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  "body": "\n## Summary\n\nAhlfors theory is a geometric counterpart of [[Nevanlinna theory|1EHH63VNAD]], due to Lars Ahlfors.\nIt views a [[meromorphic function|0NSD1TPHR0]] on a disc as a surface spread over the [[Riemann sphere|0CDMXGPSES]].\nIt compares the area of this covering surface with the length of its boundary.\nWhen the boundary is short compared with the area, the surface covers each region of the sphere about equally often.\nIts best-known consequence is the Ahlfors five islands theorem.\n\n## Explanations\n\n### L1 human\n\nThink of the image of a disc under a meromorphic function as a sheet of paper laid many times over a sphere, with folds and branch points.\nThe average number of layers is the area of the sheet divided by the area of the sphere.\nAhlfors showed that, apart from a contribution of the boundary, the number of layers over any small region is close to this average.\nHe also showed that the boundary contribution is small for a suitable sequence of radii.\nThis gives Picard-type theorems with a clear geometric meaning.\n",
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