# solution of the Levi problem

The Levi problem is the problem of characterizing domains of
holomorphy by a local condition on the boundary that does not involve
holomorphic functions^{} themselves. This condition turned out to
be pseudoconvexity.

###### Theorem.

An open set $G\mathrm{\subset}{\mathrm{C}}^{n}$ is a domain of holomorphy if and only if $G$ is pseudoconvex.

The forward direction (domain of holomorphy implies pseudoconvexity) is not hard to prove and was known for a long time. The opposite direction is really what’s called the solution to the Levi problem.

## References

- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | solution of the Levi problem |
---|---|

Canonical name | SolutionOfTheLeviProblem |

Date of creation | 2013-03-22 14:31:11 |

Last modified on | 2013-03-22 14:31:11 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 32T05 |

Classification | msc 32E40 |

Related topic | Pseudoconvex |

Related topic | DomainOfHolomorphy |

Defines | Levi problem |