# Definition:Extremal Length

## Definition

Let $\Gamma$ be a set of rectifiable curves in the complex plane $\C$.

We consider conformal metrics of the form $\map \rho z \cmod {\d z}$, where:

$\rho: \C \to \hointr 0 \to$ is Borel measurable

and:

the area $\ds \map A \rho := \iint \rho^2 \paren {x + i y} \rd x \rd y$ is finite and positive.

Every $\gamma \in \Gamma$ has a distance function with respect to such a metric, defined by:

$\ds \map L {\gamma, \rho} := \int_\gamma \map \rho z \size {\d z}$

We define:

$\ds \map L {\Gamma, \rho} := \inf_{\gamma \mathop \in \Gamma} \map L {\gamma, \rho}$

and:

$\ds \map \lambda \Gamma := \sup_\rho \frac {\map L {\Gamma, \rho}^2} {\map A \rho}$

The quantity $\map \lambda \Gamma$ is called the extremal length of the curve family $\Gamma$.

Its reciprocal:

$\mod \Gamma := \dfrac 1 {\map \lambda \Gamma}$

is called the modulus of $\Gamma$.

## Extensions of the concept

The definition generalizes immediately to curve families on arbitrary Riemann surfaces.

It is also sometimes convenient not to require the elements of $\Gamma$ to be connected, and rather require them only to be unions of rectifiable curves.

## Normalizations

Scaling the metric $\rho \size {\d z}$ by a constant does not change the quotient in the definition of extremal length.

Therefore it is often convenient to restrict to metrics that have been normalized in a certain manner.

For example, $\map \lambda \Gamma$ is the supremum of $\map L {\Gamma, \rho}$, where $\rho$ is subject to the condition $\map A \rho = 1$.

Similarly, we can consider only metrics for which $\map A \rho = \map L {\Gamma, \rho}$.

## Significance

Extremal length is a conformal invariant.

As such, it is an essential tool of geometric function theory.

Important special cases are provided by the moduli of annuli and quadrilaterals.