Why do holomorphic functions on Stein manifolds "essentially behave like" holomorphic functions open subsets of $\Bbb C^n$?

Question

I gathered from various books and conversations that the philosophy is that holomorphic functions on open subsets of Stein manifolds "essentially behave like" holomorphic functions on open subsets of $\mathbb{C}^n$ and that this is due to the embedding theorem by Remmert & others (which states that Stein manifolds can be proberly embedded by a holomorphic function into some $\mathbb{C}^n$).

However, I'm not able to grasp the concept. At first I thought it'd be easy, since if $K$ is a Stein manifold and $\Phi: K \rightarrow \mathbb{C}^n $ is a prober holomorphic embedding, $\Omega \subseteq K$ open and $f: \Omega \rightarrow X$ ($X$ a Banach space) is a holomorphic function, you can consider $f \circ \Phi^{-1}: \Phi(\Omega) \rightarrow X$ which is holomorpic in a more classical sense. Then, I noticed that $\Phi(\Omega) \subseteq \mathbb{C}^n$ is (at least in general) not open and only $\Phi(\Omega) \subseteq \Phi(K)$ is open. So the function $f \circ \Phi^{-1}$ is also only holomorphic as a function on the open subset $\Phi(\Omega)$ of the submanifold $\Phi(K) \subseteq \mathbb{C}^n$.

And thus, if I want to understand why $f$ behaves essentially like a holomorphic function on an open subset of $\mathbb{C}^n$ by looking at $f \circ \Phi^{-1}$, I have to understand why holomorphic functions on open subsets of submanifolds of $\mathbb{C}^n$ "behave essentially like" functions on open subsets of $\mathbb{C}^n$. Is that true and if yes why?

Or else, is my whole approach wrong? Or did I misinterpet the philosophy I sketched above?

Answer 1

Suppose $X$ is a complex manifold of complex dimension $n$ and let $ \mathcal{O}_X $ denote the ring of holomorphic functions on $X$. Recall that we say $X$ is a Stein manifold if the following conditions hold:

1. $X$ is is holomorphically convex, i.e. for every compact subset $K\subseteq X$, the so-called holomorphically convex hull, ${\bar {K}}=\left\{z\in X\,\left|\,|f(z)|\leq \sup _{w\in K}|f(w)|\ \forall f\in {\mathcal {O}}(X)\right.\right\}$ is also a compact subset of $X$.
2. $X$ is holomorphically separable, i.e. if $x\neq y$ are two points in X, then there exists $f(x)\in\mathcal{O}(X)$ $f(x)\neq f(y)$.

From the definition we can see that the Stein manifold is the natural generalization of holomorphically convex domain in $\mathbb{C}^n$.

So the philosophy you say is not right since not every open set of $\mathbb{C}^n$ is holomorphically convex domain.