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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C$ be an [[open set|01ZQQDCERP]], and let $\\Gamma$ be a family of locally [[rectifiable curves|0C28RT5BG7]] in $\\Omega$.\nA *metric* on $\\Omega$ is a [[Borel measurable|0BVHZXWKCD]] function $\\rho \\colon \\Omega \\to [0, \\infty]$ with area\n$$A(\\rho) = \\int_\\Omega \\rho(x + iy)^2 \\, dx\\, dy$$\nsatisfying $0 \u003c A(\\rho) \u003c \\infty$.\nThe $\\rho$-length of $\\Gamma$ is\n$$L_\\rho(\\Gamma) = \\inf_{\\gamma \\in \\Gamma} \\int_\\gamma \\rho \\, \\abs{dz},$$\nthe integral being with respect to arc length.\nThe *extremal length* of $\\Gamma$ is\n$$\\lambda(\\Gamma) = \\sup_{\\rho} \\frac{L_\\rho(\\Gamma)^2}{A(\\rho)} \\in [0, \\infty],$$\nthe supremum taken over all metrics $\\rho$ on $\\Omega$.\n\n## Remarks\n\nThe value does not change if $\\Omega$ is replaced by a larger open set and $\\rho$ is extended by $0$.\nIf $f \\colon \\Omega \\to \\Omega'$ is a conformal map, then $\\lambda(f(\\Gamma)) = \\lambda(\\Gamma)$: the metric $\\rho$ on $\\Omega'$ pulls back to $(\\rho \\circ f)\\abs{f'}$ with the same lengths and area.\nExample: let $R = (0, a) \\times (0, b)$ and let $\\Gamma$ be the curves in $R$ that join the left side to the right side.\nFor $\\rho = 1$, $L_\\rho = a$ and $A(\\rho) = ab$, so $\\lambda(\\Gamma) \\ge a/b$.\nFor any metric, each horizontal segment gives $\\int_0^a \\rho(x + iy)\\, dx \\ge L_\\rho$, so by integrating in $y$ and the Cauchy–Schwarz inequality, $b L_\\rho \\le \\int_R \\rho \\le \\sqrt{ab\\, A(\\rho)}$, and $\\lambda(\\Gamma) = a/b$.\nNon-example: $\\rho = 0$ is not a metric, since its area is $0$; it would give the meaningless ratio $0/0$.\nIf every curve of $\\Gamma_1$ contains a curve of $\\Gamma_2$, then $\\lambda(\\Gamma_1) \\ge \\lambda(\\Gamma_2)$.\n",
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