Bernstein's problem

Remarks

The answer is yes for $m \le 7$ and no for $m \ge 8$. Sergei Bernstein proved the case $m = 2$: an entire minimal surface graph over the plane is a plane. Wendell Fleming gave a new proof for $m = 2$ based on area-minimizing cones. This approach led to the cases $m = 3$ by Ennio De Giorgi, $m = 4$ by Frederick Almgren and $m \le 7$ by James Simons. Enrico Bombieri, Ennio De Giorgi and Enrico Giusti found a non-affine entire solution for $m = 8$, and so for every $m \ge 8$. The dividing line comes from Simons' cone $\set{x \in \R^8 : x_1^2 + x_2^2 + x_3^2 + x_4^2 = x_5^2 + x_6^2 + x_7^2 + x_8^2}$, which is area-minimizing. For $m = 1$ the statement is elementary: the equation says that $u'/\sqrt{1 + u'^2}$ is constant, so $u'$ is constant.