# exhaustion function

###### Definition.

Let $G\subset{\mathbb{C}}^{n}$ be a domain and let $f\colon G\to{\mathbb{R}}$ is called an exhaustion function whenever

 $\{z\in G\mid f(z)

is relatively compact in $G$ for all $r\in{\mathbb{R}}$.

For example $G$ is pseudoconvex if and only if $G$ has a continuous plurisubharmonic exhaustion function.

We can also define a bounded version.

###### Definition.

Let $G\subset{\mathbb{C}}^{n}$ be a domain and let $f\colon G\to(-\infty,c]$ for some $c\in{\mathbb{R}}$, is called a bounded exhaustion function whenever

 $\{z\in G\mid f(z)

is relatively compact in $G$ for all $r.

A domain which has a bounded plurisubharmonic exhaustion function is usually referred to as a hyperconvex domain. Note that not all pseudoconvex domains have a bounded plurisubharmonic exhaustion function. For example the Hartogs’s triangle does not, though it does have an unbounded one.

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title exhaustion function ExhaustionFunction 2013-03-22 14:32:41 2013-03-22 14:32:41 jirka (4157) jirka (4157) 5 jirka (4157) Definition msc 32U10 msc 32T35 Pseudoconvex bounded exhaustion function hyperconvex