Series Law for Extremal Length/Rho is Well Defined

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Lemma for Series Law for Extremal Length

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Proof

To see that $\rho$ is a well-defined metric, we need to check that it transforms correctly when changing local coordinates.

Let $z = \map z t$ and $w = \map w t$ be charts on the Riemann surface $X$.

Let $\map {\rho_1^z} t$ and $\map {\rho_1^w} t$ be the coefficient functions when $\rho_1$ is expressed in the local coordinates $z$ and $w$, respectively.

We use the analogous notation for $\rho_2$ and $\rho$.

Since $\rho_j$ is a metric for $j \in \set {1, 2}$, we have:

$\map {\rho_j^w} t = \map {\rho_j^z} t \cdot \size {\dfrac {\d z} {\d w} }$

where $\dfrac {\d z} {\d w}$ denotes, the derivative of the coordinate change $z \circ w^{-1}$.

Thus we have:

\(\ds \map {\rho^w} t\)

\(=\)

\(\ds \max \set {\map {\rho_1^w} t, \map {\rho_2^w} t}\)

\(\ds \)

\(=\)

\(\ds \max \set {\map {\rho_1^z} t, \map {\rho_2^z} t} \cdot \size {\frac {\d z} {\d w} }\)

\(\ds \)

\(=\)

\(\ds \map {\rho^z} t \cdot \size {\dfrac {\d z} {\d w} }\)

This means that $\rho$ transforms correctly and is a metric, as desired.

$\blacksquare$