In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves
makes them useful tools in the study of conformal and quasi-conformal mappings.
One also works with extremal length in dimensions greater than two and certain other metric spaces, but the following deals primarily with the two dimensional setting.
To define extremal length, we need to first introduce several related quantities.
In this section the extremal length is calculated in several examples.
It should be pointed out that the extremal length of the family of curves
It is natural to refer to this as a duality property of extremal length, and a similar duality property occurs in the context of the next subsection.
, while the upper bound involves proving a statement about all possible
On the other hand, We conclude that We now see that this inequality is really an equality by employing an argument similar to the one given above for the rectangle.
and apply the Cauchy-Schwarz inequality, to obtain: Squaring gives This implies the upper bound
When combined with the lower bound, this yields the exact value of the extremal length: Let
be the collection of all curves that wind once around the annulus, separating
Using the above methods, it is not hard to show that This illustrates another instance of extremal length duality.
and gave the extremal length corresponded to a flat metric.
In other words, when the Euclidean Riemannian metric of the corresponding planar domain is scaled by
We now discuss an example where an extremal metric is not flat.
The projective plane with the spherical metric is obtained by identifying antipodal points on the unit sphere in
denote the set of closed curves in this projective plane that are not null-homotopic.
is obtained by projecting a curve on the sphere from a point to its antipode.)
Then the spherical metric is extremal for this curve family.
[1] (The definition of extremal length readily extends to Riemannian surfaces.)
is any collection of paths all of which have positive diameter and containing a point
This follows, for example, by taking The extremal length satisfies a few simple monotonicity properties.
So suppose that this is not the case and with no loss of generality assume that the curves in
be a conformal homeomorphism (a bijective holomorphic map) between planar domains.
This conformal invariance statement is the primary reason why the concept of extremal length is useful.
The reverse inequality holds by symmetry, and conformal invariance is therefore established.
The notion of extremal length adapts to the study of various problems in dimensions 3 and higher, especially in relation to quasiconformal mappings.
To define the edge extremal length, originally introduced by R. J. Duffin,[2] consider a function
is interpreted as a resistor network, where each edge has unit resistance, then the effective resistance between two sets of vertices is precisely the edge extremal length of the collection of paths with one endpoint in one set and the other endpoint in the other set.