# Comparison Principle for Extremal Length

## 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:

\(\displaystyle L \left({\Gamma_1, \rho}\right)\) | \(=\) | \(\displaystyle \inf_{\gamma \mathop \in \Gamma_1} L \left({\gamma, \rho}\right)\) | $\quad$ by definition | $\quad$ | |||||||||

\(\displaystyle \) | \(\ge\) | \(\displaystyle \inf_{\gamma \mathop \in \Gamma_2} L \left({\gamma, \rho}\right)\) | $\quad$ since every curve of $\Gamma_1$ contains a curve of $\Gamma_2$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle L \left({\Gamma_2, \rho}\right)\) | $\quad$ by definition | $\quad$ |

This proves the second claim.

The second claim implies the first by definition.

$\blacksquare$