## Loewner chains and parametric representation in several complex variables.(English)Zbl 1029.32004

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.

### MSC:

 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 32A17 Special families of functions of several complex variables 32A19 Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.)
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