Uniformization Of Metric Surfaces Using Isothermal Coordinates

Abstract

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

On the duality of moduli in arbitrary codimension

“…For general p the identity (1) follows from the results of [20]. It has found applications in connection with uniformization theorems [15,10] and Sobolev extension domains [19].…”

Quasisymmetric Koebe uniformization with weak metric doubling measures

“…The second inequality in (5) follows directly from (3) and the definition of H 2 q . Also, ( 6) follows directly from (3) and the first inequality in (5).…”

“…Our strategy for proving Theorem 2.2 is to apply the existence of a quasiconformal homeomorphism f from (X, q) to a circle domain Ω. This is guaranteed by a recent result of Ikonen [5] and the classical Koebe uniformization of finitely connected Riemann surfaces. We will show in Sections 6 and 7 that f is in fact quasisymmetric, with respect to the original metric d, under a suitable normalization.…”

“…To prove this, we first apply the weak metric doubling measure to suitably deform the metric on X. We show that the deformed space is reciprocal in the sense of [9], and therefore admits a quasiconformal map into S 2 by a recent result of Ikonen [5]. We then apply geometric estimates to show that this map, when suitably normalized, is quasisymmetric.…”

The branch set of minimal disks in metric spaces

“…The prototypical result of this type is a theorem of Bonk-Kleiner that an Ahlfors 2regular metric sphere is quasisymmetrically equivalent to the standard 2-sphere if and only if it is linearly locally connected [BK02]. This result has since been extended in various directions, including in the papers [Wil08,Wil10,Raj17,Iko19].…”