arXiv math/06108632006-10-27math.MGmath.CV

Quasisymmetric parameterizations of two-dimensional metric planes

Kevin Wildrick

备注：: 28 pages
DOI：: 10.1112/plms/pdn023

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Quasisymmetric embeddings of slit Sierpiński carpets

We study the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpiński carpet into the plane. In the case of so called dyadic slit carpets, several characterizations are obtained. One characterization is in terms of a Transboundary Loewner Property (TLP) which is a transboundary analogue of the Loewner property of Heinonen and Koskela. We show that a dyadic slit carpet can be quasisymmetrically embedded into the plane if and only if it is TLP. Moreover, every dyadic slit carpet $X$ can be associated to a "pillowcase sphere" $\widehat{X}$ which is a metric space homeomorphic…

2021-02-24math.CVmath.MG

Quasisymmetric structures on surfaces

We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces in some Euclidean space that are locally bi-Lipschitz equivalent to an open subset of the plane.

2007-09-06math.CVmath.MG

Transitive bi-Lipschitz group actions and bi-Lipschitz parameterizations

We prove that Ahlfors 2-regular quasisymmetric images of the Euclidean plane are bi-Lipschitz images of the plane if and only if they are uniformly bi-Lipschitz homogeneous with respect to a group. We also prove that certain geodesic spaces are bi-Lipschitz images of Carnot groups if they are inversion invariant bi-Lipschitz homogeneous with respect to a group.

2022-07-08math.MG

Quasiconformal mappings on the Grushin plane

We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if and only if it is metrically quasiconformal and if and only if it is geometrically quasiconformal. As the main step in our argument, we show that a quasisymmetric parametrization of the Grushin plane by the Euclidean plane must also be geometrically quasiconformal. We also discuss some aspects of the Euclidean theory of quasiconformal maps, such as absolute continuity on almost every compact curve, not satisfied in the Grushin case.

2017-02-07math.MG

An Approach to Studying Quasiconformal Mappings on Generalized Grushin Planes

We demonstrate that the complex plane and a class of generalized Grushin planes $G_r$, where $r$ is a function satisfying specific requirements, are quasisymmetrically equivalent. Then using conjugation we are able to develop an analytic definition of quasisymmetry for homeomorphisms on $G_r$ spaces. In the last section we show our analytic definition of quasisymmetry is consistent with earlier notions of conformal mappings on the Grushin plane. This leads to several characterizations of conformal mappings on the generalized Grushin planes.

2014-10-20math.CVmath.MG

Thin Loewner carpets and their quasisymmetric embeddings in $S^2$

A carpet is a metric space which is homeomorphic to the standard Sierpiński carpet in $\mathbb{R}^2$, or equivalently, in $S^2$. A carpet is called thin if its Hausdorff dimension is $<2$. A metric space is called Q-Loewner if its $Q$-dimensional Hausdorff measure is Q-Ahlfors regular and if it satisfies a $(1,Q)$-Poincaré inequality. As we will show, $Q$-Loewner planar metric spaces are always carpets, and admit quasisymmetric embeddings into the plane. In this paper, for every pair $(Q,Q')$, with $1<Q<Q'< 2$ we construct infinitely many pairwise quasi-symmetrically distinct $Q$-Loewner carpe…

2020-04-08math.CAmath.DGmath.MG

更多相关论文收起更多论文

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…

2016-08-26math.CVmath.MG

平面树的拟共形单值化与延拓

Quasiconformal uniformization and extension for planar trees

The University of Tennessee（田纳西大学）

该研究针对平面树证明了类似Ahlfors的拟共形单值化定理，还给出了保持分支方向的同胚延拓结果，并构造了含Vicsek分形的四价拟共形树类，其为ℝ²上一致拟正则映射的Julia集。

2026-10-15math.MG

Poincaré-type Inequalities and Finding Good Parameterizations

A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. In 1960, E. R. Reifenberg proved that if a set is well approximated by planes at every point and at every scale, then the set is a bi-Hölder image of a plane. It is known today that Carleson-type conditions on these approximating planes guarantee a bi-Lipschitz parameterization of the set. In this paper, we consider an $n$-Ahlfors regular rectifiable set $M \subset \mathbb{R}^{n+d}$ that satisfies a Poincaré-type inequality involving the tangential derivative. …

2016-05-24math.MG

解析一维平面与组合 Loewner 性质

Analytically one-dimensional planes and the Combinatorial Loewner Property

本文构造了一个新例子，使得 Kleiner 猜想的结论要么在该例中失败从而给出新型反例，要么导致存在解析一维平面，从而解决 Kleiner–Schioppa 问题。

2024-08-30math.CAmath.MG

Conformal Grushin spaces

We introduce a class of metrics on $\mathbb{R}^n$ generalizing the classical Grushin plane. These are length metrics defined by the line element $ds = d_E(\cdot,Y)^{-β}ds_E$ for a closed nonempty subset $Y \subset \mathbb{R}^n$ and $β\in [0,1)$. We prove that, assuming a Hölder condition on the metric, these spaces are quasisymmetrically equivalent to $\mathbb{R}^n$ and can be embedded in some larger Euclidean space under a bi-Lipschitz map. Our main tool is an embedding characterization due to Seo, which we strengthen by removing the hypothesis of uniform perfectness. In the two-dimensional c…

2016-02-15math.MG