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

Question

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)?

Answer 1

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

In several complex variables, we no longer have a Riemann mapping theorem. Even for domains homeomorphic to an open ball, there are uncountably many equivalence classes of biholomorphic domains. So function theory in several complex variables is highly domain dependent, often depending in subtle ways on the geometry of the boundary.

Probably the most important open problem in several complex variables is whether every biholomorphism between smoothly bounded domains must extend smoothly to the boundary. There are no known counterexamples, and the result has been established for many different classes of domains. This would allow us, in principle, to use differential-geometric invariants of the boundary to study the biholomorphism problem.