Kai Rajala, Martti Rasimus
We give a characterization of metric spaces quasisymmetrically equivalent to a finitely connected circle domain. This result generalizes the uniformization of Ahlfors 2-regular spaces by Merenkov and Wildrick.
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Koebe uniformization of nondegenerate domains with bounded gap-ratio
本文利用跨界极值长度方法，证明了具有有界间隙比的非退化不可数连通域均可共形同胚于圆域，解决了Koebe均匀化问题。
Koebe uniformization, John domains, and transboundary modulus
本文通过证明平面 John 域共形等价于圆域，证明了 Koebe 猜想，并利用跨界模量给出充要条件。
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 domain in the standard 2-sphere is quasisymmetrically equivalent to a circle domain if and only if X is linearly locally connected and its completion is compact. We also give a counterexample in the countably connected case.
Koebe uniformization for infinitely connected attracting Fatou domains
本文针对有理映射的无穷连通Fatou域因边界复杂现有单值化结果难适用的问题，通过Koebe单值化方法证明几何有限有理映射的该类域共形同胚于圆域。
Quasisymmetric Koebe Uniformization of metric surfaces
针对度量曲面拟对称映射到圆域的问题，提出2-TLP条件作为充要条件，回答了Merenkov和Wildrick的问题，并将Bonk的平面结果推广到非平面情形。
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 orientable surfaces.
Rigidity and quasisymmetric uniformization of Thurston-type maps
本文研究图林型映射的刚性与准等距统一化，证明了无不变线场猜想并建立了相关定理。
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 Ahlfors 2-regular, linearly locally connected metric planes; instead of conformal equivalence, we are concerned with quasisymmetric equivalence.
For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which…
We study the method of finding conformal maps onto circle domains by approximating with finitely connected subdomains. Every domain $D \subset \hat{C}$ admits exhaustions, i.e., increasing sequences of finitely connected subdomains $D_j$ whose union is $D$. By Koebe's theorem, each $D_j$ admits a conformal map $f_{D_j}$ from $D_j$ onto a circle domain $f_{D_j}(D_j)$. Assuming $f_{D_j} \to f$, our goal is to find out if $f(D)$ is also a circle domain. We present a countably connected $D$ with an exhaustion $(D_j)$ so that $(f_{D_j})$ has a limit whose image is not a circle domain, and a domain …
Uniformization of metric surfaces: A survey
本文综述了度量曲面均匀化的最新进展，从经典共形均匀化到分形曲面上的拟对称与拟共形均匀化，最终介绍任意局部有限面积度量曲面的弱拟共形均匀化结果。
Uniformization of Gromov hyperbolic domains by circle domains
本文证明了黎曼球面上的域为Gromov双曲当且仅当共形等价于均匀圆域，解决了Bonk-Heinonen-Koskela猜想并验证了Koebe猜想，且均匀化映射在Möbius变换下唯一。
Simultaneous uniformization of chord-arc curves and BMO Teichmüller space
本文综述同时均匀化方法，证明弦弧曲线上BMO函数的Cauchy变换由双全纯映射导数表达且全纯依赖曲线变化，连接奇异积分与Teichmüller空间结构。
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…
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…
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.
The real analytic structure of the Teichmüller space of circle diffeomorphisms with Zygmund continuous derivatives
本文研究了具有 Zygmund 连续导数的圆 diffeomorphisms 的 Teichmüller 空间的真实解析结构，通过统一化和 Besov 空间上的 composition 操作方法，得出了该空间与 Zygmund 连续函数空间的等价性结论。
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 …
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