Series Law for Extremal Length

Theorem
Let $X$ be a Riemann surface.

Let $\Gamma_1$, $\Gamma_2$ and $\Gamma$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$.

Let every $\gamma \in \Gamma$ contain a $\gamma_1 \in \Gamma_1$ and a $\gamma_2 \in \Gamma_2$ such that $\gamma_1 \cap \gamma_2 = \varnothing$.

Then the extremal lengths of $\Gamma_1$, $\Gamma_2$ and $\Gamma$ satisfy:
 * $\lambda \left({\Gamma}\right) \ge \lambda \left({\Gamma_1}\right) + \lambda \left({\Gamma_2}\right)$

Proof
Let $\rho_1 = \rho_1 \left({z}\right) \left\vert{\mathrm d z}\right\vert$ and $\rho_2 = \rho_2 \left({z}\right) \left\vert{\mathrm d z}\right\vert$ be conformal metrics as in the definition of extremal length.

It can be assumed that these are normalized:
 * $A \left({\rho_j}\right) = L \left({\Gamma_j, \rho_j}\right)$ for $j \in \left\{ {1, 2}\right\}$.

We define another metric $\rho = \rho \left({z}\right) \left\vert{\mathrm d z}\right\vert$ by:
 * $\rho \left({z}\right) := \max \left({\rho_1 \left({z}\right), \rho_2 \left({z}\right)}\right)$

Note that this is a well-defined metric.

By definition, the area form $\rho^2 \left({z}\right) \left\vert{\mathrm d z}\right\vert$ satisfies:
 * $\rho^2 \left({z}\right) \left\vert{\mathrm d z}\right\vert^2 = \max \left({\rho_1 \left({z}\right)^2, \rho_2 \left({z}\right)^2}\right) \left\vert{\mathrm d z}\right\vert^2 \le \left({\rho_1 \left({z}\right)^2 + \rho_2 \left({z}\right)^2}\right) \left\vert{\mathrm d z}\right\vert^2$

Hence:
 * $A \left({\rho}\right) \le A \left({\rho_1}\right) + A \left({\rho_2}\right) = L \left({\Gamma_1, \rho_1}\right) + L \left({\Gamma_2, \rho_2}\right)$

On the other hand, let $\gamma \in \Gamma$.

Let $\gamma_1$, $\gamma_2$ be as in the assumption.

Then:

Thus
 * $ L \left({\Gamma, \rho}\right) \ge L \left({\Gamma_1, \rho_1}\right) + L \left({\Gamma_2, \rho_2}\right)$

Combining this with the inequality for the area:
 * $\dfrac {L \left({\Gamma, \rho}\right)^2} {A \left({\rho}\right)} \ge \dfrac {\left({L \left({\Gamma_1, \rho_1}\right) + L \left({\Gamma_2, \rho_2}\right)}\right)^2} {L \left({\Gamma_1, \rho_1}\right) + L \left({\Gamma_2, \rho_2}\right)} = L \left({\Gamma_1, \rho_1}\right) + L \left({\Gamma_2, \rho_2}\right)$

Taking the supremum over all metrics $\rho_1$ and $\rho_2$ as above:
 * $ L \left({\Gamma}\right) \ge L \left({\Gamma_1}\right) + L \left({\Gamma_2}\right)$

as claimed.

Also known as
The series law and the parallel law are also referred to collectively as the composition laws of extremal length.