Extremal quasiconformal polygon mappings for arbitrary subdomains of compact Riemann surfaces.
We study the extreme and exposed points of the convex set consisting of representing measures of the disk algebra, supported in the closed unit disk. A boundary point of this set is shown to be extreme (and even exposed) if its support inside the open unit disk consists of two points that do not lie on the same radius of the disk. If its support inside the unit disk consists of 3 or more points, it is very seldom an extreme point. We also give a necessary condition for extreme points to be exposed...
The aim of the paper is to discuss the extreme points of subordination and weak subordination families of harmonic mappings. Several necessary conditions and sufficient conditions for harmonic mappings to be extreme points of the corresponding families are established.