Past exam of the complex analysis: conformal maps and geometry course of the University of Oxford / 2023 / 2 / b / Solution

The symmetry rule for extremal length says that if a domain and its curve family are invariant under reflection, the supremum defining extremal length may be taken over reflection-invariant conformal metrics. More generally, for a conformal or anticonformal involution $T$, use metrics satisfying $T^{\ast }\rho =\rho$.

To see this, replace $\rho$ by ${\rho }_s=(\rho +T^{\ast }\rho )/2$. Every curve in the invariant family and its image have length at least $L_{\rho }$, so $L_{{\rho }_s}\ge L_{\rho }$. Convexity of the square and change of variables give $A_{{\rho }_s}\le A_{\rho }$. Therefore symmetrizing cannot reduce the length-squared to area ratio.