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arXiv · 1909.10619

Carath\'eodory-type extension theorem with respect to prime end boundaries

Joshua Kline·Jeff Lindquist·Nageswari Shanmugalingam
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Abstract

We prove a Carath\'eodory-type extension of BQS homeomorphisms between two domains in proper, locally path-connected metric spaces as homeomorphisms between their prime end closures. We also give a Carath\'eodory-type extension of geometric quasiconformal mappings between two such domains provided the two domains are both Ahlfors $Q$-regular and support a $Q$-Poincar\'e inequality when equipped with their respective Mazurkiewicz metrics. We also provide examples to demonstrate the strengths and weaknesses of prime end closures in this context.

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Joshua Kline, Jeff Lindquist, Nageswari Shanmugalingam. 2019-09-23. Carath\'eodory-type extension theorem with respect to prime end boundaries. https://arxiv.org/abs/1909.10619

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