More polynomial-growth harmonic functions than Euclidean space

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Family 361 · Differential geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

A harmonic function models a steady temperature with no internal heat sources. In Euclidean space it satisfies Laplace’s equation. If its growth is bounded by a polynomial of integer degree $k$, then the function itself is a harmonic polynomial of degree at most $k$. There are only finitely many independent possibilities.

What happens on a complete curved space with nonnegative Ricci curvature? Such curvature imposes strong volume and analytic restrictions. Yau asked whether Euclidean space gives the largest possible dimension of the space of harmonic functions with a prescribed polynomial growth bound.

Family 361 claims that the exact comparison fails even when the growth degree is an integer. M1 gives a higher-dimensional construction; M2 gives counterexamples on $\mathbb R^3$, with a fixed-factor excess over the Euclidean count. M2 also claims these metrics can be arbitrarily close to the Euclidean metric in global bi-Lipschitz distance.

The mechanism is not to find one cone with too many slow angular modes. Instead, the geometry changes gradually with radius, letting harmonic functions cycle through different growth rates whose averages remain below the chosen degree.

2. Background and essential definitions

Counting harmonic functions by growth

For a complete Riemannian manifold $(M,g)$, a real-valued function $u$ is harmonic if

$$\Delta_g u=0.$$

The Laplace–Beltrami operator $\Delta_g$ is the geometric version of the ordinary sum of second derivatives. Let $H_d(M,g)$ be the real vector space of smooth harmonic functions satisfying

$$|u(x)|\le C_u\bigl(1+d_g(o,x)\bigr)^d \qquad\text{for every }x\in M,$$

where $o$ is a base point and the finite constant $C_u$ may depend on the function. Changing $o$ does not change the space. Write $h_d=\dim_{\mathbb R}H_d$.

For an integer $k\ge1$, the Euclidean count is

$$h_k(\mathbb R^n,g_E) =\binom{n+k-1}{k}+\binom{n+k-2}{k-1}.$$

In three dimensions, homogeneous harmonic polynomials of degree $l$ have dimension $2l+1$, so

$$h_k(\mathbb R^3,g_E) =\sum_{l=0}^k(2l+1)=(k+1)^2.$$

For example, degree at most one gives the constant function and the three coordinate functions. The conjectured comparison would bound the corresponding dimension on a Ricci-nonnegative three-manifold by the same square for every integer $k$.

Geometry at infinity

Ricci curvature sums sectional curvature contributions in directions through a tangent vector. Nonnegative Ricci curvature does not require every individual sectional curvature to be nonnegative.

The asymptotic volume ratio is

$$\operatorname{AVR}(g) =\lim_{R\to\infty}\frac{\operatorname{vol}_g B_g(o,R)}{\omega_nR^n},$$

where $\omega_n$ is the volume of the Euclidean unit ball. A positive ratio means the manifold retains Euclidean-order volume growth at large distances. Under nonnegative Ricci curvature, volume comparison supplies the limit.

A tangent cone at infinity is a limit obtained by looking at the whole space from farther and farther away, rescaling lengths by $1/R_j$. Different sequences can produce different limits. A metric cone has the form

$$g_C=dr^2+r^2h,$$

where $h$ is a metric on its angular cross-section, called the link.

If the link has dimension $m=n-1$, an angular Laplacian eigenfunction with eigenvalue $\lambda$ gives a homogeneous harmonic mode $r^d\varphi$ when

$$d(d+m-1)=\lambda.$$

Thus angular spectral data determine growth rates on a fixed cone.

3. Development history and previous partial results

Yau’s questions distinguished finite dimensionality from the sharper Euclidean dimension bound. Li and Tam proved the sharp linear-growth estimate $h_1\le n+1$ under nonnegative Ricci curvature. Colding and Minicozzi established finite dimensionality at every fixed degree and later bounds with the optimal order $d^{n-1}$ as the degree grows.

Those results do not force the exact Euclidean coefficient or count at each integer. A space can have the correct order of growth in the number of functions and still exceed Euclidean space at a particular degree.

Donnelly constructed counterexamples for a real growth degree strictly between one and two in dimensions at least five. This does not settle degree two: Euclidean harmonic quadratics enter the comparison space at that endpoint, making the allowable count jump.

There are also positive results with additional hypotheses. Cai and Lai obtain the Euclidean bound for locally conformally flat manifolds with nonnegative Ricci curvature. Lin, Wang and Xu establish a three-dimensional integer-degree comparison under nonnegative sectional curvature and positive asymptotic volume ratio. These assumptions are stronger than Ricci nonnegativity alone.

