Extremal length ($\lambda (\Gamma ,X)={\sup }_{\rho }{L}_{\rho }(\Gamma {)}^2/A_{\rho }$)

For a path family $\Gamma$ on a Riemann surface, take the supremum of $L_{\rho }(\Gamma {)}^2/A_{\rho }$ over measurable conformal metrics of finite positive area. This is unchanged under conformal equivalence, and a quasiconformal map of maximal dilatation $K$ changes it by a factor between $K^{-1}$ and $K$.

Let $\alpha$ be a real closed differential form on a Riemann surface, of finite positive conformal energy $E=\int |\alpha {|}^{2}dA$. If its period on a loop homotopy class $\gamma$ is $p$, then $\lambda (\gamma ,X)\ge p^2/E$. Indeed the conformal metric $\rho =|\alpha |$ has area $E$ and each loop in the class has length at least $|\int \alpha |=|p|$. Energy is independent of the auxiliary smooth conformal metric used to compute the pointwise norm: the inverse scaling of the squared norm cancels the area scaling.

The height divided by circumference in a conformal Euclidean cylinder model of an annulus. For $r<|z|<R$, it is $(2\pi {)}^{-1}\log (R/r)$. The extremal length of winding-one core curves is $1/M$, whereas that of curves joining the two boundary components is $M$.

The quotient of a horizontal strip by a horizontal translation. The model $(\mathbb{R}/\mathbb{Z})\times (0,M)$ has conformal modulus of an annulus $M$. Averaging horizontal loop lengths and applying Cauchy-Schwarz inequality proves that the core-loop extremal length equals $1/M$.