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 ?
Best Answer
We have that there are no non-constant bounded functions on $\mathbb C^*=\mathbb C\setminus\{0\}$. The easiest way to see that is to notice that such a function has a removable singularity at the origin and hence comes from a bounded function on $\mathbb C$ (which incidentally has a removable singularity at $\infty$ and hence extends to the Riemann sphere and therefore is constant).
As for the higher-dimensional problem it is no doubt hopeless to get anything like a complete description: There are as you say domains in $\mathbb C^n$ (and there are many of those), the compact manifolds but also compact manifolds minus a codimension $2$ closed analytic subspace, the blowing up of some space that has the property, any product of two manifolds with the property and so on. Complex manifolds in higher dimension are simply too varied.
Finally, there are non-compact manifolds with only constant holomorphic functions. If $L\rightarrow X$ is an analytic line bundle over a complex manifold $X$ and $f\colon L\rightarrow\mathbb C$ is a holomorphic function, then we may Taylorexpand $f$ along the zero section of $L$: First we just look at the restriction of $f$ to the zero section which gives a function on $X$. Then we may take any local section of $L$, think of that as a tangent vector at the zero section and take the derivative of $f$ along this tangent vector. This glues together to give the first derivative as a global section of $L^{-1}$ and similarly the $n$'th derivative of $f$ along the zero section will be a section of $L^{-n}$. If $X$ is compact and $L^{-n}$ for $n>0$ only has the zero section as global section, then $f$ is constant along the zero section and all its higher derivatives along it are zero so that $f$ is constant in a neighbourhood of the zero section and hence constant. There are lots of such examples $(X,L)$.