---
tag: M8611CV76A
kind: concept
name: Ahlfors theory
msc: ["30D35", "30D30"]
sources:
  - url: https://en.wikipedia.org/wiki/Ahlfors_theory
same_as:
  wikidata: Q4695186
  wikipedia: Ahlfors theory
---

## Summary

Ahlfors theory is a geometric counterpart of [[Nevanlinna theory|1EHH63VNAD]], due to Lars Ahlfors.
It views a [[meromorphic function|0NSD1TPHR0]] on a disc as a surface spread over the [[Riemann sphere|0CDMXGPSES]].
It compares the area of this covering surface with the length of its boundary.
When the boundary is short compared with the area, the surface covers each region of the sphere about equally often.
Its best-known consequence is the Ahlfors five islands theorem.

## Explanations

### L1 human

Think 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.
The average number of layers is the area of the sheet divided by the area of the sphere.
Ahlfors showed that, apart from a contribution of the boundary, the number of layers over any small region is close to this average.
He also showed that the boundary contribution is small for a suitable sequence of radii.
This gives Picard-type theorems with a clear geometric meaning.