Related papers: Quasiconformal extension for harmonic mappings on …
We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended…
This note examines sufficient conditions for the quasiconformal extendibility of harmonic mappings defined in the unit disk. It is demonstrated that a harmonic strongly starlike mapping admits a quasiconformal extension to the entire plane,…
If $\Omega$ is a simply connected domain in $\overline{{\mathbb C}}$ then, according to the Ahlfors-Gehring theorem, $\Omega$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in…
We prove that for harmonic quasiconformal mappings $\alpha$-H\"older continuity on the boundary implies $\alpha$-H\"older continuity of the map itself. Our result holds for the class of uniformly perfect bounded domains, in fact we can…
We provide two new formulas for quasiconformal extension to $\overline{\mathbb{C}}$ for harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of the Ahlfors-Weill extension…
A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$ of ${\Bbb R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:{\Bbb R}^N\to{\Bbb R}^N$ . We give sufficient conditions in terms of the weak quasisymmetry…
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…
The conformal deformations are contained in two classes of mappings: quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every $K$ quasiconformal harmonic mapping between…
We prove the following theorem: every quasiconformal harmonic mapping between two plane domains with $C^{1,\alpha}$ ($\alpha<1$), respectively $C^{1,1}$ compact boundary is bi-Lipschitz. The distance function with respect to the boundary of…
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…
Suppose that $f: D\to D'$ is a quasiconformal mapping, where $D$ and $D'$ are domains in ${\mathbb R}^n$, and that $D$ is a broad domain. Then for every arcwise connected subset $A$ in $D$, the weak quasisymmetry of the restriction $f|_A:…
We study the Dirichlet problem for harmonic maps between hyperbolic planes, under the assumption that the Euclidean harmonic extension of the boundary map is quasiconformal.
We derive a quasiconformal extension to 3-space of the Weierstrass-Enneper lifts of a class of harmonic mappings defined in the unit disk. The extension is based on fibrations of space by circles in domain and image that correspond to each…
The behavior of a class of mappings of a domain of Euclidean space is studied. It is established that the indicated class is equicontinuous both at the inner and at the boundary points of the domain if the mappings contained in it satisfy…
Quasiconformal maps in the complex plane are homeomorphisms that satisfy certain geometric distortion inequalities; infinitesimally, they map circles to ellipses with bounded eccentricity. The local distortion properties of these maps give…
We prove the following. If $f$ is a harmonic quasiconformal mapping between the unit ball in $\mathbb{R}^n$ and a spatial domain with $C^{1,\alpha}$ boundary, then $f$ is Lipschitz continuous in $B$. This generalizes some known results for…
We prove that every quasiconformal mapping from the harmonic $\beta$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $\alpha$-H\"older continuous for $\alpha<1-\beta$, with the H\"older coefficient…
We demonstrate the existence of quasiconformal mappings on closed manifolds that cannot be decomposed as a composition of mappings with arbitrarily small conformal distortion.
We give improved bounds for the distortion of the Hausdorff dimension under quasisymmetric maps in terms of the dilatation of their quasiconformal extension. The sharpness of the estimates remains an open question and is shown to be closely…
We prove that every sense-preserving harmonic $K$--quasiconformal homeomorphism $f\colon D\to\Omega$ between Lyapunov domains (equivalently, bounded $C^{1,\alpha}$ domains) in $\mathbb{R}^n$, $\alpha\in(0,1]$, is globally Lipschitz on…