Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains

The conformal mapping $$f(z)=(z+1)^2 $$f(z)=(z+1)2 from $${\mathbb {D}}$$D onto the standard cardioid has a homeomorphic extension of finite distortion to entire $${\mathbb {R}}^2 .$$R2. We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from $${\mathbb {D}}$$D onto cardioid-type domains.

Prime ends in theory of mappings with finite distortion in the plane

In the present paper, it is studied the boundary behavior of the so-called lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec, also to solutions of the Beltrami equations, as well as to finitely bi-Lipschitz mappings that a far-reaching extension of the known classes of isometric and quasiisometric mappings.

On Homeomorphisms between Extension Spaces

In this note, conditions are obtained which will ensure that two topological spaces are homeomorphic when they have homeomorphic extension spaces of a certain kind. To discuss this topic in suitably general terms, an unspecified extension procedure, assumed to be applicable to some class of topological spaces, is considered first, and it is shown that simple conditions imposed on the extension procedure and its domain of operation easily lead to a condition of the desired kind. After the general result has been established it is shown to be applicable to a number of particular extensions, such as the Stone-Čech compactification and the Hewitt Q-extension of a completely regular Hausdorff space, Katětov's maximal Hausdorff-closed extension of a Hausdorff space, the maximal zero-dimensional compactification of a zero-dimensional space, the maximal Hausdorff-minimal extension of a semi-regular space, and Freudenthal's compactification of a rim-compact space. The case of the Hewitt Q-extension was first discussed by Heider (6).