Higher-dimensional Chern Bernstein conjecture for affine maximal hypersurfaces

Determine whether every Euclidean-complete, locally uniformly convex entire graph solving the affine maximal type equation with parameter a=-(n+1)/(n+2) in dimension n≥3 must be an elliptic paraboloid.

Background

The paper studies locally uniformly convex solutions of the affine maximal type equation, a fourth-order nonlinear PDE associated with the affine area functional. In the case a=-(n+1)/(n+2), the equation governs affine maximal hypersurfaces, and the Bernstein problem asks whether suitable complete entire solutions must be quadratic, equivalently whether their graphs must be elliptic paraboloids.

The two-dimensional case corresponds to Chern’s conjecture with a=-3/4 and was resolved by Trudinger and Wang. The authors explicitly state that the higher-dimensional version remains unresolved, including dimension n=3. The paper constructs non-quadratic Euclidean-complete examples for other parameter ranges, but does not resolve this higher-dimensional Chern conjecture.