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  "body": "\n## Definition\n\nAhlfors theory studies a [[meromorphic function|0NSD1TPHR0]] $f$ on $\\C$ through the covering surfaces $f(\\set{\\abs{z} \u003c r})$ over the [[Riemann sphere|0CDMXGPSES]].\nThese surfaces are measured in the spherical metric $\\abs{dw}/(1 + \\abs{w}^2)$, for which the whole sphere has area $\\pi$.\nIts basic quantities, for $r \u003e 0$, are:\n\n1. the *mean sheet number*\n$$S(r) = \\frac{1}{\\pi} \\iint_{\\abs{z} \u003c r} \\left(\\frac{\\abs{f'(z)}}{1 + \\abs{f(z)}^2}\\right)^2 dx \\, dy,$$\nthe spherical area of the image counted with multiplicity, divided by the area of the sphere;\n2. the *boundary length*\n$$L(r) = \\int_{\\abs{z} = r} \\frac{\\abs{f'(z)}}{1 + \\abs{f(z)}^2} \\, \\abs{dz},$$\nthe spherical length of the image of the circle $\\abs{z} = r$.\n\n## Remarks\n\nExample: for $f(z) = z$, $S(r) = \\frac{1}{\\pi}\\int_0^r \\frac{2\\pi t}{(1+t^2)^2} dt = \\frac{r^2}{1+r^2}$, which tends to $1$, the sheet number of a map that covers the sphere once.\nHere $L(r) = 2\\pi r/(1 + r^2)$ tends to $0$, so the boundary becomes negligible.\nThe Ahlfors–Shimizu characteristic $T_0(r) = \\int_0^r S(t)/t \\, dt$ differs from the [[Nevanlinna characteristic|YT6Z0GWS0W]] $T(r, f)$ by a bounded function.\nThe main results compare, for disjoint regions $D_1, \\dots, D_q$ of the sphere, the number of times the surface covers each $D_j$ with $S(r)$, with an error proportional to $L(r)$.\nFor a transcendental meromorphic function on $\\C$ the ratio $L(r)/S(r)$ tends to $0$ along a sequence of radii.\nThis leads to the Ahlfors five islands theorem.\nLet $f$ be a transcendental meromorphic function on $\\C$, and let $D_1, \\dots, D_5$ be Jordan domains on the sphere with pairwise disjoint closures.\nThen $f$ maps some bounded simply connected domain in $\\C$ conformally onto one of the $D_j$.\n",
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