Browse › Differential geometry › 353 Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample
A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten
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Abstract
We construct a smooth nonquadratic entire graph in dimension ten that solves the classical affine maximal equation and has positive-definite Hessian everywhere. This gives a smooth counterexample to the entire-graph affine Bernstein assertion in dimension ten. Hessian positivity is pointwise; no global uniform lower bound or completeness of the Berwald–Blaschke metric is asserted.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Neil S. Trudinger, Xu-Jia Wang, The higher dimensional affine Bernstein and Plateau problems (arXiv:2610.09385, v1 7 Oct 2026). Gives a smooth entire convex nonquadratic solution of the affine maximal equation in every dimension n≥10 (Theorem 2.1, via a fourth-order ODE in Section 3), and proves interior regularity for the affine Plateau problem and an affine Bernstein theorem under a growth condition in all dimensions. A "Notes added on October 7th, 2026" section names this paper, states that its dimension-ten counterexample "is already included in Section 3 of this paper", and reports submission on 2026-02-02 and conference presentations in March and May 2026.
Formalization
- OpenAI math release, Lean scope notes for family 353 (lean/docs/353.md). The release's Lean development covers only the companion theorem for dimensions 3 to 9; the dimension-ten construction in this paper is not part of the stated scope. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Shi-Zhong Du, Bernstein Problem of Affine Maximal Type Hypersurfaces on Dimension N>=3 (arXiv:2103.08921, 2021; corrigendum to J. Differential Equations 269 (2020) 7429-7469). Constructs nonquadratic Euclidean-complete affine maximal type hypersurfaces for N≥3 and θ in (1/2,(N-1)/N); relevant to the introduction's discussion of determinant-power variants, which cites Du's later 2024 paper but not this one.
Digests and expositions
- Gradient Damage, Calabi Flow, Affine Bernstein Problem and 7 More in Differential Geometry (video companion page, 2026-10-15). The chapter beginning at 1:23 introduces the affine Bernstein problem and presents the family's claimed dimension threshold, with the page distinguishing its illustrations from proofs. [family-level]
Elsewhere
- Emergent Mind: family 353
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026,
author = {{OpenAI}},
title = {{A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf}{OAI:Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/353.md. The Lean statements have not been checked by people against the paper's claims.
Family 353: Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample
Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric.
Other papers in this family:
- The affine Bernstein theorem in dimensions three through nine · September 24, 2026