The Euclidean-complete affine Bernstein problem in dimensions three through nine

Let $u$ be smooth with positive-definite Hessian on an open convex $\Omega\subset\mathbb R^n$ and solve the affine maximal equation $U^{ij}w_{ij}=0$ , $U^{ij}=\det(D^2u)(D^2u)^{-1}_{ij}$ , $w=(\det D^2u)^{-(n+1)/(n+2)}$ , the Euler equation of affine area. Chern asked whether an entire such surface ( $n=2$ ) must be an elliptic paraboloid; Calabi proved it under Euclidean and affine completeness, and Trudinger and Wang proved the Euclidean-complete surface case in 2000, reduced higher dimensions to a uniform strict-convexity estimate, and exhibited a singular nonquadratic weak solution in dimension ten. For $n\ge3$ , if the graph of $u$ is complete for the induced Euclidean metric, must $\Omega=\mathbb R^n$ and $u$ be quadratic, and in which dimensions?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Geometric PDE; affine maximal hypersurfaces
Posed by: Shiing-Shen Chern (Affine minimal hypersurfaces, 1977 seminar, published 1979) for surfaces; the higher-dimensional Euclidean-complete form follows Trudinger and Wang (Invent. Math. 2000)
Year posed: 1979
Years open: 47y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 32 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims: for $3\le n\le9$ , every smooth affine maximal graph with positive-definite Hessian over an open convex domain whose graph is complete for the induced Euclidean metric is entire and quadratic, so an elliptic paraboloid; no growth or affine-completeness assumption. Via Trudinger-Wang it extends to affine-complete immersed hypersurfaces. The companion constructs a smooth nonquadratic entire solution with positive-definite Hessian in dimension ten, so the range cannot extend to $n=10$ . Dimensions $n\ge11$ are not addressed, and the dimension-ten example asserts no uniform Hessian bound or affine completeness.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The dimension-ten counterexample is a separate manuscript dated 5 October 2026.

Verification

No independent mathematician has checked this yet. Checked here: abstracts, introductions and main theorems of both manuscripts, read against the problem as framed there. The proofs were not refereed. Lean-checked on the release's own Comparator challenge AffineBernstein together with its solution module OAI.Analysis.AffineBernstein.Main, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.AffineBernstein.affine_bernstein) takes 3 <= n <= 9, a nonempty open convex domain, a smooth u with positive-definite coordinate Hessian solving the classical affine maximal equation, and sequential completeness of the induced Euclidean path metric, and concludes that the domain is all of R^n and u is a positive-definite quadratic plus affine term, the graph an affine image of the standard paraboloid. That is the headline claim of this entry. Not formalized: the affine-complete immersed-hypersurface corollary and the dimension-ten counterexample.

The Euclidean-complete affine Bernstein problem in dimensions three through nine

What was actually shown

What the AI did

Verification

Sources

Discussion