Liouville's theorem states that all bounded holomorphic functions on $\mathbb{C}^n$ are constant.

I'm wondering which connected complex manifolds have this property ?

Connected compact complex manifolds have it since all holomorphic functions there are constant.

There are no simply connected proper open subsets of $\mathbb{C}$ satisfying this because of the Riemann Mapping Theorem. But are there some other open subsets satisfying it ?

In higher dimension there are some, such as the Fatou–Bieberbach domains (open subsets of $\mathbb{C}^n$ biholomorphic to it).

I would be interested in references on this property on complex manifolds.

Incidentally, is it true that any complex manifold where all holomorphic functions are bounded is compact ?