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  "body": "\n## Problem\n\nLet $m \\ge 1$ and let $u \\colon \\R^m \\to \\R$ be a twice continuously differentiable function that solves the [[minimal surface equation|3GV2E2YTZX]]\n$$\\operatorname{div} \\left( \\frac{\\nabla u}{\\sqrt{1 + \\abs{\\nabla u}^2}} \\right) = 0$$\non all of $\\R^m$.\nMust $u$ be an affine function, so that its graph is a hyperplane in $\\R^{m+1}$?\n\n## Remarks\n\nThe answer is yes for $m \\le 7$ and no for $m \\ge 8$.\nSergei Bernstein proved the case $m = 2$: an entire [[minimal surface|41X0YW2R9N]] graph over the plane is a plane.\nWendell Fleming gave a new proof for $m = 2$ based on area-minimizing cones.\nThis approach led to the cases $m = 3$ by Ennio De Giorgi, $m = 4$ by Frederick Almgren and $m \\le 7$ by James Simons.\nEnrico Bombieri, Ennio De Giorgi and Enrico Giusti found a non-affine entire solution for $m = 8$, and so for every $m \\ge 8$.\nThe 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.\nFor $m = 1$ the statement is elementary: the equation says that $u'/\\sqrt{1 + u'^2}$ is constant, so $u'$ is constant.\n\n## Explanations\n\n### L1 human\n\nA soap film that spreads over the whole plane as the graph of a function must be flat.\nBernstein proved this, and the question was whether the same holds in higher dimensions.\nIt does up to dimension seven, but in dimension eight and beyond there are curved minimal graphs over the whole space.\nThe change at dimension eight is one of the surprising facts of geometric analysis.\n",
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