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  "body": "\n## Statement\n\nWrite points of $\\R^8$ as $(x, y)$ with $x, y \\in \\R^4$, and let $E = \\set{(x, y) : \\abs{x} \u003c \\abs{y}}$.\n\n1. The set $E$ minimizes [[perimeter|Y68BHXZJBV]] in $\\R^8$: $P(E, B) \\le P(F, B)$ for every open ball $B$ and every measurable $F$ with $F \\setminus B = E \\setminus B$.\n2. The [[Simons cone|ED3DAZVQZ4]] $\\partial E = \\set{\\abs{x} = \\abs{y}}$ is therefore an area-minimizing hypersurface with a singular point at the origin.\n3. There is a smooth function $u \\colon \\R^8 \\to \\R$ that is not affine and solves the [[minimal surface equation|3GV2E2YTZX]]\n$$\\operatorname{div} \\left( \\frac{\\nabla u}{\\sqrt{1 + \\abs{\\nabla u}^2}} \\right) = 0$$\non all of $\\R^8$.\n\n## Remarks\n\nEnrico Bombieri, Ennio De Giorgi and Enrico Giusti proved these results in 1969.\nFor parts 1 and 2 they compared $E$ with a family of smooth minimal hypersurfaces that fill each side of the cone.\nSuch a family forces any competitor to have at least as much perimeter.\nFor part 3 they built a subsolution and a supersolution of the minimal surface equation, and found an entire solution between them.\nAt infinity both behave like a homogeneous function that vanishes on the Simons cone.\nThe function $u$ of part 3 gives, for every $m \\ge 8$, a non-affine solution on $\\R^m$ that does not depend on the last $m - 8$ variables.\nSo the answer to [[Bernstein's problem|NP948D6FR0]] is negative in all dimensions $m \\ge 8$, while Simons had shown it is positive for $m \\le 7$.\nPart 2 shows that the bound $n - 8$ on the dimension of the singular set of area-minimizing hypersurfaces in $\\R^n$ is sharp.\n\n## Explanations\n\n### L1 human\n\nA minimal graph is a soap-film-like surface that lies over the whole of a flat space as the graph of a function.\nIn low dimensions such a graph must be flat.\nThis theorem shows that in eight dimensions there are curved minimal graphs.\nIt also shows that least-area surfaces in eight dimensions can have a sharp point, the tip of the Simons cone.\n\n### L2 human\n\nThe minimizing cone and the minimal graph are linked through a blow-down argument.\nIf a non-affine entire minimal graph exists over $\\R^m$, a blow-down argument of Fleming and De Giorgi produces a non-flat area-minimizing cone in $\\R^m$.\nSimons showed that there is no such cone in $\\R^m$ for $m \\le 7$, which gives Bernstein's theorem for $m \\le 7$.\nThe theorem shows that this obstruction disappears in $\\R^8$, and constructs the corresponding graph.\nIt is a standard example of how the regularity theory of minimal surfaces and the Liouville-type theorems for the minimal surface equation depend on the dimension.\n",
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