At infinity, Huang relates harmonic dimensions to the spectrum of a unique tangent cone under nonnegative Ricci curvature and maximal volume growth. Earlier constructions of Perelman and Colding–Naber showed that nonunique cones are possible even with favorable Ricci and volume properties. Such flexibility is relevant here.

M1 uses slowly varying Berger-sphere links and exact control in fixed finite-dimensional harmonic spaces. M2 is an independent three-dimensional construction whose angular eigenspaces move and can interact with infinitely many higher modes. It does not invoke M1’s theorem as an input.

4. What the new manuscripts claim

M1: an integer-degree violation in higher dimension

M1 claims a complete smooth metric $g$ on $\mathbb R^n$, for some even $n\ge8$, and an integer $k\ge2$, such that

$$\operatorname{Ric}_g\ge0, \qquad h_k(\mathbb R^n,g)>h_k(\mathbb R^n,g_E).$$

The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. Its finite spectral-selection appendix states that one may take $n=16$ and $k=50000$. That numerical selection is a stated ingredient, not a calculation independently certified here.

M2: a fixed-factor excess in dimension three

For every

$$\frac49<v<1, \qquad 1<c<\frac{9v}{4},$$

M2 claims a threshold $k_0(v,c)$ such that, for each integer $k\ge k_0$, there is a complete smooth metric $g_k$ on $\mathbb R^3$ with

$$\operatorname{Ric}_{g_k}\ge0, \qquad\operatorname{AVR}(g_k)=v,$$

$$h_k(\mathbb R^3,g_k)\ge c(k+1)^2>(k+1)^2.$$

The metric is Euclidean near the origin and has positive Ricci curvature outside a compact set. It has uncountably many pairwise nonisometric pointed tangent cones at infinity. Some radial sectional curvatures are negative at points arbitrarily far out, so it does not satisfy the stronger sectional-curvature hypothesis of the positive comparison theorem above.

The order of quantifiers is important: choose a sufficiently large degree, then construct its metric. M2 does not assert one fixed metric with this excess for all large degrees.

For every $\varepsilon>0$ and $1<c<9/4$, M2 further claims such metrics for all sufficiently large $k$ with

$$(1+\varepsilon)^{-2}g_E\le g_k\le(1+\varepsilon)^2g_E.$$

The identity map is therefore globally $(1+\varepsilon)$-bi-Lipschitz. The volume ratio can be prescribed arbitrarily close to one. The value $9/4$ comes from the construction; no endpoint or optimality assertion is made.

5. Why the advance matters

Importance: 5/5, conditional on the claims. The papers would answer the integer-degree form of a central comparison question negatively, with M2 placing the phenomenon in dimension three and showing a substantial multiplicative excess.

The result would not undermine the established finite-dimensionality or optimal-order bounds. It would show that the stronger intuition “Euclidean space has the most functions at every integer degree” needs additional hypotheses.

The almost-Euclidean examples are particularly instructive. Controlling all distances within a factor close to one does not control how small geometric changes accumulate over infinitely many scales. Harmonic functions solve a global equation, and their long-range growth can record this accumulation.

The method also changes the role of spectral information at infinity. A single frozen cone assigns one growth rate to each mode. A varying geometry can transfer a function between modes, so its eventual exponent reflects a carefully arranged average rather than one fixed eigenvalue.

6. The main idea

Average logarithmic growth rates

On a fixed cone, a mode growing like $r^d$ grows exponentially like $e^{dt}$ in logarithmic radius $t=\log r$. If a mode spends comparable stretches of logarithmic radius at several rates, its total growth can be governed by their mean.

An elementary analogy is alternating between multiplication factors $e^{at}$ and $e^{bt}$: over equal durations, the logarithm accumulates at average rate $(a+b)/2$. In the manuscripts, the difficulty is arranging exact harmonic continuation while changing angular directions. Tiny contamination by a faster-growing mode can eventually dominate, so approximate cycling alone is inadequate.

M1: exchange directions inside fixed harmonic blocks

M1 uses Berger metrics on odd-dimensional spheres. These deform the metric in a distinguished Hopf circle direction while preserving a normalized volume form. The direction itself can rotate.

Each angular Laplacian preserves the same finite-dimensional space of round spherical harmonics of degree $l$. A spectral selection finds enough directions whose mean selected growth rate lies below $k$, even though some individual rates may exceed $k$.

