De Giorgi’s conjecture in dimension eight

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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.

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What the papers state

Theorem 1.1 (Monotone rigidity in dimension eight)

In A positive resolution of De Giorgi’s conjecture in dimension eight, 26 September 2026

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Let u\in C^2(\R^8,(-1,1)) satisfy

\Delta u=u^3-u,\qquad \partial_8u>0\quad\text{in }\R^8.

Then there are e\in\mathbb S^7 with e_8>0 and c\in\R such that

u(x)=\tanh\!\left(\frac{e\cdot x-c}{\sqrt2}\right) \qquad\text{for every }x\in\R^8.

Theorem 1.2 (Stable rigidity in dimension seven)

In A positive resolution of De Giorgi’s conjecture in dimension eight, 26 September 2026

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

Let v\in C^2(\R^7,[-1,1]) be a stable solution of \Delta v=v^3-v. Then either v\equiv1, v\equiv-1, or

v(x)=\tanh\!\left(\frac{a\cdot x-b}{\sqrt2}\right) \qquad\text{for some }a\in\mathbb S^6,\ b\in\R.

No energy-growth assumption is required.

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One more possible match

Theorem 1.3 (Finite-density stable rigidity) in A positive resolution of De Giorgi’s conjecture in dimension eight, 26 September 2026.

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Papers

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A positive resolution of De Giorgi’s conjecture in dimension eight

26 September 2026, current edition.

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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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Explanations by others

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mo271, Paper/DeGiorgi: De Giorgi’s conjecture holds in dimension at most 8 (issue 6915), 7 October 2026. Source
The release proves De Giorgi’s conjecture in dimension eight.

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26 references in A positive resolution of De Giorgi’s conjecture in dimension eight

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