Quasiconformal geometry and removable sets for conformal mappings

2020/06/04 by Ikonen, Toni, Romney, Matthew
#52A38 #53B40 #FOS: Mathematics #Metric Geometry (math.MG) #Primary 30L10. Secondary 30C35

paper · doi:10.48550/arxiv.2006.02776

Abstract

We study metric spaces defined via a conformal weight, or more generally a measurable Finsler structure, on a domain Ω⊂ ℝ² that vanishes on a compact set E ⊂ Ω and satisfies mild assumptions. Our main question is to determine when such a space is quasiconformally equivalent to a planar domain. We give a characterization in terms of the notion of planar sets that are removable for conformal mappings. We also study the question of when a quasiconformal mapping can be factored as a 1-quasiconformal mapping precomposed with a bi-Lipschitz map.

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