A Survey On Loewner Chains And Related Problems For Bounded Balanced Pseudoconvex Domains In C-N

Let Omega subset of C-n be a bounded balanced pseudoconvex domain with C-1 plurisubharmonic defining functions. In this paper, we study properties of the Caratheodory family M(Omega), and prove that it is compact with respect to the topology of locally uniform convergence on H(Omega). Then we survey some results in the theory of Loewner chains on Omega, which are extensions to bounded balanced pseudoconvex domains of corresponding results on the Euclidean unit ball B-n in C-n. A special interest in our discussion is played by the family S-0(Omega) of normalized univalent mappings which have parametric representation on Omega. Certain questions and conjectures related to the family S-0(Omega) are also considered.