Quasiconformal Reflection Research Articles

Moduli of quadrilaterals and quasiconformal reflection

We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in terms of the Lauricella hypergeometric function. This result enables us to understand the dissimilarities of the exterior and interior of a convex polygonal quadrilateral. We also give a Mathematica algorithm for the computation. Then we study the special case of an isosceles trapezoidal polygon L and obtain two-sided estimates for the coefficient of quasiconformal reflection over L in terms of special functions and geometric parameters of L.

Ahlfors-regular curves and Carleson measures

We study the relation between the boundary of a simply connected domain $\Omega$ being Ahlfors-regular and the invariance of Carleson measures under the push-forward operator induced by a conformal mapping from the unit disk $\Delta$ onto the domain $\Omega$. As an application, we characterize the chord-arc curves with small norms and the asymptotically smooth curves in terms of the complex dilatation of some quasiconformal reflection with respect to the curve.

Extremal function for capacity and estimates of QED constants in [formula omitted

This paper is devoted to the study of some fundamental problems on modulus and extremal length of curve families, capacity, and n-harmonic functions in the Euclidean space Rn. One of the main goals is to establish the existence, uniqueness, and boundary behavior of the extremal function for the conformal capacity cap(A,B;Ω) of a capacitor in Rn. This generalizes some well known results and has its own interests in geometric function theory and potential theory. It is also used as a major ingredient in this paper to establish a sharp upper bound for the quasiextremal distance (or QED) constant M(Ω) of a domain in terms of its local boundary quasiconformal reflection constant H(Ω), a bound conjectured by Shen in the plane. Along the way, several interesting results are established for modulus and extremal length. One of them is a decomposition theorem for the extremal length λ(A,B;Ω) of the curve family joining two disjoint continua A and B in a domain Ω.

Transformation of the quasiconformal reflection coefficient and of the Fredholm eigenvalue

Abstract. The behavior of the quasiconformal reflection coefficient and of the Fredholm eigenvalue of a plane analytic Jordan curve is studied under the square root transformation.

On Conformal Mappings onto Quasidisks

In this paper, we prove a distortion theorem for conformal mappings whose images are quasidisks. As an application, we construct a quasicon-formal reflection with bounded partial derivatives near the reflection boundary based on the Douady-Earle extension.

Duality, uniformity, and linear local connectivity

Using Väisälä’s metric duality on joinability of sets, we show, among other things, that in $\mathbb{R}^3$ if the complementary domains of a surface are LLC, they are also uniform. As an application, we show that an Ahlfors regular topological sphere that admits a quasiconformal reflection is quasisymmetrically equivalent to the standard sphere.

Conformal invariants of QED domains

Given a Jordan domain $\Omega$ in the extended complex plane $\overline{\kern-1.5pt\Bbb C}$, denote by $M_b(\Omega), M(\Omega)$ and $R(\Omega)$ the boundary quasiextremal distance constant, quasiextremal distance constant and quasiconformal reflection constant of $\Omega$, respectively. It is known that $M_b(\Omega)\le M(\Omega)\le R(\Omega)+1$. In this paper, we will give some further relations among $M_b(\Omega), M(\Omega)$ and $R(\Omega)$ by introducing and studying some other closely related constants. Particularly, we will give a necessary and sufficient condition for $M_b(\Omega)=R(\Omega)+1$ and show that $M(\Omega)<R(\Omega)+1$ for all asymptotically conformal extension domains other than disks. This gives an affirmative answer to a question asked by Yang, showing that the conjecture $M(\Omega)=R(\Omega)+1$ by Garnett and Yang is not true for all asymptotically conformal extension domains other than disks. Our discussion relies heavily on the theory of extremal quasiconformal mappings, which in turn gives some interesting results in the extremal quasiconformal mapping theory as well.

Quasiconformal reflection coefficient and Fredholm eigenvalue of an ellipse of hyperbolic geometry

We consider extremal quasiconformal reflections at a Jordan curve C, and related questions. Then we examine in more detail e.g., the special case of an ellipse С of hyperbolic geometry.

A Bound for Reflections Across Jordan Curves

Abstract The properties of logarithmic derivative of the Riemann mapping function of a quasidisk are applied to quantitative estimation of quasiconformal reflections across its boundary.

Quasiconformal Reflection for Generalized Spiral-Shaped Domains

We discuss a method of constructing a generalized spiral domain such that the quasiconformal reflection with respect to the boundary of this domain can be described explicitly. Bibliography: 7 titles.

Some Estimates for the Derivative of a Quasiconformal Reflection Spiral-Shaped Domains

Some estimates of derivatives are sharpened for quasiconformal reflections of a special class of simply connected plane domains. Bibliography: 2 titles.

Extremal distance and quasiconformal reflection constants of domains in ℝ N

Extremal distance and quasiconformal reflection constants of domains in ℝ N

Quasiconformal reflection, uniform and quasiconformal extension domains

In this paper we prove that a quasiconformal reflection domain is a uniform and quasiconformal exten-sion domain. We also prove that a quasiconformal extension domain which is quasiconformally equivalent to a ball is uniform.

The Schwarzian derivative and quasiconformal reflections on S^n

The Schwarzian derivative and quasiconformal reflections on S^n

Injectivity, quasiconformal reflections and the logarithmic derivative

Injectivity, quasiconformal reflections and the logarithmic derivative