Boundary values versus dilatations of harmonic mappings

Abstract

This article is divided into two parts. In the first part, we consider univalent harmonic mappings from the unit diskU onto a Jordan domain Ω whose dilatation functions$a = \bar f_{\bar z} /f_z$ have modulus one on an interval of the unit circle. The boundary values off depend very strongly on the values ofa(e ^it). A complete characterization of the inverse imagef ^-1 (q) of a pointq on ∂Ω is given. We then consider the case where the dilatation functiona(z) is a finite Blaschke product of degreeN. It is shown that in this case, Ω can have at mostN+2 points of convexity. Finally, we give a complete characterization of simply connected Jordan domains Ω with the property that there exists a nonparametric minimal surface over Ω such that the image of its Gaussian map is the upper half-sphere covered exactly once.