Complex structures on punctured disks.

Question

Let $X$ be a smooth surface diffeomorphic to the punctured unit disk $\{(x,y)\in \mathbb{R}^2 \ | \ 0<x^2+y^2<1\}$ in the plane. It admits a lot of non equivalent complex structures, for example those induced by the diffeomorphisms with

• punctured unit disk $\mathbb{D}^*=\{z \in \mathbb{C} \ | \ 0< |z|<1\}$
• punctured unit plane $\mathbb{C}^*=\{z \in \mathbb{C} \ | \ z \neq 0 \}$
• open annuli $A(1,R)=\{ z \in \mathbb{C} \ | \ 1<|z|<R\}$

These structures are pairwise non-biholomorphic. Is this list exhaustive or am I missing some class of complex structures?

Answer 1

The following is (an adaptation of) theorem 10 of section 6.5 in Ahlfors "Complex Analysis".

Suppose $\Omega \subset \Bbb C$ is an $n$-connected domain for which none of the connected components of $S^2 \setminus \Omega$ are a single point. Then for some real numbers $\lambda_1, ..., \lambda_{n-1}$, there is a one-to-one conformal mapping of $\Omega$ onto the annulus $1 < |w| < e^{\lambda_1}$ minus $n-2$ concetric arcs situated on the circles $|w| = e^{\lambda_i}$.

Reducing to the 2-connected case, this gives the theorem you want.