Plurisubharmonic exhaustion function

Remarks

Example: on $\C^n$, $\varphi(z) = \abs{z}^2$ is a smooth strictly plurisubharmonic exhaustion function. Its Levi form is $\abs{w}^2$, and its sublevel sets are open balls. Non-example: on the unit ball $B \subseteq \C^n$, $\varphi(z) = \abs{z}^2$ is still strictly plurisubharmonic but is not an exhaustion, since $\set{\varphi < 2} = B$ is not relatively compact in $B$. On $B$, the function $-\log(1 - \abs{z}^2)$ is a smooth strictly plurisubharmonic exhaustion function. For a domain $\Omega \subsetneq \C^n$, an equivalent form of clause 2 for continuous $\varphi$ is that $\varphi(z) \to +\infty$ as $z$ tends to $\partial\Omega$ or, when $\Omega$ is unbounded, as $\abs{z} \to \infty$.