Bounded holomorphic functions on which domains are only constants Let $\Omega\subset{\mathbb{C}}^n$ be an open connected set. Let $H^\infty(\Omega)$ denote all bounded holomorphic functions on $\Omega$. 
Can you tell for what choices of $\Omega$ does $H^\infty(\Omega)$ consists of only constant functions?
I know that when $\Omega=\mathbb{C}$, then the above holds by the Lioville’s theorem. But can you tell any other domains, preferably in $\mathbb{C}$ and bounded but not equal to $\mathbb{C}$?
 A: In the plane this is the case if and only if the analytic capacity of the complement $K = \mathbb{C} \setminus \Omega$ is zero. For some simple examples choose $K$ as a finite (or, more generally, a countably infinite discrete set), so that every singularity of a bounded holomorphic function is removable by Riemann's removable singularities theorem. The extension is then a bounded entire, hence constant function. The general question which sets have zero analytic capacity is more complicated, see https://en.wikipedia.org/wiki/Analytic_capacity
A: I don't have a full answer to your question, but I think this is a good lead in the case $n=1$. As always, the difficulties come with the unit disc :)
For a start, if $\Omega$ is simply connected, it it isomorphic to the unit disc, which has many automorphisms, so $H^\infty(\Omega)$ has many non constant functions.
Then, if $\Omega$ is not simply connected, its universal cover can only be $\mathbb{C}$ or the unit disc $\Delta$, since the Riemann sphere is compact and $\Omega$ is not (see uniformization theorem).
If the cover is $\mathbb{C}$, then any bounded function on $\Omega$ will lift to a bounded function of $\mathbb{C}$, showing that $H^\infty(\Omega)=\mathbb{C}$.
So, we are left with the case where the open set $\Omega$ is a quotient of $\Delta$ by some subgroup $\Gamma \simeq \pi_1(\Omega)$ of $PSL_2(\mathbb{R})$, and we are wondering, if there exist non constant $\Gamma$-invariant bounded functions on $\Delta$.
We know that $\Gamma$ must be a free group, and a discrete subgroup of $PSL_2$.
I lack some knowledge on $PSL_2$ and complex analysis to finish this argument, but this is a well studied subject, so I am hoping that we can conclude this argument with some help. Maybe there is a distinction to be made, depending on wether $\Gamma$ is finitely generated or not ?
I hope this helps a bit.
