Conformally invariant metrics and prime ends

Abstract:

Let $R$ be a Riemann surface such that the group of conformal self-mappings of $R$ acts transitively on $R$. If $d$ is a metric on $R$ which is invariant under all conformal automorphisms of $R$ and which induces the given topology on $R$, then it is shown that the metric space $\left \langle {R,d} \right \rangle$ is complete. This result is used to show that the prime end compactification of a simply connected Riemann surface $R$ cannot be obtained by completion of a metric space $\left \langle {R,d} \right \rangle$, where $d$ defines the given topology on $R$ and is conformally invariant.