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)<r\}

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)<r\}

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

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
Canonical name	ExhaustionFunction
Date of creation	2013-03-22 14:32:41
Last modified on	2013-03-22 14:32:41
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Definition
Classification	msc 32U10
Classification	msc 32T35
Related topic	Pseudoconvex
Defines	bounded exhaustion function
Defines	hyperconvex