Bilipschitz Approximations of Quasiconformal Maps

2005

The interaction between hyperbolic geometry and conformal analysis is a beautiful and fruitful aspect of the fields of analysis and low-dimensional geometry-topology. In particular, the study of hyperbolic geometry intertwines complex analysis, geometric function theory (especially in the guise of the study of quasiconformal mappings), and topology in a way that allows one to study a fixed object from diverse perspectives.

1996

We study boundary properties of quasiconformal self-mappings depending on complex dilatations. We give some new conditions for the corresponding quasisymmetric function to be asymptotically symmetric and obtain an explicit asymptotical representation for the distortion ratio of boundary correspondence when the complex dilatation has directional limits.

Journal d'Analyse Mathématique, 2009

We study the local growth of quasiconformal mappings in the plane. Estimates are given in terms of integral means of the pointwise angular dilatations. New sufficient conditions for a quasiconformal mapping f to be either Lipschitz or weakly Lipschitz continuous at a point are given.

2003

In this note we give some geometrical and analytical properties which caracterise the quasiconformal maps in Riemannian manifolds.

Mathematica Moravica

We give a new glance to the theorem of Wan (Theorem 1.1) which is related to the hyperbolic bi-Lipschicity of the K-quasiconformal, K 1, hyperbolic harmonic mappings of the unit disk D onto itself. Especially, if f is such a mapping and f (0) = 0, we obtained that the following double inequality is valid 2|z|/(K + 1) |f (z)| √ K|z|, whenever z ∈ D.

2016

Journal d'Analyse Mathématique, 1986

In this paper we investigate the geometry of the quasihyperbolic metric of domains in R". This metric arises from the conformally flat generalized Riemannian metric d(x, aD)-'ldx I. Due to the fact that the density cl(x, aD) -t is not necessarily differentiable, the classical theories of Riemannian geometry do not apply to this metric. The quasihyperbolic metric has been found to have many interesting and varied applications in geometric function theory. In particular, quasiconformal mappings are quasi-isometries of this metric for sufficiently far-lying points, also bounds on the quasihyperbolic metric in terms of other metrics imply that a domain is uniform which then implies certain injectivity criteria for locally-Lipschitz mappings, amongst others. In fact there is quite a strong relationship between uniform domains and the quasihyperbolic metric. Most of these basic results on the quasihyperbolic metric can be found in [3], [2] and . We note here that the quasihyperbolic metric is complete and generates the usual topology on a proper subdomain of R". Further, geodesics (length minimizing curves) always exist for this metric and these geodesics have Lipschitz continuous first derivatives, which is in fact best possible.

2010

We prove in dimension n > 2 that a K-quasiconformal harmonic mapping u of the unit ball B n onto itself is Euclidean bi-Lipschitz if u(0) = 0 and K < 2 n−1. This is an extension of a similar result of Tam and Wan for hyperbolic harmonic mappings with respect to a hyperbolic metric. The proof uses Möbius transformations on the related space and a recent result of the first author, which states that harmonic quasiconformal self-mappings of the unit ball are Lipschitz continuous.

Mathematical Surveys and Monographs, 2017

AMS: Functions of a complex variable -Geometric function theory -Quasiconformal mappings in R n . msc | Functions of a complex variable -Geometric function theory -Quasiconformal mappings in the plane. msc Classification: LCC QA360 .G437 2016 | DDC 515/.93-dc23 LC record available at . loc.gov/2016029235 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Proceedings of the American Mathematical Society, 1995

We give the sharp constants in the area distortion inequality for quasiconformal mappings in the plane. Astala [I] proved the following theorem conjectured by Gehring and Reich in [3]: Theorem A. Let f be a K-quasiconformal mapping of D = { z : lzl < 1) onto itself with f ( 0 )= 0 . Then for any measurable E c D we have where I I stands for the area. The first author [2] obtained a shorter proof which did not make use of the elaborate Thermodynamic Formalism and Holomorphic Motion Theory of the original proof of Astala. In late 1992 the second author [4] circulated a minimal proof which gives sharp bounds for the constants under the normalization f E C ( K ), i.e. f is a K-quasiconformal mapping of the plane which is conformal on C\o and f ( z )= z +o(1) near oo . In the interests of having a short sharp proof we combined our efforts.