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 $K$.

If a conformal map or anticonformal map $T$ is an involution preserving a curve family, its extremal length can be computed using invariant conformal metrics. Replace any density by ${\rho }_s=(\rho +T^{\ast }\rho )/2$. Every curve and its image have length at least $L_{\rho }$, hence $L_{{\rho }_s}\ge L_{\rho }$. Convexity of the square and change of variables give $A_{{\rho }_s}\le A_{\rho }$, proving the assertion. For reflection in the real axis, invariant means $\rho (z)=\rho (\bar{z})$.^[ref]

For a conformal map $f$ and a conformal metric $\rho$ on its image, the pullback density is $\rho (f(z))|f^{\prime }(z)|$. The change of variables formula preserves its metric area, and the chain rule preserves the length of each corresponding curve. Applying this to $f$ and its inverse proves the title equality. This allows extremal length computations on rectangles, discs and annuli to be transported to other domains.

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 $M$.

For an inner circle of real center $c$ and radius $d$ lying strictly inside the unit circle, choose the real root $|a|<1$ of $c{a}^2-(1+c^2-d^2)a+c=0$. When $c=0$, take $a=0$. The automorphism of the unit disk $T_a$ sends the inner circle to a concentric circle. Indeed it sends the two real endpoints $c-d,c+d$ to opposite real numbers; preservation of circles and reflection symmetry then give an image centered at zero. Its radius is $r=T_a(c+d)\in (0,1)$.

The conformal invariance of harmonic measure and harmonic measure of an annulus give outer-boundary measure $\log (|T_a(z)|/r)/\log (1/r)$. For $c=d=2/5$, the quadratic gives $a=1/2$ and $r=1/2$, an explicit eccentric-annulus example.

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 $1/M$.