Uniformization of metric surfaces: A survey

TL;DR

This survey traces the evolution of uniformization for metric surfaces from classical conformal theory to modern non-smooth settings. It highlights key tools—modulus, circle packing, metric Sobolev spaces, and geometric-analytic methods—that enable parametrizations of 2D manifolds by standard models such as $\widehat{\mathbb{C}}$, $\mathbb{D}$, or $\mathbb{C}$ under various regularity assumptions. The core contributions include the Bonk–Kleiner quasisymmetric uniformization for LLC, Ahlfors-regular spheres; Rajala’s reciprocal-quasiconformal uniformization; and Ntalampekos–Romney’s weakly quasiconformal uniformization under minimal hypotheses, together with canonical parametrizations and polyhedral approximation techniques. Collectively, these results connect geometric group theory, dynamics, and analysis on metric spaces, providing quantitative and structural understandings of when metric surfaces admit well-behaved uniformizations. The framework thus offers robust pathways to study non-smooth geometries via canonical mappings to smooth models, with implications for curvature, rigidity, and rectifiability.

Abstract

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 Koebe and Poincaré. Then we discuss the Bonk-Kleiner theorem on the quasisymmetric uniformization of metric spheres, which marks the beginning of the study of the uniformization problem on fractal surfaces. The next result presented is Rajala's theorem on the quasiconformal uniformization of metric spheres. We conclude with the final result in this series of works, due to Romney and the author, on the weakly quasiconformal uniformization of arbitrary metric surfaces of locally finite area under no further assumption.

Figures (8)

• Figure 1: Illustration of a conformal map from the plane onto a Riemannian surface embedded in Euclidean space.
• Figure 2: Illustration of a conformal map from the standard sphere onto a polyhedral sphere.
• Figure 3: Surfaces with a cusp, a thin bottleneck, and dense wrinkles, respectively.
• Figure 4: The definition of a quasisymmetric map.
• Figure 5: The first stage of the construction of the snowsphere.

Theorems & Definitions (61)

• Conjecture 1.2
• Theorem 2.1: Classical uniformization
• Theorem 2.2: Classical Riemannian uniformization
• Theorem 2.3: Existence of isothermal coordinates
• Theorem 2.4
• Theorem 4.1: Bonk--Kleiner theorem
• proof : Outline of proof
• Theorem 4.2: LytchakWenger:parametrizations*Theorem 6.2
• proof : Outline of proof
• Theorem 4.3: CreutzRomney:branch*Theorem 1.2