Affine Bernstein rigidity through dimension nine
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Must every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete in its induced Euclidean metric, be an elliptic paraboloid? Euclidean completeness and completeness in the affine metric are separate hypotheses.
Why it matters and what counts as progress
Why it matters: This rigidity problem seeks higher-dimensional control without the extra uniform strict-convexity assumption used in earlier theorems.
What counts as progress: Prove rigidity without an added uniform strict-convexity modulus, or exhibit a smooth counterexample satisfying the same completeness condition.
How the problem developed
- 2000 · Conjecture [1]
Neil S. Trudinger and Xu-Jia Wang Proved the smooth two-dimensional affine Bernstein result and conjectured paraboloid rigidity for Euclidean-complete, locally uniformly convex affine-maximal hypersurfaces through dimension nine.
- 24 Sept 2026 · Reported claim [2]
OpenAI The preprint claims the conjectured rigidity in dimensions three through nine under the stated completeness and convexity assumptions.
AI contribution
The affine Bernstein theorem in dimensions three through nine
The OpenAI manuscript claims that, for dimensions 3 through 9, every smooth locally uniformly convex affine-maximal graph complete in its induced Euclidean metric is an elliptic paraboloid. The manuscript also derives the affine-metric-complete immersed-hypersurface corollary. The companion smooth nonquadratic entire example in dimension 10 is a separate contribution and does not supply affine-metric completeness.
Sources
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Publication dates and editorial changes.
OpenAI reports affine Bernstein rigidity through dimension nine