Series Law for Extremal Length
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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 $\gamma_1 \cap \gamma_2 = \O$.
Then the extremal lengths of $\Gamma_1$, $\Gamma_2$ and $\Gamma$ satisfy:
- $\map \lambda \Gamma \ge \map \lambda {\Gamma_1} + \map \lambda {\Gamma_2}$
Proof
Let $\rho_1 = \map {\rho_1} z \size {\d z}$ and $\rho_2 = \map {\rho_2} z \size {\d z}$ be conformal metrics as in the definition of extremal length.
It can be assumed that these are normalized:
- $\map A {\rho_j} = \map L {\Gamma_j, \rho_j}$ for $j \in \set {1, 2}$.
We define another metric $\rho = \map \rho z \size {\d z}$ by:
- $\map \rho z := \map \max {\map {\rho_1} z, \map {\rho_2} z}$
Note that this is a well-defined metric.
By definition, the area form $\map {\rho^2} z \size {\d z}$ satisfies:
- $\map {\rho^2} z \size {\d z}^2 = \map \max {\map {\rho_1} z^2, \map {\rho_2} z^2} \size {\d z}^2 \le \paren {\map {\rho_1} z^2 + \map {\rho_2} z^2} \size {\d z}^2$
Hence:
- $\map A \rho \le \map A {\rho_1} + \map A {\rho_2} = \map L {\Gamma_1, \rho_1} + \map L {\Gamma_2, \rho_2}$
On the other hand, let $\gamma \in \Gamma$.
Let $\gamma_1$, $\gamma_2$ be as in the assumption.
Then:
| \(\ds \map L {\gamma, \rho}\) | \(\ge\) | \(\ds \map L {\gamma_1, \rho} + \map L {\gamma_2, \rho}\) | as $\gamma_1$ and $\gamma_2$ are disjoint | |||||||||||
| \(\ds \) | \(\ge\) | \(\ds \map L {\gamma_1, \rho_1} + \map L {\gamma_2, \rho_2}\) | Definition of $\rho$ | |||||||||||
| \(\ds \) | \(\ge\) | \(\ds \map L {\Gamma_1, \rho_1} + \map L {\Gamma_2, \rho_2}\) | Definition of $\map L {\Gamma_j, \rho_j}$ |
Thus
- $\map L {\Gamma, \rho} \ge \map L {\Gamma_1, \rho_1} + \map L {\Gamma_2, \rho_2}$
Combining this with the inequality for the area:
- $\dfrac {\map L {\Gamma, \rho}^2} {\map A \rho} \ge \dfrac {\paren {\map L {\Gamma_1, \rho_1} + \map L {\Gamma_2, \rho_2} }^2} {\map L {\Gamma_1, \rho_1} + \map L {\Gamma_2, \rho_2} } = \map L {\Gamma_1, \rho_1} + \map L {\Gamma_2, \rho_2}$
Taking the supremum over all metrics $\rho_1$ and $\rho_2$ as above:
- $\map L \Gamma \ge \map L {\Gamma_1} + \map L {\Gamma_2}$
as claimed.
$\blacksquare$
Also known as
The series law and the parallel law are also referred to collectively as the composition laws of extremal length.