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)
Manuscripts (2)
A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten
October 5, 2026 · source · cite as \cite{OAI:Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026}
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.
The affine Bernstein theorem in dimensions three through nine
September 24, 2026 · source · cite as \cite{OAI:The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026}
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.
Result 353 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md