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.

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