Loewner chains and parametric representation in several complex variables

Let B be the unit ball of Cn with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f (z, t) and the transition mapping v(z, s, t) associated to f (z, t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z ∈ B. Moreover, we prove that a mapping f ∈ H(B) has parametric representation if and only if there exists a Loewner chain f (z, t) such that the family {e−tf (z, t)}t 0 is a normal family on B and f (z) = f (z, 0) for z ∈ B. Also we show that univalent solutions f (z, t) of the generalized Loewner differential equation in higher dimensions are unique when {e−tf (z, t)}t 0 is a normal family on B. Finally we show that the set S0(B) of mappings which have parametric representation on B is compact.  2003 Elsevier Science (USA). All rights reserved.