Result 353, Differential geometry

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.

Lean formalization Proof

The bigger picture

Why it matters

Can a curved surface satisfying a geometric equilibrium equation have any shape beyond a quadratic bowl? These manuscripts claim a sharp dimensional boundary: under specified convexity and completeness assumptions, rigidity holds through dimension nine but fails in dimension ten.

What changes?

In dimensions three through nine, the manuscript reports that every smooth locally uniformly convex affine-maximal graph complete for its induced Euclidean metric is an elliptic paraboloid, a quadratic bowl. Local uniform convexity requires positive curvature in every direction; affine maximality imposes a particular geometric equation. The classification also covers connected smooth immersed noncompact hypersurfaces without boundary, with the same convexity and equation, when complete for the affine Berwald–Blaschke metric. This second statement needs no initial graph assumption.

What does that help mathematicians do?

The dimension-ten manuscript reports a smooth nonquadratic graph defined over the whole space, solving the affine maximal equation with positive-definite Hessian everywhere. Such an entire graph is Euclidean-complete, making the upper dimension nine sharp for the stated graph classification. Researchers therefore cannot extend that classification to dimension ten under unchanged assumptions. Crucially, the example asserts neither a global uniform lower Hessian bound nor Berwald–Blaschke completeness, so it does not settle those stronger variants.

Are there practical applications?

The immediate value is foundational: the classification reduces whole classes of geometric solutions to quadratic models, rather than leaving researchers to analyze arbitrary curved shapes. The counterexample identifies where that simplification breaks down. Together, the claims clarify why dimension and the choice of completeness metric must be tracked separately when formulating further rigidity theorems.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten

October 5, 2026 7 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

The affine Bernstein theorem in dimensions three through nine

September 24, 2026 23 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/353.md.

Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves the Euclidean-complete affine Bernstein theorem for graph dimensions 3≤n≤93\le n\le9. If a smooth function on a nonempty open convex domain has positive-definite Hessian, satisfies the affine maximal equation, and its graph is complete in the induced Euclidean metric, then the domain is all of Rn\mathbb R^n and the function is a positive-definite quadratic polynomial plus an affine term. Its graph is therefore an elliptic paraboloid.

The paper's extension to immersed hypersurfaces without an initial graph assumption is outside this selected statement.

Comparator links

Result Comparator statement
Affine Bernstein classification for complete graphs in dimensions three through nine AffineBernstein.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.