Harmonic Mappings And Juhn Surfaces

The concept of John domain was introduced by F.John when he studied the elastic theory in plane. John domain is an important object of study in complex dynamic system, approximation theory and elastic theory because of the close relationship in John domain and various measurements of the region. According to the image of the unit disk under the canonical lift of a harmonic mapping, Willy proposed the concept of John surface in 2013. Because harmonic mappings and John surfaces have much to do with the geometries of 3-manifolds, the study of the Schwarzian derivative of a planar harmonic mapping and John surfaces associated with the Weierstrass-Enneper lift of a class of harmonic mappings are important subjects of geometric function theory.There are three parts in this article.The first part is the preface. In this part, we introduce the concept of John domain as well the development and the research situation of harmonic mapping in the plane and its Schwarzian derivative. The main results of this article are briefly introduced in this chapter.In part 2, on the basis of Pokornyi’s univalence criteria, we prove the image of the unit disk under the canonical lift of a harmonic mapping to be a John surface when the Schwarzian derivative and the conformal factor of the harmonic mapping satisfy certain condition by using a comparison theorem of second order differential equation.In part 3, we introduce a definition of the conformal factor of a unanalytic harmonic mapping in the unit disc, and discuss the univalence property of lift of the harmonic mapping, and also obtain the two-point distortion theorem for the harmonic mapping.