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 of the family.