Quasiconformal geometry and removable sets for conformal mappings

广义Grushin平面上拟共形映射的研究

An Approach to Studying Quasiconformal Mappings on Generalized Grushin Planes

证明了复平面与一类广义Grushin平面$G_r$（其中$r$是满足特定要求的函数）是拟对称等价的.然后利用共轭，我们能够发展出$G_r$空间上同胚的拟对称的解析定义。在最后一节中，我们表明我们的准对称性的解析定义是一致的，与早期的概念共形映射上的Grushin平面。由此得到了广义Grushin平面上共形映射的几个特征。

不可分解的准共形映射 manifold

Indecomposable Quasiconformal Maps of Manifolds

本文研究了闭合manifold上不可分解的准共形映射的存在性，核心方法是构造具有任意小共形扭曲的映射组合，主要贡献是证明了此类映射的不可分解性。

Sublinear quasiconformality and the large-scale geometry of Heintze groups

This article analyzes sublinearly quasisymmetric homeo-morphisms (generalized quasisymmetric mappings), and draws applications to the sublinear large-scale geometry of negatively curved groups and spaces. It is proven that those homeomorphisms lack analytical properties but preserve a conformal dimension and appropriate function spaces, distinguishing certain (nonsymmetric) Riemannian negatively curved homogeneous spaces, and Fuchsian buildings, up to sublinearly biLipschitz equivalence (generalized quasiisometry).

使用离散曲率流计算Riemann曲面上的拟共形映射

Computing Quasiconformal Maps on Riemann surfaces using Discrete Curvature Flow

本文提出了一种在一般Riemann曲面上求解Beltrami方程以计算离散拟共形映射的新方法，通过定义离散辅助度量并结合离散Yamabe流算法，实现了映射的准确计算与收敛性验证。

Non-removability of Sierpinski spaces

We prove that all Sierpiński spaces in ${\mathbb{S}}^n$, $n\geq 2$, are non-removable for (quasi)conformal maps, generalizing the result of the first named author arXiv:1809.05605. More precisely, we show that for any Sierpiński space $X\subset \mathbb{S}^n$ there exists a homeomorphism $f\colon \mathbb{S}^n\to \mathbb{S}^n$, conformal in $\mathbb{S}^n\setminus X$, that maps $X$ to a set of positive measure and is not globally (quasi)conformal. This is the first class of examples of non-removable sets in higher dimensions.

Intrinsic metrics under conformal and quasiregular mappings

The distortion of six different intrinsic metrics and quasi-metrics under conformal and quasiregular mappings is studied in a few simple domains $G\subsetneq\mathbb{R}^n$. The already known inequalities between the hyperbolic metric and these intrinsic metrics for points $x,y$ in the unit ball $\mathbb{B}^n$ are improved by limiting the absolute values of the points $x,y$ and the new results are then used to study the conformal distortion of the intrinsic metrics. For the triangular ratio metric between two points $x,y\in\mathbb{B}^n$, the conformal distortion is bounded in terms of the hyperb…

Remarks on conformal modulus in metric spaces

We give an example of an Ahlfors $3$-regular, linearly locally connected metric space homeomorphic to $\mathbb{R}^3$ containing a nondegenerate continuum $E$ with zero capacity, in the sense that the conformal modulus of the set of nontrivial curves intersecting $E$ is zero. We discuss this example in relation to the quasiconformal uniformization problem for metric spaces.

复伸张的仿射平衡：调和拟共形映射的仿射不变量与最优畸变

Affine balancing of complex dilatations: affine invariants and optimal distortion of harmonic quasiconformal mappings

本文针对调和拟共形映射，引入仿射外接半径与仿射直径作为仿射不变量，刻画最优仿射平衡下的最小畸变，并建立锐利双边界及二阶估计，统一极值畸变问题。

极大收敛群与秩一对称空间

Maximal Convergence Groups and Rank One Symmetric Spaces

证明了非紧型秩一对称空间（除双曲平面外）边界的共形同胚群是极大收敛群。此外，我们还证明了一族一致拟共形同胚具有收敛性。我们的定理推广了Gehring和Martin在实双曲情形下的结果。因此，这表明$d$ -球面的自同胚群的极大收敛子群在共轭下不是唯一的。最后，我们讨论了极大性的一些含义。

次黎曼流形上的拟正则映射

Quasiregular mappings on sub-Riemannian manifolds

研究了次黎曼流形上关于Carnot-Caratheodory距离拟正则的映射，并讨论了几个相关的概念.在H-型卡诺群上，拟正则映射已经在早期用解析定义引入，但是到目前为止，在一般的次黎曼流形的情况下，还没有一个好的工作定义。因此，在本文中，我们采用了度量而不是分析的观点。作为第一个主要结果，我们证明了次黎曼透镜空间中存在非平凡一致拟正则（UQR）映射，即所有迭代的畸变有一致界的拟正则映射。在这样做的过程中，我们还获得了新的例子UQR映射的标准次黎曼球面。证明是基于一种方法，用于建设共形陷阱上使用准共形流的子黎曼领域，并适应这种方法的concentients的领域。然后可以研究由UQR映射生成的拟正则半群。在本文的第二部分中，我们遵循Tukia证明了存在一个可测的共形结构，这是不变的，在这样的半群。在这里，共形结构仅在水平分布上指定，并且回撤使用Margulis-Mostow导数（其推广了经典和Pansu导数）来定义。

解析函数与拟正则映射的Nagata维度变形

Analytic and quasiregular distortion of Nagata dimension

本文研究了解析函数和拟正则映射对Nagata维度的变形，揭示了紧致性和非紧致性条件下不同映射对Nagata维度的影响，并提出了未来研究方向。