In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves is an open set in the complex plane. One also works with the conformal modulus, the reciprocal of the extremal length. The fact that extremal length and conformal modulus are conformal invariants 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.

Definition of extremal length

is a collection of rectifiable curves is Borel-measurable we let

denote the denotes the Euclidean with respect to the Borel measure. In other words, it is the Lebesgue-Stieltjes integral. Also set

The area is defined as

and the extremal length is

.

The term (conformal) modulus.

The extremal distance with one endpoint in one set and the other endpoint in the other set.

Examples

In this section the extremal length is calculated in several examples. The first three of these examples are actually useful in applications of extremal length.

Extremal distance in rectangle

has finite length.) We will now prove that in this case

.

. The latter inequality may be written as

implies

1. .

and an application of the Cauchy–Schwarz inequality give

1. .

, as required.

.

.

Extremal distance in annulus

.

On the other hand,

We conclude that

. Then

and apply the Cauchy-Schwarz inequality, to obtain:

Squaring gives

. When combined with the lower bound, this yields the exact value of the extremal length:

Extremal length around an annulus

. Using the above methods, it is not hard to show that

This illustrates another instance of extremal length duality.

Extremal length of topologically essential paths in projective plane

and gave the extremal length corresponded to a flat metric. In other words, when the Euclidean Riemannian metric, the resulting metric is flat. In the case of the rectangle, this was just the original metric, but for the annulus, the extremal metric identified is the metric of a cylinder. 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 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..

Extremal length of paths containing a point

. This follows, for example, by taking

1. .

Elementary properties of extremal length

to a subinterval of its domain). Another sometimes useful inequality is

, which proves the inequality.

Conformal invariance of extremal length

be a conformal homeomorphism (a bijective holomorphic map. This conformal invariance statement is the primary reason why the concept of extremal length is useful.

is Borel-measurable. Define

A change of variables gives

. Formally, we may use a change of variables again:

To justify this formal calculation, as required. The above equalities give,

. It remains to handle the non-rectifiable curves.

. Then a change of variable as above gives

, we have

1. .

.

Using the results of the previous section, we have

1. .

. The reverse inequality holds by symmetry, and conformal invariance is therefore established. ==Some applications of extremal length== By the calculation.

Extremal length in higher dimensions

The notion of extremal length adapts to the study of various problems in dimensions 3 and higher, especially in relation to quasiconformal mappings.

June 2008

Discrete extremal length

is some graph. There are two variants of extremal length in this setting. To define the edge extremal length, originally introduced by R. J. Duffin, over all edges in the path, counted with multiplicity. The "area 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. Thus, discrete extremal length is useful for estimates in discrete potential theory.

Another notion of discrete extremal length that is appropriate in other contexts is vertex extremal length over the vertices visited by the path, with multiplicity.

Notes

References