Some properties of quasiconformal mappings in Riemannian manifolds (original) (raw)

The theory of quasiconformal mappings in higher dimensions, I

Handbook of Teichmüller Theory, Volume IV, 2014

This chapter presents 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.

On the quasisymmetry of quasiconformal mappings and its applications

2012

Abstract: Suppose that $ D $ is a proper domain in IRn\ IR^ n IRn and that $ f $ is a quasiconformal mapping from $ D $ onto a John domain $ D'$ in IRn\ IR^ n IRn. First, we show that if $ D $ and $ D'$ are bounded, and $ D $ is a broad domain, then for an arcwise connected subset $ A $ in $ D ,, , f (A) $ is $ LLC_2 $ with respect to deltaD′\ delta_ {D'} deltaD′ in $ D'$ if and only if the restriction $ f| _A: A\ to f (A) $ is quasisymmetric in the metrics deltaD\ delta_D deltaD and deltaD′\ delta_ {D'} deltaD′.

Quasiregular mappings and 𝒲𝒯 -classes of differential forms on Riemannian manifolds

Pacific Journal of Mathematics, 2002

The purpose of this paper is to study the relations between quasiregular mappings on Riemannian manifolds and differential forms. Four classes of differential forms are introduced and it is shown that some differential expressions connected in a natural way to quasiregular mappings are members in these classes.

A Quasi–Linear Manifolds and Quasi–Linear Mapping Between Them

Turkish Journal of …, 2004

We further develop in this article the theory of QL-mappings, which was started by AI Shnirelman ([5]), continued by MAEphendiev ([3]) and also by myself ([1]). As was proved in [1], the classes FQL and FSQL-mappings coincide; however the latter class is more adapted to ...

Hyperbolic Geometry and Quasiconformal Mappings

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.

On Boundary Correspondence Under Quasiconformal Mappings

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.

On lipschitz continuity of quasiconformalmappings in space

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.

Criteria of convergence for quasiconformal mappings and their generalizations

Ukrainian Mathematical Journal, 1996

We establish necessary and sufficient conditions for the convergence of normalized homeomorphisms of Sobolev class in terms of the Fourier transforms of complex characteristics in the case where the upper bound of dilations is exponentially bounded in measure. This allows us to construct various metrics generating locally uniform convergence of mappings.