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  "body": "\n## Definition\n\nLet $S \\subset \\R^3$ be a [[regular surface|6080X3FAD4]] with [[mean curvature|PA78NFEPCF]] $H \\colon S \\to \\R$.\nThe surface $S$ is *minimal* if $H(p) = 0$ for every point $p \\in S$.\n\n## Remarks\n\nThe [[catenoid|6NN1HP54GB]] $x(u, v) = (\\cosh v \\cos u, \\cosh v \\sin u, v)$ is minimal.\nIts principal curvatures are $\\pm 1/\\cosh^2 v$, so $H = 0$.\nThe sphere of radius $r$ is not minimal: both principal curvatures equal $1/r$ for the inward normal, so $H = 1/r \\neq 0$.\n\nEquivalent forms:\n\n1. $S$ is minimal if and only if, for every compact region $R \\subset S$ and every smooth normal variation of $R$ fixed on $\\partial R$, the first variation of area is zero.\n2. If $x$ is a conformal parametrization of $S$, then $S$ is minimal if and only if each coordinate function of $x$ is harmonic.\n3. The graph of a smooth function $u(x, y)$ is minimal if and only if $(1 + u_y^2) u_{xx} - 2 u_x u_y u_{xy} + (1 + u_x^2) u_{yy} = 0$, the minimal surface equation.\n\nA minimal surface need not minimize area among all surfaces with the same boundary; it is only a critical point.\n\n## Explanations\n\n### L1 human\n\nA minimal surface is a surface whose mean curvature vanishes everywhere.\nAt every point the bending in one principal direction cancels the bending in the other.\n",
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