Comparison Principle for Extremal Length
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Theorem
Let $X$ be a Riemann surface.
Let $\Gamma_1$ and $\Gamma_2$ be families of rectifiable curves (or, more generally, families of unions of rectifiable curves) on $X$.
Let every element of $\Gamma_1$ contain some element of $\Gamma_2$.
Then the extremal lengths of $\Gamma_1$ and $\Gamma_2$ are related by:
- $\lambda \left({\Gamma_1}\right) \ge \lambda \left({\Gamma_2}\right)$
More precisely, for every conformal metric $\rho$ as in the definition of extremal length, we have:
- $L \left({\Gamma_1, \rho}\right) \ge L \left({\Gamma_2, \rho}\right)$
Proof
We have:
| \(\ds L \left({\Gamma_1, \rho}\right)\) | \(=\) | \(\ds \inf_{\gamma \mathop \in \Gamma_1} L \left({\gamma, \rho}\right)\) | by definition | |||||||||||
| \(\ds \) | \(\ge\) | \(\ds \inf_{\gamma \mathop \in \Gamma_2} L \left({\gamma, \rho}\right)\) | since every curve of $\Gamma_1$ contains a curve of $\Gamma_2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds L \left({\Gamma_2, \rho}\right)\) | by definition |
This proves the second claim.
The second claim implies the first by definition.
$\blacksquare$