Unit ball in C^n

Remarks

Its boundary is the unit sphere $S^{2n-1} = \set{z : \abs{z} = 1}$. It is a Reinhardt domain: if $z \in \mathbb{B}^n$ and $\theta_j \in \R$, then $(e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n)$ has the same norm, so it lies in $\mathbb{B}^n$. Every unitary map $U$ of $\C^n$ preserves the norm, so it maps $\mathbb{B}^n$ onto itself. For $n = 1$ it is the unit disc, and for $n \ge 2$ it is not biholomorphic to the unit polydisc, by Poincaré's theorem.