Chern's affine Bernstein conjecture

Let $u\colon\mathbb{R}^2\to\mathbb{R}$ be smooth and locally uniformly convex, and suppose its graph is an affine maximal graph, meaning that $u$ satisfies the affine maximal surface equation

$$\sum_{i,j=1}^{2}\frac{\partial^2}{\partial x_i\partial x_j}(U^{ij}w)=0, \qquad w=[\det D^2u]^{-\frac{3}{4}},$$

where $U=(U^{ij})$ is the cofactor matrix of $D^2u$. Chern's conjecture. The graph of $u$ must be an elliptic paraboloid. This is the two-dimensional affine Bernstein problem; it was proved by Trudinger and Wang in 2000, so the conjecture is solved.