Conformally invariant metrics and prime ends
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- by Carl David Minda
- Proc. Amer. Math. Soc. 44 (1974), 315-317
- DOI: https://doi.org/10.1090/S0002-9939-1974-0338379-6
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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.References
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Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 44 (1974), 315-317
- MSC: Primary 30A72
- DOI: https://doi.org/10.1090/S0002-9939-1974-0338379-6
- MathSciNet review: 0338379