Recently I faced a problem, which I realized has a close connection with the following problem. $\{ f_{n} \}_{n=1}^{\infty}$ is analytic map from $C^{n}$ to $C^{n} $\ $U$ where $U$ is open neighborhood of 0 and $f$ is a normal family.
I know when n=1, this is really the Montel Normal family criterion. However, I did not know whether it is true for higher dimensions. Also, I heard that for any two topological equivalent simple connected domains in $C^{n}$ $(n\geq 2)$, the probability for these two domains to be holomorphically equivalent is 0. I want to know what is the precise statement for this theorem.
Any advice and comments will be appreciated.