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A positive resolution of De Giorgi's conjecture in dimension eight
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Abstract
We resolve De Giorgi's conjecture positively in dimension eight: every entire C2 solution $u:\mathbb R^8\to(-1,1)$ of $\Delta u=u^3-u$ with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution $v:\mathbb R^7\to[-1,1]$ of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.
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Follow-up works
- Zhouyu Long and Wenming Zou, On the Allen–Cahn equation (arXiv:2610.28080, 20 October 2026). Proves that finite Morse index implies $O(R^{n-1})$ energy growth for bounded entire solutions in dimensions $4\le n\le 9$; the introduction cites this paper as [12] and says its positive normal equation, penalized reach, root-sum-squares gluing and radial comparison "follow the approach of [12]".
Formalization
- mo271, Paper/DeGiorgi: De Giorgi's conjecture holds in dimension at most 8 (google-deepmind/formal-conjectures issue #6915) (GitHub issue, opened 7 October 2026). Proposes marking
DeGiorgi_le_eight,DeGiorgi_eightandDeGiorgi_four–DeGiorgi_seveninFormalConjectures/Paper/DeGiorgi.leanas solved on the basis of this paper's Theorem 1.1; the issue says its comparison of the statements was AI-assisted, asks for a check before a PR, and carries the label "solution found". [AI-generated per source]
Possibly relevant literature (found by the agent, not checked)
- Yong Liu, Kelei Wang and Juncheng Wei, Global minimizers of the Allen-Cahn equation in dimension n≥8 (arXiv:1606.05315, 2016). Constructs nonplanar global minimizers in $\mathbb R^8$ whose nodal sets are asymptotic to strictly area-minimizing Lawson cones; relevant to Part I, where the paper shows the monotone solution is a global minimizer and then uses directional monotonicity of the blow-down cone to rule out nonplanar behaviour.
- Enric Florit-Simon, Phase transitions with bounded index: Parallels to De Giorgi's conjecture (arXiv:2602.03136, 3 February 2026). Shows that finite-index solutions with bounded energy density are one-dimensional in dimension four, and in dimensions 4–7 conditional on the classification of stable solutions; related to the paper's Theorems 1.2 and 1.3 (stable and finite-density stable rigidity for $n\le7$).
- Yong Liu, Tianci Luo, Kelei Wang, Juncheng Wei, Yong Wei and Ke Wu, Sheeting property and rigidity for stable solutions to the Allen–Cahn equation (arXiv:2610.08499, 6 October 2026). Proves a sheeting theorem in dimensions 2–10 and deduces that stable entire solutions in $\mathbb R^N$, $N\le7$, with $O(R^{N-1})$ energy growth are one-dimensional, overlapping with the paper's Theorem 1.3; it does not cite the paper.
Digests and expositions
- Gradient Damage, De Giorgi's Conjecture in Dimension 8: Must the Wall Be Flat? (video and accompanying description, 2026-10-15). Illustrates the claimed dimension-eight monotone rigidity and dimension-seven stable classification, explicitly describing the animation as an illustration rather than a proof.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 375
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:De-Giorgis-conjecture-in-dimension-eight-September-26-2026,
author = {{OpenAI}},
title = {{A positive resolution of De Giorgi's conjecture in dimension eight}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf}{OAI:De-Giorgis-conjecture-in-dimension-eight-September-26-2026}},
year = {2026}
}Family 375: De Giorgi's conjecture in dimension eight
Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution $u:\mathbb R^8\to(-1,1)$ of $\Delta u=u^3-u$ that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions $v:\mathbb R^7\to[-1,1]$ as constant wells or planar transitions, without an energy-growth assumption.
This is the only paper in this family.