Related papers: Uniformization of two-dimensional metric surfaces

We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result,…

We prove that any length metric space homeomorphic to a 2-manifold with boundary, also called a length surface, is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. As an application, using the classical…

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…

The classical uniformization theorem states that any simply connected Riemann surface is conformally equivalent to the disk, the plane, or the sphere, each equipped with a standard conformal structure. We give a similar uniformization for…

We establish the following uniformization result for metric spaces $X$ of finite Hausdorff 2-measure. If $X$ is homeomorphic to a smooth 2-manifold $M$ with non-empty boundary, then we show that $X$ admits a quasiconformal almost…

We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly…

We investigate the quasisymmetric uniformization of a special class of metric surfaces known as paper surfaces, constructed as quotients of planar multipolygons via segment pairings, including infinite Type W identifications. These spaces,…

We give an alternate proof to the following generalization of the uniformization theorem by Bonk and Kleiner. Any linearly locally connected and Ahlfors 2-regular closed metric surface is quasisymmetrically equivalent to a model surface of…

Bonk and Kleiner showed that any metric sphere which is Ahlfors 2-regular and linearly locally contractible is quasisymmetrically equivalent to the standard sphere, in a quantitative way. We extend this result to arbitrary metric compact…

In this survey we present the most recent developments in the uniformization of metric surfaces, i.e., metric spaces homeomorphic to two-dimensional topological manifolds. We start from the classical conformal uniformization theorem of…

The main purpose of the note is to explore the invariant properties of sphericalization and flattening and their applications in quasi-metric spaces. We show that sphericalization and flattening procedures on a quasimetric spaces preserving…

We use the recently established existence and regularity of area and energy minimizing discs in metric spaces to obtain canonical parametrizations of metric surfaces. Our approach yields a new and conceptually simple proof of a well-known…

We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely connected…

We prove that every two-dimensional quasisphere is the limit of a sequence of smooth spheres that are uniform quasispheres. In the case of metric spheres of finite area we provide necessary and sufficient geometric conditions for a…

We verify a conjecture of Rajala: if $(X,d)$ is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain $\Omega \subset \mathbb{R}^2$, then there…

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $\mu$ on such a space, we introduce $\mu$-quasiconformal maps $f:X \to \mathbb…

A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved…

We look for minimal conditions on a two-dimensional metric surface $X$ of locally finite Hausdorff $2$-measure under which $X$ admits an (almost) parametrization with good geometric and analytic properties. Only assuming that $X$ is locally…

We study metric spheres Z obtained by gluing two hemispheres of the Euclidean sphere along an orientation-preserving homeomorphism mapping the equator onto itself, where the distance on Z is the canonical distance that is locally isometric…

We show that for all $n \geq 2$, there exists a doubling linearly locally contractible metric space $X$ that is topologically a $n$-sphere such that every weak tangent is isometric to $\R^n$ but $X$ is not quasisymmetrically equivalent to…