# Extremal Length of Union

This page has been identified as a candidate for refactoring of basic complexity.In particular: There are two results here - they need to be in separate pages.Until this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## 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 $X$.

Then the extremal length of their union satisfies:

- $\dfrac 1 {\map \lambda {\Gamma_1 \cup \Gamma_2} } \le \dfrac 1 {\map \lambda {\Gamma_1} } + \dfrac 1 {\map \lambda {\Gamma_2} }$

Suppose that additionally $\Gamma_1$ and $\Gamma_2$ are disjoint in the following sense: there exist disjoint Borel subsets:

- $A_1, A_2 \subseteq X$ such that $\ds \bigcup \Gamma_1 \subset A_1$ and $\ds \bigcup \Gamma_2 \subset A_2$

Then

- $\dfrac 1 {\map \lambda {\Gamma_1 \cup \Gamma_2} } = \dfrac 1 {\map \lambda {\Gamma_1} } + \dfrac 1 {\map \lambda {\Gamma_2} }$

## Proof

Set $\Gamma := \Gamma_1\cup \Gamma_2$.

Let $\rho_1$ and $\rho_2$ be conformal metrics as in the definition of extremal length, normalized such that:

- $\map L {\Gamma_1, \rho_1} = \map L {\Gamma_2, \rho_2} = 1$

We define a new metric by:

- $\rho := \map \max {\rho_1, \rho_2}$

This article, or a section of it, needs explaining.In particular: Prove that $\rho$ is a metricYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

Then:

- $\map L {\Gamma, \rho} \ge 1$

and:

- $\map A \rho \le \map A {\rho_1} + \map A {\rho_2}$

This article, or a section of it, needs explaining.In particular: What is $A$?You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

Hence:

\(\ds \frac 1 {\map \lambda \Gamma}\) | \(\le\) | \(\ds \frac {\map A \rho} {\map L {\Gamma, \rho} }\) | ||||||||||||

\(\ds \) | \(\le\) | \(\ds \map A \rho\) | ||||||||||||

\(\ds \) | \(\le\) | \(\ds \map A {\rho_1} + \map A {\rho_2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac 1 {\map L {\Gamma_1, \rho_1} } + \frac 1 {\map L {\Gamma_2, \rho_2} }\) |

Taking the infimum over all metrics $\rho_1$ and $\rho_2$, the claim follows.

Now suppose that the disjointness assumption holds, and let $\rho$ again be a Borel-measurable conformal metric, normalized such that $\map L {\Gamma, \rho} = 1$.

We can define $\rho_1$ to be the restriction of $\rho$ to $A_1$, and likewise $\rho_2$ to be the restriction of $\rho$ to $A_2$.

By this we mean that, in local coordinates, $\rho_j$ is given by

- $\map {\rho_j} z \size {\d z} = \begin {cases} \map \rho z \size {\d z} & : z \in A_j \\ 0 \size {\d z} & : \text {otherwise} \end {cases}$

This article, or a section of it, needs explaining.In particular: The above section from "By this we mean" needs considerably more explanation, as none of the concepts introduced here can be understood without reference to links from elsewhere.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

Then:

- $\map A \rho = \map A {\rho_1} + \map A {\rho_2}$

and:

- $\map L {\Gamma_1, \rho_1}, \map L {\Gamma_2, \rho_2} \ge 1$

This article, or a section of it, needs explaining.In particular: How do these two statements follow from what went before?You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

Hence:

\(\ds \map A \rho\) | \(=\) | \(\ds \map A {\rho_1} + \map A {\rho_2}\) | ||||||||||||

\(\ds \) | \(\ge\) | \(\ds \frac {\map A {\rho_1} } {\map L {\Gamma_1, \rho} } + \frac {\map A {\rho_2} } {\map L {\Gamma_2, \rho} }\) | ||||||||||||

\(\ds \) | \(\ge\) | \(\ds \frac 1 {\map \lambda {\Gamma_1} } + \frac 1 {\map \lambda {\Gamma_2} }\) |

Taking the infimum over all metrics $\rho$, we see that:

- $\dfrac 1 {\map \lambda {\Gamma_1 \cup \Gamma_2} } \ge \dfrac 1 {\map \lambda {\Gamma_1} } + \dfrac 1 {\map \lambda {\Gamma_2} }$

Together with the first part of the Proposition, this proves the claim.

$\blacksquare$