De Giorgi's conjecture in dimension eight

Claims De Giorgi's conjecture in dimension eight: monotone Allen–Cahn solutions there are flat one-dimensional profiles.

1 paper September 26, 2026 Explained by AI

The Allen–Cahn equation models a mixture of two phases, like oil and water, with a thin layer between them. De Giorgi asked in 1979 whether, up to dimension eight, every solution that rises steadily in one direction must be flat. Flat means it is just a single one-dimensional profile, the same along parallel planes. It was known up to dimension three, and through eight with an extra limit assumption. This paper claims the full dimension-eight case with no extra assumption.

First step

1979

Ennio De Giorgi

Significance

Landmark

It claims the dimension-eight case of De Giorgi's well-known conjecture without the extra limit assumption earlier proofs needed.

Trust

Claimed, not checked

Field

Partial differential equations

1 paper

The explanation

Start with the curve g(s) = tanh(s/√2). It rises from −1 to 1. It solves g″ = g³ − g. At s = 1, g ≈ 0.6089. Both g″ and g³ − g are then ≈ −0.3831. At s = 2 both are ≈ −0.1872. Now spread this curve across the plane. Set u(x, y) = g(0.6x + 0.8y). Every line 0.6x + 0.8y = constant is a level line, so the level sets are parallel. Because 0.8 > 0, u rises as y increases. This is a planar transition, also called a planar heteroclinic.

The equation Δu = u³ − u is the Allen–Cahn equation. Here Δ is the Laplacian, the sum of second derivatives in each direction. It models two phases, at −1 and +1, separated by a thin layer. A solution is entire if it is defined on all of space. De Giorgi asked whether, in dimension eight or less, every bounded entire solution that rises strictly in one direction is planarSource 1.

The question parallels minimal surfaces, which are soap-film shapes that locally minimize area. Nonflat area-minimizing cones exist in dimension eight. Nonflat minimal graphs over all of space first appear in dimension nineSource 2. Ghoussoub and Gui proved the conjecture in dimension 2Source 3. Ambrosio and Cabré proved it in dimension 3Source 4. Savin proved it through dimension 8 assuming u tends to ±1 along each line in the rising directionSource 5. Del Pino, Kowalczyk and Wei built nonplanar monotone solutions in dimension nine and upSource 6.

The paper claims the dimension-eight case with no limit assumptionSource 7. It needs no energy bound or minimizing property eitherSource 8. The key is a second theorem about stable solutions. Stable means small local changes cannot lower the energy to second order. The paper claims every stable solution in seven dimensions is a constant, −1 or 1, or a planar transitionSource 9. This is sharp: Pacard and Wei built stable nonplanar solutions in dimension eightSource 10. The limits of a monotone solution along its rising direction are stable in seven dimensions, so the second theorem feeds the first.

The main obstacle was a stable solution whose energy might grow too fast in large balls. The remaining task is to show this density stays finiteSource 11. The paper does this with new curvature estimates on the level set u = 0.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. The proof is long and joins many delicate estimates. No specialist review of the paper has been published.

What it doesn't show. This does not cover dimension nine and above, where nonflat monotone solutions are known to exist. It has no Lean check.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

A landmark-level claim if the proof holds: it would settle the dimension-eight case of a famous conjecture. The proof is long, there is no Lean check, and no specialist review is published.

Words used

Allen–Cahn equation: The equation Δu = u³ − u, a model of two phases separated by a thin layer.
Laplacian (Δ): The sum of the second derivatives of a function in each direction.
Entire solution: A solution defined on all of space.
Planar heteroclinic: A solution of the form g(e·x − c) that rises from −1 to 1 and is constant along parallel planes.
Monotone: Strictly increasing in one fixed direction.
Stable solution: A solution whose energy cannot be lowered, to second order, by small local changes.
Minimal surface: A surface, like a soap film, that locally has the least possible area.
Energy density: The energy in a large ball divided by the size of a flat layer crossing it.

The original paper

A positive resolution of De Giorgi's conjecture in dimension eight (PDF, opens in a new tab)
September 26, 2026

All quoted lines (11)

De Giorgi's conjecture asks whether every bounded entire solution that is strictly increasing in a fixed direction has this form in ambient dimension at most eight.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
The graph condition in the minimal-surface problem is important: nonflat minimizing cones occur in ambient dimension eight, whereas nonaffine entire minimal graphs first occur in ambient dimension nine.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
Ghoussoub and Gui established the conjecture in dimension two, and Ambrosio and Cabr\'e in dimension three.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
Ghoussoub and Gui established the conjecture in dimension two, and Ambrosio and Cabr\'e in dimension three.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
Savin proved the result through dimension eight under the additional assumption $\lim_{t\to-\infty}u(x',t)=-1,\qquad \lim_{t\to+\infty}u(x',t)=1 \quad\text{for every }x'\in\R^{n-1}$ .
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
Del Pino, Kowalczyk and Wei constructed nonplanar strictly monotone solutions in every ambient dimension at least nine, near large dilations of nonaffine minimal graphs.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
We resolve De Giorgi's conjecture positively in dimension eight: every entire $C^2$ solution $u:\R^8\to(-1,1)$ of $\Delta u=u^3-u$ with an everywhere positive directional derivative is a planar heteroclinic.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
Neither energy growth nor minimization is assumed.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
We prove that every stable solution $v:\R^7\to[-1,1]$ of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
This stability endpoint is sharp: Pacard and Wei constructed bounded stable nonplanar solutions in dimension eight.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text
For stable solutions in $\R^7$ , the remaining task is to prove that $M_\infty$ is finite.
A positive resolution of De Giorgi's conjecture in dimension eight Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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