We establish a uniformization result for metric surfaces -metric spaces that are topological surfaces with locally finite Hausdorff 2-measure.Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct suitable isothermal coordinates. Contents 1. Introduction 1.1. Preliminaries 1.2. Main results 1.3. Structure of the paper Acknowledgements 2. Newtonian-Sobolev spaces 2.1. Paths 2.2. Modulus 2.3. Upper gradients 3. Analytic properties of quasiconformal maps 3.1. Basic properties 3.2. Pointwise dilatations 3.3. Approximate metric differentials 4. Uniformization 4.1. Isothermal parametrizations 4.2. Analytic properties of parametrizations 4.3. Existence of isothermal parametrizations 4.4. Conformal surfaces 5. Quasiconformal maps between reciprocal surfaces 5.1. Norm fields 5.2. Quasiconformal maps

We establish a uniformization result for metric surfaces – metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct an atlas of suitable isothermal coordinates.

We study metric spaces defined via a conformal weight, or more generally a measurable Finsler structure, on a domain Ω ⊂ R 2 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.

We construct an isometric embedding from Gigli's abstract tangent module into the concrete tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when the embedding is an isomorphism. Together with arguments from [3], this equivalence is used to show that theWe also provide a direct proof of a result in [14] that, for a space with a strongly rectifiable decomposition, Gigli's tangent module admits an isometric embedding into the so-called Gromov-Hausdorff tangent module, without any a priori reflexivity assumptions.

We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y, dY ). We say that a metric space (Y, dY ) is a quasiconformal Jordan domain if the completion ̄Y of (Y, dY ) has finite Hausdorff 2-measure, the boundary ∂Y = ̄Y \ Y is homeomorphic to 𝕊1, and there exists a homeomorphism ϕ: 𝔻 →(Y, dY ) that is quasiconformal in the geometric sense. We show that ϕ has a continuous, monotone, and surjective extension Φ: 𝔻 ̄ → Y ̄. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for Φ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of Φ to 𝕊1 being a quasisymmetry and to ∂Y being bi-Lipschitz equivalent to a quasicircle in the plane.