Conformally invariant metrics and prime ends

Abstract

Let R R be a Riemann surface such that the group of conformal self-mappings of R R acts transitively on R R . If d d is a metric on R R which is invariant under all conformal automorphisms of R R and which induces the given topology on R R , then it is shown that the metric space ⟨ R , d ⟩ \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 R cannot be obtained by completion of a metric space ⟨ R , d ⟩ \left \langle {R,d} \right \rangle , where d d defines the given topology on R R and is conformally invariant.