On the uniqueness problem of harmonic quasiconformal mappings

On Harmonic Quasiconformal Self-mappings of the Unit Ball

It is proved that any family of harmonic K-quasiconformal mappings {u = P [f ], u(0) = 0} of the unit ball onto itself is a uniformly Lipschitz family providing that f ∈ C. Moreover, the Lipschitz constant tends to 1 as K → 1.

Harmonic Quasiconformal Mappings of Riemannian Manifolds

Quasiconformal Extension of Harmonic Mappings in the Plane

Let f be a sense-preserving harmonic mapping in the unit disk. We give a sufficient condition in terms of the pre-Schwarzian derivative of f to ensure that it can be extended to a quasiconformal map in the complex plane. Introduction A well-known criterion due to Becker [5] states that if a locally univalent analytic function φ in the unit disk D satisfies (1) sup z∈D ∣∣∣∣φ′′(z) φ′(z) ∣∣∣∣ (1− ...

On the Maximal Dilation of Quasiconformal Mappings

1. Let G and G' be two plane open sets and w(z) a topological map.ping of G onto G'. By Q we denote any quadrilateral in G, i.e. the topological image of a closed square with a distinguished pair of opposite sides. The conformal modulus m of Q is the ratio m=a/b of the sides of a conformally equivalent rectangle 22, the distinguished sides of Q corresponding to the sides of length b. We call th...

Mappings with Convex Potentials and the Quasiconformal Jacobian Problem

This paper concerns convex functions that arise as potentials of quasiconformal mappings. Several equivalent definitions for such functions are given. We use them to construct quasiconformal mappings whose Jacobian determinants are singular on a prescribed set of Hausdorff dimension less than 1.