Harmonic Quasiconformal Mappings Research Articles

On quasiconformal harmonic mappings lifting to minimal surfaces

We prove a growth theorem for a function to belong to the class \sum(\mu;a) and generalize a Weierstrass-Enneper representation type theorem for the minimal surfaces given in [5] to spacelike minimal surfaces which lie in 3-dimensional Lorentz-Minkowski space L^3. We also obtain some estimates of the Gaussian curvature of the minimal surfaces in 3-dimensional Euclidean space R^3 and of the spacelike minimal surfaces in L^3.

(K,K')-Quasiconformal harmonic mappings of the upper half plane onto itself

In this paper we show that (K;K 0 )-quasiconformal mappings with unbounded image domains are not Holder continuous, which is dierent from the case with bounded image domains given by Kalaj and Mateljevi¢. For a (K;K 0 )-quasiconformal harmonic mapping of the upper half plane onto itself, we prove that it is Lipschitz and hyperbolically Lipschitz continuous. Moreover, we get four equivalent conditions for a harmonic mapping of the upper half plane onto itself to be a (K;K 0 )-quasiconformal mapping.

Hyperbolically Bi-Lipschitz Continuity for -Harmonic Quasiconformal Mappings

We study the class of -harmonic -quasiconformal mappings with angular ranges. After building a differential equation for the hyperbolic metric of an angular range, we obtain the sharp bounds of their hyperbolically partial derivatives, determined by the quasiconformal constant . As an application we get their hyperbolically bi-Lipschitz continuity and their sharp hyperbolically bi-Lipschitz coefficients.

Quasiconformal harmonic mappings between $${{\fancyscript{C}}^{1,\mu}}$$ Euclidean surfaces

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 \({{\fancyscript{C}}^{1,\mu} (0 < \mu\le 1)}\) surfaces with \({{\fancyscript{C}}^{1,\mu}}\) boundary is a Lipschitz mapping. This extends some recent results of several authors where the same problem has been considered for plane domains. As an application it is given an explicit Lipschitz constant of normalized isothermal coordinates of a disk-type minimal surface in terms of boundary curve only. It seems that this kind of estimates are new for conformal mappings of the unit disk onto a Jordan domain as well.

On boundary correspondences under quasiconformal harmonic mappings between smooth Jordan domains

AbstractA quantitative version of an inequality obtained in [8, Theorem 2.1] is given. More precisely, for normalized K quasiconformal (q.c.) harmonic mappings of the unit disk onto a Jordan domain Ω ∈ C1, μ (0 &lt; μ ≤ 1), we give an explicit Lipschitz constant depending on the structure of Ω and on K. In addition, we give a characterization of q.c. harmonic mappings of the unit disk onto an arbitrary Jordan domain with C2, α boundary in terms of the boundary function using the Hilbert transform. Moreover, a sharp explicit quasiconformal constant is given in terms of the boundary function.

Hölder continuity of harmonic quasiconformal mappings

We prove that for harmonic quasiconformal mappings α-Holder continuity on the boundary implies α-Holder continuity of the map itself. Our result holds for the class of uniformly perfect bounded domains, in fact we can allow that a portion of the boundary is thin in the sense of capacity. The problem for general bounded domains remains open.

Doubly Connected Minimal Surfaces and Extremal Harmonic Mappings

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.

Invertible harmonic mappings beyond the Kneser theorem and quasiconformal harmonic mappings

Invertible harmonic mappings beyond the Kneser theorem and quasiconformal harmonic mappings

A Schwarz–Pick inequality for harmonic quasiconformal mappings and its applications

The main result of this paper is the sharp generalized Schwarz–Pick inequality for euclidean harmonic quasiconformal mappings with convex ranges, which generalizes a result given by Mateljević. As its applications, we obtain the property of quasi-isometry with respect to the Poincaré distance and an analogue of the Koebe theorem for this class of mappings.

Quasiconformal harmonic mappings and close-to-convex domains

Let f = h + ? be a univalent sense preserving harmonic mapping of the unit disk U onto a convex domain ?. It is proved that: for every a such that |a| &lt; 1 (resp. |a| = 1) the mapping ?a = h + a? is an |a| quasiconformal (a univalent) close-to-convex harmonic mapping. This gives an answer to a question posed by Chuaqui and Hern?ndez (J. Math. Anal. Appl. (2007)). 2010 Mathematics Subject Classifications. Primary 30C55, Secondary 31C05. .

Bi-lipschicity of quasiconformal harmonic mappings in the plane

We show that quasiconformal harmonic mappings on the proper domains in R2 are bi-Lipschitz with respect to the quasihyperbolic metric. .

Inner estimate and quasiconformal harmonic maps between smooth domains

We prove a type of “inner estimate” for quasi-conformal diffeomorphisms, which satisfies a certain estimate concerning their Laplacian. This, in turn, implies that quasiconformal harmonic mappings between smooth domains (with respect to an approximately analytic metric), have bounded partial derivatives; in particular, these mappings are Lipschitz. We discuss harmonic mappings with respect to (a) spherical and Euclidean metrics (which are approximately analytic) (b) the metric induced by a holomorphic quadratic differential.

On the Douglas–Dirichlet functional and harmonic quasiconformal mappings

In this article, we discuss the minimal mappings of Douglas–Dirichlet functional and harmonic quasiconformal mappings, and solve the uniqueness problem of harmonic quasiconformal mappings posed by Shibata.

Distance and volume decreasing theorems for quasiconformal mappings

The method used by the author in deriving a Picard type theorem for quasiconformal mappings [Proc.. Amer. Math. Soc. 61 (1976), 265-27], improved by a proposition of S.-T. Yau [Amer. J. Math. 100 (1978), 197-203] is employed here to extend the Schwarz-Ahlfors lemma to harmonic quasiconformal mappings. The target space is not necessarily hyperbolic, not even negatively curved.

ON THE THEORY OF EXTREMAL QUASICONFORMAL MAPPINGS

In this work criteria are established for the extremality of both quasiconformal deformations of open Riemann surfaces of hyperbolic type and Kleinian groups of the mappings in homotopy classes modulo ideal boundaries. The existence of generalized harmonic quasiconformal mappings in these classes is shown. Bibliography: 17 titles.

Harmonic Quasiconformal Mappings of Riemannian Manifolds

Harmonic Quasiconformal Mappings of Riemannian Manifolds

Harmonic quasiconformal mappings of Riemannian manifolds

Harmonic quasiconformal mappings of Riemannian manifolds