Let $D$ be an unbounded pseudoconvex domain in $\mathbb{C}^2$. I would like to study the peak set of $D$.


1)Can the peak set of $D$ be empty? Or


2) Does $D$ always admit a nonconstant bounded holomorphic function?


Any reference or idea or example will be appreciated. Thanks.


Here is an example.


Let $f\in O(\mathbb{C})$ be holomorphic. Set $D=\{(z,w)\in\mathbb{C}^2, |w|< e^{-\Re(f(z))}\}$. It is well known that $D$ is unbounded and pseudoconvex. Set $F(z,w)=we^{f(z)}$ then $|F|<1$ and is holomorphic in $D$.



Bounded holomorphic functions in unbounded domain
See MR0549981 (81b:32005) 
Diederich, Klas; Sibony, Nessim
Strange complex structures on Euclidean space.
J. Reine Angew. Math. 311/312 (1979), 397–407. 


In this paper, the authors construct various examples of Stein (hence pseudoconvex) open subsets of $\mathbb{C}^2$ diffeomorphic to $\mathbb{R}^4$ and not isomorphic with $\mathbb{C}^2$. One such example $\Omega\_3$ has the property that there is no nonconstant bounded holomorphic function on $\Omega\_3$.


