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The affine Bernstein theorem in dimensions three through nine

OpenAI · September 24, 2026

Lean

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Abstract

We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, for connected smooth open (noncompact and without boundary) affine-complete locally uniformly convex immersed hypersurfaces that are classically affine maximal in these dimensions.

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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.

Formalization

  • OpenAI math release, Lean scope notes for family 353 (lean/docs/353.md). States that the Lean development proves the Euclidean-complete graph classification for 3≤n≤9, including that the domain is all of Euclidean space and the solution is quadratic, and explicitly excludes the extension to affine-complete immersed hypersurfaces. [family-level]

Possibly relevant literature (found by the agent, not checked)

  • Neil S. Trudinger, Xu-Jia Wang, The higher dimensional affine Bernstein and Plateau problems (arXiv:2610.09385, v1 7 Oct 2026). Theorem 2.2 proves quadratic rigidity in all dimensions under a growth condition (dilation-type, or x·Du ≥ (1+δ)u at infinity), a comparison for this paper's claim without growth assumptions in dimensions 3 to 9; its Remark 3.2 reads "For dimension n≤9, there is no such function and we cannot rule out the scenario of Case 2)". Its added note discusses only the companion dimension-ten paper, not this one.
  • 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 remark that determinant-power generalizations admit nonquadratic examples, where only Sun–Xing–Xu is cited.

Digests and expositions

Elsewhere

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Cite

@misc{OAI:The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026,
  author = {{OpenAI}},
  title = {{The affine Bernstein theorem in dimensions three through nine}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf}{OAI:The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026}},
  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.

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