Contributed Talk: Extremal results, approximation properties, and related problems associated with the Loewner differential equation in Cn
In the first part of this talk we survey various results related to Loewner chains, the generalized Loewner differential equation, and Herglotz vector fields on the Euclidean unit ball Bn in Cn. Extremal problems and related results for the family S0A(Bn) of univalent mappings with A-parametric representation on Bn will be also discussed, where A∈L(Cn) such that k+(A)<2m(A). Here k+(A) is the Lyapunov index of the operator A and m(A)=min‖z‖=1ℜ⟨A(z),z⟩. Next, we present recent results on approximation properties of various families of normalized univalent mappings f on Bn, with Runge image, by automorphisms of Cn and smooth quasiconformal diffeomorphisms of Cn onto itself, whose restrictions to Bn have the same geometric property as the initial mappings f. Open problems and conjectures will be also considered. \bigskip Joint work with Ian Graham (Toronto), Hidetaka Hamada (Fukuoka), Gabriela Kohr (Cluj-Napoca), and Mihai Iancu (Cluj-Napoca).