For each integer n≥ 2, let u∈ C^∞(R^n) be an entire locally uniformly convex solution of the affine maximal equation U^ijw_ij=0, where U^ij is the cofactor matrix of D^2u and w=(det D^2u)^-(n+1)/(n+2). Must u be a quadratic polynomial, equivalently, must its graph be an elliptic paraboloid?

Affine Bernstein problem

The affine Bernstein problem asks whether entire, smooth, locally uniformly convex affine-maximal graphs must be elliptic paraboloids. Trudinger and Wang established the affirmative result in dimension two, but the unrestricted higher-dimensional problem remains unsettled.

Known results

Dimension $n=2$ : affirmative result for the classical exponent (Trudinger--Wang, 2000).
Dimensions $n\ge 10$ : nonsmooth counterexamples are known (Trudinger--Wang).
Under uniform strict convexity or suitable completeness and growth assumptions, higher-dimensional Bernstein theorems are known.
The smooth conjecture is recorded as positive for $n\le 9$ and allowing counterexamples for $n\ge 10$ , but this remains a conjectural boundary in the unrestricted setting.

2026 high-dimensional counterexample claim

Neil S. Trudinger and Xu-Jia Wang's preprint The higher dimensional affine Bernstein and Plateau problems claims a dimension threshold where the unrestricted Bernstein statement fails, while preserving a positive theorem under a growth condition. The preprint is unrefereed and has no independent mathematical assessment in the retrieved sources.

Current status (as of October 2026): The two-dimensional theorem and several conditional higher-dimensional results are established; a new high-dimensional failure is claimed but unverified, so the unrestricted smooth problem remains open.

Affine Bernstein problem

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Additional references

Progress summary

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