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.



Normal Family for several complex variable (from $C^n$ to $C^n$ \ $U$)
The key phrase to keep in mind is "taut manifold": 
 a complex manifold X which is taut has the property that Hol(Y,X) is normal for every complex
manifold Y. A fact that may be of interest to you is that a taut domain in $\mathbb{C}^n$ is necessarily pseudoconvex.


More information, including criteria of tautness and the relation with the notion of hyperbolicity can be found in the following books: by Marco Abate,


<http://www.dm.unipi.it/~abate/libri/libriric/files/IterationThTautMan2-1.pdf>


and by Shoshichi Kobayashi,


Hyperbolic Complex Spaces, Springer, 1998


As for lack of biholomorphic equivalence for domains in $\mathbb{C}^n$, I am not sure if your "probabilistic" statement can be put on rigorous footing. It is true that an euclidean ball and a polydisk in $\mathbb{C}^n$ are not biholomorphically equivalent when $n \geq 2$. The result goes back to Poincare and has several proofs. You can learn more from the text by S. Krantz in several comlex variables; there is also a good sketch of proof among the exercises in the text by Grauert and Fritzsche, "From holomorphic functions to complex manifolds." 


