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

By conformal invariance of extremal length, work first on $1<|w|<R$. For a conformal metric $\rho$, let $L_{\rho }$ be the infimum of lengths of curves joining the two circles, and $A_{\rho }$ its area. Each radial segment is admissible, so the Cauchy-Schwarz inequality gives, for every angle,Integrating in angle gives $2\pi L_{\rho }^2\le A_{\rho }\log R$. Conversely, $\rho (w)=1/|w|$ has shortest joining length $\log R$ and area $2\pi \log R$, attaining the bound. Thus the conformal modulus of an annulus givesThe second formula is the pullback of $c/|w|$ by $\psi$. Pullback preserves curve lengths and metric area, so every displayed positive multiple is an extremal metric in $\Omega$.