我们提出了一个调查的许多和各种元素的现代高维理论的拟共形映射及其广泛而多样的应用。它是由作者的兴趣主题统一（和限制）的。因此，我们将讨论的基本理论，因为它在20世纪60年代发展的早期工作的F.W. Gehring和Yu G. Reshetnyak和随后探索的连接与几何函数理论，非线性偏微分方程，微分和几何拓扑和动力学，因为他们随后在未来几十年。我们给一些证据，因为我们试图概述该地区的主要成果和当前的研究主题。我们并不试图以最大的一般性来呈现这些结果，因为要实现这一点，读者需要相当多的技术知识。我们试图让人感觉到这个领域在哪里，中心思想和问题是什么，以及目前与其他领域研究人员的主要互动在哪里。我们还在这里和那里添加了一些历史。我们还没有能够涵盖许多最近的进展概括的理论映射的有限失真和退化椭圆Beltrami系统的理论密切联系的变分法和非线性弹性，非线性霍奇理论和相关领域，虽然读者可能会看到阴影这方面的部分。
英文摘要：
We present a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author's interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the major results of the area and current research themes. We do not strive to present these results in maximal generality, as to achieve this considerable technical knowledge would be necessary of the reader. We have tried to give a feel of where the area is, what are the central ideas and problems and where are the major current interactions with researchers in other areas. We have also added a bit of history here and there. We have not been able to cover the many recent advances generalising the theory to mappings of finite distortion and to degenerate elliptic Beltrami systems which connects the theory closely with the calculus of variations and nonlinear elasticity, nonlinear Hodge theory and related areas, although the reader may see shadows of this aspect in parts.
In this paper, we extend Ahlfors's univalent criteria and Ahlfors's quasiconformal extension for analytic functions to harmonic mappings defined in the unit disk. Moreover, we give a general quasiconformal extension of harmonic Teichmüller mappings, whose maximal dilatation estimate is asymptotically sharp.
We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains.
We consider quasiconformal mappings of the unit disk that have a planar extension which have $p$-integrable distortion. In this paper, we establish a bound for the modulus of continuity for the inverse mapping and show sharpness of this bound. Furthermore, we also obtain bounds for the compression multifractal spectra of such mappings.
Harmonic K-quasiconformal Koebe functions: construction and application to Pavlovic's problem
Hunan First Normal University（湖南第一师范学院）；Liaocheng University（聊城大学）；Guangdong Technion-Israel Institute of Technology（广东以色列理工学院）；Technion-Israel Institute of Technology（以色列理工学院）；Changsha University of Science and Technology（长沙理工大学）
We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities.
We construct quasiconformal mappings $f\colon \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ for which there is a Borel set $E \subset \mathbb{R}^2 \times \{0\}$ of positive Lebesgue $2$-measure whose image $f(E)$ has Hausdorff $2$-measure zero. This gives a solution to the open problem of inverse absolute continuity of quasiconformal mappings on hypersurfaces, attributed to Gehring. By implication, our result also answers questions of Väisälä and Astala--Bonk--Heinonen.
In this article, we establish the Bohr inequalities for the sense-preserving $K$-quasiconformal harmonic mappings defined in the unit disk $\mathbb{D}$ involving classes of Ma-Minda starlike and convex univalent functions, usually denoted by $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ respectively, and for $\log (f(z)/z)$ where $f$ belongs to the Ma-Minda classes or satisfies certain differential subordination. We also estimate Logarithmic coefficient's bounds for the functions in $\mathcal{C}(ψ)$ for the case $ψ(\mathbb{D})$ be convex.
In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk $ \overline{B}(1/(1-k^4),k^2/(1-k^4))$. This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension $1$. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a $1$-dimensional subset of a line under a…
Let $\overline{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\overline{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings $f$ satisfying the Poisson's equation $Δf=g$ in $\mathbb{D}$, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results.
If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension …
On Koebe-type functions for harmonic quasiconformal mappings
Hunan First Normal University（湖南第一师范学院）；Changsha University of Science and Technology（长沙理工大学）；Guangdong Technion-Israel Institute of Technology（广东以色列理工学院）；Technion-Israel Institute of Technology（以色列理工学院）
In this paper, we provide partial solutions to a problem raised by Väisälä on local properties of free quasiconformal mappings. In particular, we show that a locally free quasiconformal mapping is globally free quasiconformal under the condition of locally relative quasisymmetry.
Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability
在他著名的论文面积失真的拟共形映射，Astala显示了最佳面积失真界和维数失真估计平面拟共形映射。他问（问题4.4）是否更精细的结果举行，即绝对连续性的豪斯多夫措施下推进拟共形映射。这在Astala、Clop、Mateu、Orobitg和作者的联合工作（“Distortion of Hausdorff measures and improved Painlev 'e removability for bounded quasirregular mappings”，Duke Math J.，出现[ACMOU]）（定理1.1），其他情况保持开放。一个相关的问题，我们离开开放的[ACMOU]（问题4.2）（这是问Astala的作者在[ACMOU]之前，在一个等价的形式在个人通信）是是否BMO可移动的K-拟正则映射和（$L^{\infty}$）可移动的K-拟正则映射确实是不同的问题。本文给出了[ACMOU]中正问题4.2的一系列例子，同时证明了[ACMOU]中定理1.1在两种不同意义上的清晰性，并给出了在拟共形映射推进下Hausdorff测度的绝对连续性在这两种不同意义上的清晰性的例子.
It is known for some time that there exists an energy-minimal diffeomorphism between two doubly-connected domains $Ω$ and $D$ provided that $\mathrm{Mod}(Ω)\le \mathrm{Mod}{D}$ and that diffeomorphism is harmonic \cite{tedi}. In this note we give a short proof of the fact that for given annuli $Ω$ and $D$ satisfying the condition $\mathrm{Mod}(Ω)\le \mathrm{Mod}(D)$ there exist a $K-$quasiconformal harmonic diffeomorphism $f:Ω\onto D$, where $K=K(τ)$, $τ=\mathrm{Mod}(Ω)/ \mathrm{Mod}(D)$ and $\lim_{τ\to 1}K(τ)=1$.
We recall the notions of conformal and quasiconformal mappings \textit{in the sense of Gromov}, extending the classical notions of conformal and quasiconformal mappings, and prove the following theorem. {\em If the mapping $ F: \mathbb{R}^{n} \to \mathbb{R}^{2} $, where $ n \geq 2 $, quasiconformal in the sense of Gromov, omits more than one value on the plane $\mathbb{R}^{2} $, then it is a constant mapping.}
