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Proceedings of the American Mathematical Society

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Conformally invariant metrics and prime ends

Author: Carl David Minda
Journal: Proc. Amer. Math. Soc. 44 (1974), 315-317
MSC: Primary 30A72
MathSciNet review: 0338379
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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.

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Keywords: Conformal transitivity, conformally invariant metrics, prime ends
Article copyright: © Copyright 1974 American Mathematical Society