# exhaustion function

###### Definition.

Let $G\subset {\u2102}^{n}$ be a domain and let $f:G\to \mathbb{R}$ is called an exhaustion function whenever

$$ |

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 $G\subset {\u2102}^{n}$ be a domain and let $f:G\to (-\mathrm{\infty},c]$ for some $c\in \mathbb{R}$, is called a bounded exhaustion function whenever

$$ |

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

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 |