Let $\Omega \subset \mathbb C^n$ be a bounded domain which is biholomorphic to the unit ball $B^n=\{|z|<1 \mid z\in \mathbb C^n\}$. Can we show $\Omega$ must be a Runge domain? By definition, $\Omega$ is a Runge domain if any analytic function in $\Omega$ can be approximated by polynomials.


Notice that J. Wermer gave a [counterexample](https://link.springer.com/content/pdf/10.1007/BF01420524.pdf) if $\Omega$ is biholomorphic to a bidisk.



Is a domain biholomorphic to the unit ball a Runge domain?
The answer to your question is NO. Every domain in $\mathbb{C}^n$ can be embedded into $\mathbb{C}^n$ such that its image is non-Runge. 


See:


Wold, E. F.: A Fatou–Bieberbach domain in $\mathbb{C}^2$ which is not Runge.\* Math. Ann. 340 (2008) 775–780 


