Comparison Principle for Extremal Length
Proposition
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 $X$.
If every element of $\Gamma_1$ contains some element of $\Gamma_2$, then the extremal lengths of $\Gamma_1$ and $\Gamma_2$ are related by
- $\lambda (\Gamma_1) \geq \lambda (\Gamma_2)$
More precisely, for every conformal metric $\rho$ as in the definition of extremal length, we have
- $L (\Gamma_1, \rho) \geq L (\Gamma_2, \rho)$
Proof
We have:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle L (\Gamma_1, \rho)\) | \(=\) | \(\displaystyle \inf_{\gamma \in \Gamma_1} L (\gamma, \rho)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | by definition | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\geq\) | \(\displaystyle \inf_{\gamma \in \Gamma_2} L (\gamma, \rho)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | since every curve of $\Gamma_1$ contains a curve of $\Gamma_2$ | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle L (\Gamma_2, \rho)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | by definition |
This proves the second claim. The second claim implies the first by definition.
$\blacksquare$
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