---
tag: HX72KT9D0G
kind: concept
name: Extremal length
msc: [30C75, 31A15]
sources:
  - url: https://en.wikipedia.org/wiki/Extremal_length
same_as:
  wikidata: Q5422297
  wikipedia: Extremal length
---

## Summary

The extremal length of a family of curves in a plane domain measures how long the curves are compared with the area they sweep out.
It is defined by testing all possible ways of measuring length with a weight function $\rho$, and it is invariant under conformal maps.
Introduced by Beurling and developed by Ahlfors, it is a flexible tool for estimates in geometric function theory and for the theory of quasiconformal maps.
Its reciprocal is the [[conformal modulus|2FQAKBVBX5]] of the family.

## Explanations

### L1 human

Think of a rectangle of width $a$ and height $b$, and of all the curves that cross it from left to right.
These curves are long when $a$ is large and plentiful when $b$ is large, and the ratio $a/b$ captures both effects; it is the extremal length of the family.
Stretching the rectangle by a common factor does not change $a/b$, and in fact no conformal map changes the extremal length.
The definition replaces the Euclidean length by any weighted length and takes the most favourable weight.