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  "body": "\n## Example\n\nWrite points of $\\R^8$ as $(x, y)$ with $x, y \\in \\R^4$.\nThe *Simons cone* is the hypersurface\n$$C = \\set{(x, y) \\in \\R^4 \\times \\R^4 : \\abs{x} = \\abs{y}}.$$\nIt is a [[cone|7TPBQPPN97]]: if $(x, y) \\in C$ and $t \u003e 0$, then $(tx, ty) \\in C$.\nIt is smooth away from the origin, and the origin is its only singular point.\n\nCheck that $C \\setminus \\set{0}$ is a [[minimal surface|41X0YW2R9N]], that is, has [[mean curvature|PA78NFEPCF]] zero.\nThe cone is the zero set of $u(x, y) = \\abs{x}^2 - \\abs{y}^2$, and $\\nabla u = 2(x, -y)$ vanishes only at the origin.\nThe unit normal is $\\nu = (x, -y)/r$ with $r = \\sqrt{\\abs{x}^2 + \\abs{y}^2}$, and the mean curvature is, up to a constant factor, $\\operatorname{div} \\nu$.\nSince $\\operatorname{div}(x, -y) = 4 - 4 = 0$ and $(x, -y) \\cdot \\nabla (1/r) = -(\\abs{x}^2 - \\abs{y}^2)/r^3$, one gets $\\operatorname{div} \\nu = -(\\abs{x}^2 - \\abs{y}^2)/r^3$, which is $0$ on $C$.\n\n## Remarks\n\nThe intersection of $C$ with the unit sphere $S^7$ is the product $S^3(1/\\sqrt{2}) \\times S^3(1/\\sqrt{2})$ of two $3$-spheres of radius $1/\\sqrt{2}$, a minimal hypersurface of $S^7$.\nJames Simons showed in 1968 that $C$ is stable, that is, the second variation of area is nonnegative.\nBombieri, De Giorgi and Giusti proved in 1969 that $C$ is in fact area-minimizing: the region $\\set{\\abs{x} \u003c \\abs{y}}$ minimizes perimeter among all sets that agree with it outside a ball.\nSo area-minimizing hypersurfaces in $\\R^8$ can have isolated singular points, and the dimension bound for the singular set of such hypersurfaces is sharp.\nThe analogous cones $\\set{\\abs{x} = \\abs{y}}$ with $x, y \\in \\R^k$ for $k \\le 3$ are not area-minimizing.\nThe cone is the reason why the answer to [[Bernstein's problem|NP948D6FR0]] changes at dimension $8$.\n\n## Explanations\n\n### L1 human\n\nTake two copies of four-dimensional space and look at the points where the distance from the origin in the first copy equals the distance in the second.\nThis set is a cone in eight dimensions, with its tip at the origin.\nIt balances its own surface tension perfectly away from the tip, like a soap film.\nSurprisingly, it is even the least-area surface with its boundary, so least-area surfaces in eight dimensions can have a sharp point.\n",
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