Loewner chains and parametric representation in several complex variables (Q1399330)

Let \(B\) be the unit ball of \(\mathbb{C}^n\) with respect to an arbitrary norm. The authors study certain properties of Loewner chains and their transition mappings on the unit ball \(B\). They 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 \(z\in B\). Moreover, they prove that a mapping \(f\in H(B)\) has parametric representation if and only if there exists a Loewner chain \(f(z,t)\) such that the family \(\{e^{-t}f(z,t)\}_{t\geq 0}\) is a normal family on \(B\) and \(f(z)=f(z,0)\). The authors also show that univalent solutions \(f(z,t)\) of the generalized Loewner differential equation in higher dimensions are unique when \(\{e^{-t}f(z,t)\}_{t\geq 0}\) is a normal family in \(B\). Finally, they show that the set of mappings which have parametric representation on \(B\) is compact.