The boundary correspondence under quasiconformal mappings

A characterisation of plane quasiconformal maps using triangles

“…This proposition is similar to criteria given for global homeomorphisms in § 2.2-2.5 in [2] and [4, Th. 18 and Cor.…”

Asymptoticity of grafting and Teichmüller rays

“…Then by mapping the rectangle to a unit disk and applying the AhlforsBeurling extension [2], any vertex-preserving piecewise-affine C 1 homeomorphism f W @S ! @S of dilatation of the order of 1CC can be extended to a homeomorphism f W S !…”

“…This divides y S into two parts, and on one of them we define the modified C 1 foliation to be one that (1) agrees with the horocyclic foliation for height greater than D D ln.1= / (when the leaves have width less than ), (2) interpolates between the leaf at height D and the arc a in such a way that the lengths of the leaves is a decreasing C 1 function of (nonnegative) height, (3) has each leaf orthogonal to .…”

“…The positive real number m is the modulus of the quadrilateral, denoted mod.Q/. The following are the fundamental results of Ahlfors and Beurling [2] (for the version stated here see Bishop [3]). Briefly, quasisymmetric maps of the boundary circle extend to quasiconformal maps of the unit disk, and vice versa, with the distortion constants (with K and M as above) being close to 1 if one of them is.…”

On the extremality of quasiconformal mappings and quasiconformal deformations

“…The deformation provides a way of generating a family of quasiconformal mappings, and is suitable for obtaining some distortion theorems (see Reich [6]). Now for a homemorphism h of the unit circle ∂∆ onto itself, denote by QC(h) the class of all quasiconformal mappings of the unit disk ∆ = {z : |z| < 1} with boundary values h. Then QC(h) is non-empty if and only if h is quasisymmetric in the sense of Beurling-Ahlfors [2]. A quasisymmetric function h then determines the extremal maximal dilatation K h , defined as…”