TITLE: Complex manifolds where bounded holomorphic functions are constant
QUESTION [9 upvotes]: 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 ?

REPLY [2 votes]: The reference for complex manifolds in dimention 1 (Riemann surfaces) is the 
subject known under the title "Classification theory of (open) Riemann surfaces".
It used to be very popular in the 1950-s. Some books are:
M. Tsuji, Potential theory in modern function theory,
L. Ahlfors and Sario, Riemann surfaces.
Look also at the nice paper by P. Doyle, On deciding whether a surface is parabolic or
hyperbolic.