There are a number of nice results about extending holomorphic and meromorphic functions from the complex plane $\mathbb C$ to the Riemann sphere $S^2$. See for instance
How can I adapt these (or are there other results?) for when I'm working with functions defined on proper subsets of $\mathbb C$? In these cases we lose Liouville's theorem and maybe compactness.
For example let $A \subset \mathbb C$ be the region $\mathbb C$ minus two disjoint closed disks. Suppose $f:A\rightarrow A$ is analytic and bijective. Then $f$ does not have an essential singularity at $\infty$ by injectivity and Great Picard or Cassorati-Weierstrass. Viewing this region of a subset of $S^2$, it is conformally equivalent to an annulus. Does it follow that $f$ can be "extended" to a holomorphic map from this annulus to itself, since it has a pole or removable singularity at infinity?