Some applications of variation of Loewner chains in several complex variables

In a recent paper, Bracci, Graham, Hamada, Kohr developed a new method to construct Loewner chains, by considering variations of certain Loewner chains. We use this method first, to prove a topological property of the class S0 of mappings with parametric representation on the Euclidean unit ball Bn, which for n≥2 immediately implies the density of the automorphisms of Cn that, restricted to Bn, have parametric representation, and second, to prove that every normalized univalent mapping on Bn whose image is Runge and which is C1 up to the boundary embeds into a normalized Loewner chain with range Cn.