Definition:Extremal Length

Definition

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

We consider conformal metrics of the form $\rho(z)|dz|$, where $\rho:\C\to [0,\infty)$ is Borel measurable and the area:

$\displaystyle A(\rho) := \iint \rho^2(x+iy) dx\, dy$

is finite and positive. Every $\gamma\in\Gamma$ has a length with respect to such a metric, defined by:

$\displaystyle L(\gamma,\rho) := \int_{\gamma} \rho(z)|dz|$

We define:

$\displaystyle L(\Gamma,\rho) := \inf_{\gamma\in\Gamma} L(\gamma,\rho)$

and:

$\displaystyle \lambda(\Gamma) := \sup_{\rho} \frac{L(\Gamma,\rho)^2}{A(\rho)}$

The quantity $\lambda(\Gamma)$ is called the extremal length of the curve family $\Gamma$. Its reciprocal:

$\displaystyle \operatorname{mod}(\Gamma) := \frac 1 {\lambda(\Gamma)}$

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 $\rho|dz|$ 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, $\lambda(\Gamma)$ is the supremum of $L(\gamma,\rho)$, where $\rho$ is subject to the condition $A(\rho)=1$. Similarly, we can consider only metrics for which $A(\rho)=L(\Gamma,\rho)$.

Significance

Extremal length is a conformal invariant. (See Invariance of Extremal Length under Conformal Mappings.) As such, it is an essential tool of geometric function theory.

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

Sources

• L.V.Ahlfors: Conformal Invariants: Topics in Geometric Function Theory (1973)