Quasisymmetric Koebe Uniformization

Uniformization with infinitesimally metric measures

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 $μ$ on such a space, we introduce $μ$-quasiconformal maps $f:X \to \mathbb R^2$, whose definition involves deforming lengths of curves by $μ$. We show that if $μ$ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $μ$-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinite…

Uniformization of two-dimensional metric surfaces

We establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result give…

低维随机空间的拟对称几何

Quasisymmetric geometry of low-dimensional random spaces

该研究针对随机分形的拟对称单值化问题，证明布朗运动迹、SLE_κ变体非拟弧，CLE_κ的环外点集与标准谢尔宾斯基地毯同胚但不与圆形地毯拟对称等价，推进了低维随机空间拟对称几何领域研究。

Quasiconformal almost parametrizations of metric surfaces

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 geodesic, we show that Jordan domains in $X$ of finite boundary length admit a quasiconformal almost parametrization. If $X$ satisfies some further conditions then such an almost parametrization can be upgraded to a geometrically quasiconformal homeomorphism or a quasisymmetric homeomorphism. In particular, we recover Rajala's recent quasiconformal uniformi…

Uniformization of Cantor sets with bounded geometry

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus from 2 to higher dimensions. In particular, we show that a compact subset of $\mathbb{R}^n$ is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set $\mathcal{C}$ in $\mathbb{R}^{n+1}$.

Planar relative Schottky sets and quasisymmetric maps

A relative Schottky set in a planar domain D is a subset of D obtained by removing from D open geometric discs whose closures are in D and are pairwise disjoint. In this paper we study quasisymmetric and related maps between relative Schottky sets of measure zero. We prove, in particular, that quasisymmetric maps between such sets in Jordan domains are conformal, locally bi-Lipschitz, and that their first derivatives are locally Lipschitz. We also provide a locally bi-Lipschitz uniformization result for relative Schottky sets in Jordan domains and establish rigidity with respect to local quasi…

平面中的统一化问题：综述

Uniformization problems in the plane: A survey

综述探讨了复平面上几个基础的（准）共形统一化问题的历史与最新进展，重点讨论了Koebe猜想的开放性问题及相关唯一性问题，以及Schottky集的映射问题。

On the conformal walk dimension: Quasisymmetric uniformization for symmetric diffusions

We introduce the notion of conformal walk dimension, which serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that, for a given strongly local, regular symmetric Dirichlet space in which every metric ball has compact closure (MMD space), the finiteness of the conformal walk dimension characterizes the conjunction of the metric doubling property and the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension of an MMD space is defined as the infimum over all possible values of the walk dimension with which…

Canonical parametrizations of metric surfaces of higher topology

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 the same topology. Moreover, we show that this is also true for surfaces as above with non-empty boundary and that the corresponding map can be chosen in a canonical way. Our proof is based on a local argument involving the existence of quasisymmetric parametrizations for metric discs as shown in in a paper of Lytchak and Wenger.

Weak Capacity and Critical Exponents

We investigate critical exponents relating to weak capacity in Ahlfors regular metric measure spaces. This allows a proof of a weak capacity version of a result by Bonk and Kleiner about the uniformization of metric $2$-spheres. Using our result, we promote local quasisymmetric equivalence with $\mathbb{S}^2$ to global quasisymmetric equivalence. We also use our weak capacity version to derive conditions for quasisymmetric equivalence to $\mathbb{S}^2$ in the presence of a group action. We investigate the relation between our defined critical exponents and Ahlfors regular conformal dimension, …

在几何约束下的纸面度量几何

The metric geometry of paper surfaces under geometric constraints

本文研究在几何约束下纸面的度量几何，证明其满足Ahlfors 2-正则性和线性局部可缩性，从而保证存在到标准2球面的准等距参数化。

Prime ends in metric spaces and boundary extensions of mappings

By using the inner diameter distance condition we define and investigate new, in such a generality, class $\mathcal{F}$ of homeomorphisms between domains in metric spaces and show that, under additional assumptions on domains, $\mathcal{F}$ contains (quasi)conformal, bi-Lipschitz and quasisymmetric mappings as illustrated by examples. Moreover, we employ a prime ends theory in metric spaces and provide conditions allowing continuous and homeomorphic extensions of mappings in $\mathcal{F}$ to topological closures of domains, as well as homeomorphic extensions to the prime end boundary. Domains …