Quasiconformal geometry and removable sets for conformal mappings

We study metric spaces defined via a conformal weight, or more generally a measurable Finsler structure, on a domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ that vanishes on a compact set $E\subset \mathrm{\Omega }$ 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.