufficiently large integer $k$ admits 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 $k$ . Since the Euclidean count is $(k + 1)^{2}$ , the condition $c > 1$ gives a fixed-factor excess. The repository summary also states an excess of at least $(k + 2)^{2}$ functions for every sufficiently large integer $k$ .

At the level described in the abstracts, the strategy is constructive: choose a metric satisfying the curvature and completeness requirements while supporting more slowly growing harmonic functions than the Euclidean benchmark. The abstracts do not explain the mechanism producing those functions or the detailed argument establishing their number. The companion strengthens the dimensional scope to three and states that, for each $1 < c < 9/4$ , its metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. These metrics may depend on $k$ .

Useful notions

Harmonic functions
Polynomial growth of functions
Dimension and linear independence
Riemannian metrics and Ricci curvature
Large-distance geometry

What was known before / what is new

Before

The source identifies the proposed bound as Yau’s dimension comparison question, here restricted to integer growth degrees. The benchmark is the dimension of Euclidean harmonic polynomials; in three dimensions, the count for growth at most $k$ is $(k + 1)^{2}$ . The supplied abstracts give no further account of earlier partial results.

New

The main manuscript supplies a Lean-formalized counterexample in some even dimension at least eight. The companion manuscript shows in the manuscript a three-dimensional family with a fixed-factor excess for every sufficiently large integer growth degree. It additionally states that, for each allowed excess factor below $9/4$ , the metrics can be chosen arbitrarily close to Euclidean in global bi-Lipschitz distance.

Limits and assumptions

The main statement is existential in both dimension and degree: some even $n \geq 8$ and some integer $k \geq 2$ . The companion requires $4/9 < v < 1$ , $1 < c < 9 v /4$ , and sufficiently large integer $k$ ; its metrics may depend on $k$ . Both concern complete smooth metrics with nonnegative Ricci curvature and real harmonic functions with pointwise polynomial growth. The source identifies the main result as Lean-formalized, but does not specify the formalization’s detailed coverage or identify a formalization of the companion claims. These statements do not establish a single metric giving the stated excess for every sufficiently large degree, nor do the supplied abstracts resolve other versions of the comparison question.

Mini glossary

Harmonic function: A function satisfying the Laplace equation associated with the space’s metric.
Pointwise polynomial growth at most $k$: A bound on a function’s absolute value by a constant multiple of $(1 + r)^{k}$ , where $r$ is distance from a fixed point.
Nonnegative Ricci curvature: The condition that the metric’s Ricci curvature is nonnegative in every direction.
Asymptotic volume ratio: The limiting ratio of the volume of large metric balls to that of Euclidean balls of the same radius and dimension.
Tangent cone at infinity: A limiting large-scale geometry obtained by viewing a space at successively more compressed scales.
Global bi-Lipschitz closeness: A comparison in which distances throughout two spaces differ by multiplicative factors close to one.

Keywords : Harmonic functions · Nonnegative Ricci curvature · Dimension comparison

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/361.md

Manuscripts in this family

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A counterexample to integer-degree harmonic dimension comparison

25/09/2026

Verified by Lean

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.

PDF GitHub folder Lean formalization

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

26/09/2026

To be verified

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

PDF GitHub folder