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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

Does entire function extend to the holomorphic function of Riemann sphere?

Nomenclature in complex analysis

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?

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Yes, your reasonning is correct. More generally, if $f : U \to V$ is a holomorphic map defined on a planar domain $U=\mathbb C-P$, where $P$ is a compact with non-empty interior and $V$ is a proper subdomain of $\mathbb C$, then Casatori-Weierstrass tells you that $f$ has a pole or a removable singularity at infinity, and so $f$ extends to a meromorphic function on $U \cup \{\infty\}$. There is no need for injectivity.

Also, note that here $\infty$ is not a special point : the same reasonning works for any isolated point in the boundary of $U$, by composing with a Möbius transformation.

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