Removable singularity theorems for manifolds

Question

All the theorems on removable singularities are for functions defined on open domains $\Omega \in \mathbb{C}$. But what are the corresponding theorems for functions defined on Riemann surfaces? How do they differ and are there any extra issues we need to take into account?

Answer 1

As is pointed out in the comments, for Riemann Surfaces, the following version of Riemann's Removable Singularity Theorem holds:

Theorem:Let $X$ be a Riemann surface, and $f:X-\{p\}\to \mathbb{C}$ a holomorphic function. If $f$ is bounded in some neighborhood of $p$, then $f$ extends uniquely to a holomorphic function $f:X\to\mathbb{C}$.

This is proven, as the comment suggests, by working some some coordiante disc around $p$ and applying the usual Riemann's Theorem.

A generalization of this to higher-dimensional manifolds has several possible forms. The most general that I know of is Hartog's Extension Theorem. This is a much deeper result, and has many implications. It's algebraic analogue is every useful in algebraic geometry as well.