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

48 pages, 2 figures. Fixed LaTeX compiling error