Abstract
We prove that quasiconformal maps onto domains which satisfy a suitable growth condition on the quasihyperbolic metric are uniformly continuous when the source domain is equipped with the internal metric. The obtained modulus of continuity and the growth assumption on the quasihyperbolic metric are shown to be essentially sharp. As a tool, we prove a new capacity estimate.
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Hencl, S., Koskela, P. Quasihyperbolic boundary conditions and capacity: Uniform continuity of quasiconformal mappings. J. Anal. Math. 96, 19–35 (2005). https://doi.org/10.1007/BF02787823
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DOI: https://doi.org/10.1007/BF02787823