Question

Recently I face up with a problem, which I realized that have 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, whether f is a normal family.

I know when n=1, this is really Montel Normal family criterion, However I did know whether it is true for high dimension. also I heard that the for any two topological equivalent simple connected domain in$ C^{n}$ $(n\geq 2)$, the probability of holomorphic equivalent for this two domain is 0. I want to know what is the precise statement for this theorem.

Any advice and comments will be appreciated.

Answer 1

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."