The Euclidean-complete affine Bernstein problem in dimensions three through nine

Claims: for $3\le n\le9$, every smooth affine maximal graph with positive-definite Hessian over an open convex domain whose graph is complete for the induced Euclidean metric is entire and quadratic, so an elliptic paraboloid; no growth or affine-completeness assumption. Via Trudinger-Wang it extends to affine-complete immersed hypersurfaces. The companion constructs a smooth nonquadratic entire solution with positive-definite Hessian in dimension ten, so the range cannot extend to $n=10$. Dimensions $n\ge11$ are not addressed, and the dimension-ten example asserts no uniform Hessian bound or affine completeness.

Claims: for $3\le n\le9$, every smooth affine maximal graph with positive-definite Hessian over an open convex domain whose graph is complete for the induced Euclidean metric is entire and quadratic, so an elliptic paraboloid; no growth or affine-completeness assumption. Via Trudinger-Wang it extends to affine-complete immersed hypersurfaces. The companion constructs a smooth nonquadratic entire solution with positive-definite Hessian in dimension ten, so the range cannot extend to $n=10$. Dimensions $n\ge11$ are not addressed, and the dimension-ten example asserts no uniform Hessian bound or affine completeness.