- Classify all simply connected regions in the extended complex plane up to biholomorphic equivalence. i.e, give a list of simply connected region, prove that every simply connected region in the extended complex plane is biholomorphic equivalent to a member in your list. Prove also that no two members in your list are biholomorphic equivalent.
Solution
Up to biholomorphic equivalence, the complete list is \[ \boxed{\widehat{\mathbb C},\qquad \mathbb C,\qquad \Delta.} \]
1. Every simply connected region in \(\widehat{\mathbb C}\) is equivalent to one of these three.
Let \(U\subset\widehat{\mathbb C}\) be a nonempty simply connected region.
If \(U=\widehat{\mathbb C}\), it is already the first model.
Suppose next that the complement consists of exactly one point \(p\). A Möbius transformation sending \(p\) to \(\infty\) maps \(U\) biholomorphically onto \(\widehat{\mathbb C}\setminus\{\infty\}=\mathbb C\).
Finally suppose that \(\widehat{\mathbb C}\setminus U\) contains at least two points. Choose \(p\) in the complement and a Möbius transformation \(M\) with \(M(p)=\infty\). Then \[ V=M(U)\subset\mathbb C \] is a simply connected plane domain. It is proper: a second point of the complement is carried to a finite point outside \(V\). By the Riemann mapping theorem, every nonempty proper simply connected plane domain is biholomorphic to the unit disk \(\Delta\). Hence \(U\) is biholomorphic to \(\Delta\).
These three cases exhaust the possible complements.
2. No two models in the list are biholomorphic.
The Riemann sphere is compact. A biholomorphism is in particular a homeomorphism, so it preserves compactness. Neither \(\mathbb C\) nor \(\Delta\) is compact; therefore the sphere is biholomorphic to neither of them.
It remains to distinguish \(\mathbb C\) and \(\Delta\). If there were a biholomorphism \(F:\mathbb C\to\Delta\), then \(F\) would be a bounded entire function. Liouville’s theorem would force \(F\) to be constant, contradicting bijectivity. Thus \(\mathbb C\) and \(\Delta\) are not biholomorphic.
Consequently the three displayed regions form a complete and irredundant classification.