Short geometric control pulses are designed to permute these directions simultaneously in finitely many harmonic blocks. Long nonround intervals then supply the desired growth rates. A long round rest regularizes the incoming radial derivative before the next pulse.

The source tunes the transfer across the entire pulse-plus-dwell period, not just the value at the end of the pulse. This avoids a small radial-derivative error feeding an unwanted fast mode during the following long interval. Exact cycling and negligible relative transition costs yield the average exponent.

The link changes slowly enough that a concave radial correction preserves nonnegative Ricci curvature. Estimates between scheduled endpoints then give the growth bound at every radius, not merely at selected spheres.

M2: move eigenlines through crossings on the two-sphere

In three dimensions the link is $S^2$. M2 builds a periodic path of near-round sphere metrics with a common area form. Its low eigenvalues are simple except at carefully arranged isolated double crossings.

At a crossing, there are two ways to label modes: order eigenvalues by size at each instant, or continue a smooth eigenline through the equality. Continuing the eigenline exchanges its position in the ordered spectrum. A sequence of crossings cycles a long band through many positions.

Choose $a=\sqrt v$ and a parameter $\vartheta$ with

$$\sqrt c<\vartheta<\frac{3a}{2}.$$

A band reaching round degree approximately $\vartheta k$ provides about $\vartheta^2k^2$ modes. Counting the multiplicities $2l+1$ shows that their average round degree is approximately $2\vartheta k/3$. The radial scaling contributes a factor $1/a$, leaving the average below $k$. The strict inequalities leave room for the small geometric perturbations.

This is the origin of the range $c<9v/4$. The construction does not find an individual limiting sphere with too many eigenvalues below the threshold. Each frozen link obeys the relevant comparison; the cycling is essential.

Control the high modes and the Euclidean center

Unlike M1’s fixed finite blocks, M2’s moving angular eigenspaces can leak into infinitely many higher modes. The manuscript works with exact inward maps: given boundary data on an outer sphere, extend harmonically across the whole enclosed ball and restrict to an inner sphere.

These maps incorporate smoothness at the Euclidean center from the start. They preserve constants and contract the appropriate $L^2$ norm. A gap above the selected low spectrum produces an invariant graph describing the high-mode component in terms of low-mode data.

Small controls near each crossing then match the selected invariant lines to the continued eigenbranches. One solves a scalar recurrence in opposite directions on its contracting and expanding sides; the crossing control changes the mismatch with a definite sign. Finite-dimensional fixed points and a compact limiting argument choose all controls for one resulting metric.

The radial metric has the form

$$g=dr^2+f(r)^2H_*(\log r),\qquad f(r)/r\to\sqrt v.$$

Slow angular motion and a concave radial tail control the Ricci tensor. The mixed curvature terms are absorbed quadratically using a positive tangential Ricci block, rather than being assumed absent.

Compatible traces on nested spheres yield independent entire harmonic functions. Periodic averaging bounds their logarithmic growth, inward contraction controls intermediate spheres, and local elliptic estimates convert spherical $L^2$ bounds into the required pointwise bound everywhere.

7. Scope, limitations, and open questions

The distinction between a fixed metric and a family depending on $k$ is essential. M2 proves the claimed excess degree by degree, not an asymptotic lower bound for one common three-manifold metric as $k\to\infty$.

Neither construction violates Li–Tam’s sharp linear-growth theorem. Nor do they contradict comparison results assuming nonnegative sectional curvature, local conformal flatness or a suitable unique cone at infinity. Their geometry deliberately falls outside those additional hypotheses.

Global bi-Lipschitz proximity to Euclidean space does not mean closeness of curvature or all metric derivatives. Negative radial sectional planes and changing cone limits remain possible despite the distance comparison.

The factor $9/4$ is not claimed to be best possible, and its endpoint is excluded. Questions about optimal excess, smaller integer degrees and a single fixed metric exhibiting repeated large-degree excess remain separate. The spectral selections, transfer arguments and geometric constructions are reported as stated without proof verification.

8. Sources and further reading

M1 develops fixed-block Berger-link control and gives the finite spectral-selection appendix. M2 develops crossing control and whole-ball harmonic continuation in dimension three; its theorem explicitly records dependence of the metric on the chosen degree.

Primary background includes Li–Tam’s linear-growth theorem, Donnelly’s noninteger examples, Huang’s cone-spectral comparison, Cai–Lai, and Lin–Wang–Xu.

Source manuscripts

• M1. OpenAI, A counterexample to integer-degree harmonic dimension comparison (September 25, 2026).
• M2. OpenAI, A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison (September 26, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.