Analytic embedding of complex plane in Riemann surfaces with finite complement

Question

$\mathbb{C}$ can be analytically embedded in the Riemann sphere such that the complement of the image of this embedding is only one point. Are there any other Riemann surfaces such that we can embed $\mathbb{C}$ in it such that the complement is finite?

Answer 1

Let $X$ be a connected Riemann surface that has an open subset isomorphic ( as a $1$-dim complex manifold ) to $\mathbb{C}$. Since $\mathbb{C}$ is simply connected, the simply connected cover $\bar X$ of $X$ will contain one or several copies of $\mathbb{C}$ ( as many as the multiplicity of the cover $\bar X \to X$). But now, Riemann classification theorem says that $\bar X$ is isomorphic to $\mathbb{C}$, $S^2$ (Riemann's sphere), or $D$ ( the unit disk). It cannot be the unit disk, since we cannot have an injective holomorphic map $\mathbb{C}\to D$. Therefore $\bar X$ is either $\mathbb{C}$ or $S^2$. Now use Picard's theorem to conclude that the image of $\mathbb{C}$ in $\bar X$ is the whole $\bar X$ or $\bar X$ without a point.

Conclusion: if $X$ is a connected Riemann surface containing an open subset $U$ iso to $\mathbb{C}$ then $U = X$, or $U= X\backslash\{pt\}$, and $X$ iso to $S^2$. Moreover, the imbeddings of $\mathbb{C}$ into $\mathbb{C}$ are of the form $z\mapsto a z + b, a\ne 0$, while the imbeddings of $\mathbb{C}$ into $S^2$ are of the form $z \mapsto \frac{a z + b}{c z + d}$.