# Conformal/Biholomorphism equivalence classes in $\mathbb{C}^n$

Recently I have got interested in the topic of conformal equivalence classes of complex domains, mostly one-dimensional ones.

Here by conformal map $f: U \rightarrow V$ I mean a complex holomorphic bijection which is onto (where $U,V$ are open domains in $\mathbb{C}^n$). We call $U,V$ conformally equivallent if such a map exists; this is an equivalence relation.

My knowledge in this subject can be summarazied as follows:

• By Riemann's mapping theorem, Every simply connected domain is in the class of the unit disc $D=\{|z|<1\}$.

• Two annuli $\{r_1<|z|<r_2\},\{R_1<|z|<R_2\}$ are equivalent iff $\frac{R_2}{R_1}=\frac{r_2}{r_1}$. Moreover, different infinite annuli are not equivalent.

• Every one-dim. domain with single hole (=of genus 1) is equivalent to an annulus.

My general question is - What can be said about this classification? How many classes are there? Is there a full answer? Also, what if further restrictions are taken on the domains, such as connectivity (is this necessary in the general case)?

Another question is - what happens when $n>1$? I have read that the theory is poorer, but could not quite understand why. If so, can the notion of conformal equivalence be replaced by another (complex-analysis related) concept, with more theory (maybe some local conditions)?

Can someone give a brief summary, with relevant references (books/articles)?

As Mariono says in the comments, Liouville's theorem severely limits how interesting conformal equivalence is in more than $2$ real dimensions.