Is there a classification of branched coverings of the closed unit disk $\mathbb{D} =\{z\in \mathbb{C} \ | \ |z| \leq 1 \}$? Here we consider only branched covering projections which restrict to unbranched covering projections on the boundary. I am especially interested in the following question:
Which Riemann surfaces $X$ can/cannot be obtained as branched coverings of $\mathbb{D}$?
What I can prove:
- since $\mathbb{D}$ is not closed, no closed $X$ can be realized as a branched covering of it
- some computations with Riemann-Hurwitz formula show that if there is only one branching point $p \in \mathbb{D}$ then $X$ must be homeomorphic to a disk.
- for a covering $f:X\to \mathbb{D}$ of degree $N\geq 2$ with $r$ branching points on $\mathbb{D}$ we have $$ \chi (X) = (1-r)N + \sum_{i=1}^r s_i $$ where $s_i$ is the cardinality of the fiber over the $i$-th branching point. But this formula does not give any restriction, since it allows for any value of $\chi$(X).
Thanks in advance for any reference/suggestion.