Prove that there is no conformal metric \[ \rho=\rho(z)|dz| \] on \(\widehat{\mathbb C}\) that is invariant under the full automorphism group \(\operatorname{Aut}(\widehat{\mathbb C})\).
Solution
1. Invariance of a conformal metric under a holomorphic automorphism\(g\) means\[ \rho(g(z))|g'(z)|=\rho(z) \]in any local coordinate.
The pullback of the metric \(\rho(w)|dw|\) by \(w=g(z)\) is \[ g^*\rho =\rho(g(z))|g'(z)|\,|dz|. \] Thus \(g^*\rho=\rho\) is exactly the displayed equality.
2. Apply this identity to the Möbius automorphism\[ g(z)=2z. \]
This map is an automorphism of \(\widehat{\mathbb C}\), fixes \(0\), and has \(g'(0)=2\). If the metric were invariant under the full automorphism group, then at \(z=0\) one would have \[ \rho(0) =\rho(g(0))|g'(0)| =2\rho(0). \] Hence \(\rho(0)=0\).
3. This contradicts the definition of a conformal metric.
2. Therefore no conformalmetric on \(\widehat{\mathbb C}\) can be invariant under all Möbiusautomorphisms.