For n≥ 2, let f∈ C^∞(R^n) be locally uniformly convex, so that D^2f is positive definite, and suppose its graph is complete with respect to the induced Euclidean metric. If [ ∑_i,j=1^n f^ijw_ij=0,qquad w=(det D^2f)^-fracn+1n+2, ] where (f^ij)=(D^2f)^-1, must f be a quadratic polynomial? Equivalently, must the graph of f be an elliptic paraboloid?

Bernstein problem for the affine maximal type equation

For $n\ge 2$ , let $f\in C^\infty(\mathbb{R}^n)$ be locally uniformly convex, so that $D^2f$ is positive definite, and suppose its graph is complete with respect to the induced Euclidean metric. If

\sum_{i,j=1}^n f^{ij}w_{ij}=0,\qquad w=(\det D^2f)^{-\frac{n+1}{n+2}},

where $(f^{ij})=(D^2f)^{-1}$ , must $f$ be a quadratic polynomial? Equivalently, must the graph of $f$ be an elliptic paraboloid?

References

Primary source

arXiv

Additional references

New non-quadratic Euclidean complete affine maximal type hypersurfaces via Calabi affine geometry — arXiv

affine-geometry differential-geometry partial-differential-equations

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new preprint claims broader nonquadratic counterexamples, while the main higher-dimensional rigidity question remains open.

Chern proposed the entire-graph conjecture in 1977; Trudinger–Wang later formulated its broader affine maximal-type version. It asks whether complete, locally uniformly convex solutions must be elliptic paraboloids.

Known results

Trudinger–Wang: the case $N=2$ , $\theta=3/4$ .
Jia–Li: $N=2$ , $\theta\in(3/4,1]$ .
The full Bernstein theorem holds for $N=1$ , $\theta>0$ , and $N=2$ , $\theta\in[3/4,1]$ .
Rotationally symmetric solutions satisfy Bernstein rigidity for $N\geq 3$ and $\theta>0$ .

August 2026 extension of counterexamples

The preprint <i>New non-quadratic Euclidean complete affine maximal type hypersurfaces via Calabi affine geometry</i> claims an extension of Du-attributed counterexamples to a wider parameter range. This narrows the possible rigidity regime, but the higher-dimensional Chern conjecture at its distinguished parameter remains open.

Current status (as of August 2026): Positive results are known in low dimensions and special regimes, while broader nonquadratic counterexamples are claimed but the higher-dimensional distinguished-parameter conjecture remains open.

Sources

RemarkAI-assistedClaimed by OpenAI. Claims a smooth nonquadratic entire affine-maximal graph in dimension ten with everywhere positive-definite Hessian, giving a counterexample to the unrestricted quadratic-graph assertion. A uniform Hessian lower bound and affine-metric completeness are not asserted.See full solutionHide full solution

Claimed by OpenAI. Claims a smooth nonquadratic entire affine-maximal graph in dimension ten with everywhere positive-definite Hessian, giving a counterexample to the unrestricted quadratic-graph assertion. A uniform Hessian lower bound and affine-metric completeness are not asserted.

Scope relative to this problem: This is the reported dimension-ten counterexample to the target quadratic-graph assertion. The source graph is smooth over all of Euclidean ten-space; its induced Euclidean graph metric is complete because it dominates the base Euclidean metric. This is an elementary completeness inference, distinct from the affine-metric completeness excluded by the source. The target inverse-Hessian equation is equivalent to the source cofactor equation after multiplying by the positive Hessian determinant. No uniform Hessian lower bound is inferred.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf

OpenAI-353-01-A-Smooth-Nonquadratic-Entire-Affine-Maximal-Graph-in-Dimension-Ten.pdf284,068 bytesOpen