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  "body": "\n## Example\n\nFor $(u, v) \\in \\R^2$, let\n$$x(u, v) = \\left(u - \\frac{u^3}{3} + u v^2, \\; -v + \\frac{v^3}{3} - u^2 v, \\; u^2 - v^2\\right).$$\nIts image is the Enneper surface.\nIt is a [[minimal surface|41X0YW2R9N]].\n\nCheck.\nThe partial derivatives are $x_u = (1 - u^2 + v^2, \\, -2uv, \\, 2u)$ and $x_v = (2uv, \\, -1 + v^2 - u^2, \\, -2v)$.\nA direct computation gives $E = G = (1 + u^2 + v^2)^2$ and $F = 0$, so $x$ is a conformal immersion.\nThe second derivatives are $x_{uu} = (-2u, -2v, 2)$ and $x_{vv} = (2u, 2v, -2)$, so $x_{uu} + x_{vv} = 0$.\nFor a conformal parametrization this means that the [[mean curvature|PA78NFEPCF]] vanishes.\n\n## Remarks\n\nIn the [[Weierstrass–Enneper parameterization|YWV6J23AP6]], this surface has data $f = 2$ and $g(z) = z$ on $\\C$, with $z = u + iv$.\nThe surface is complete, but it is not embedded: it intersects itself.\nIts Gauss map is the inverse stereographic projection of $g(z) = z$, which covers the sphere once except for one point, so its total curvature is $-4\\pi$.\n\n## Explanations\n\n### L1 human\n\nNear the origin the Enneper surface looks like a saddle.\nFarther out its edges fold over and pass through each other.\nIt is one of the simplest minimal surfaces after the plane, the catenoid and the helicoid, and its formula is just a polynomial.\n",
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