# 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 $G\subset{\mathbb{C}}^{n}$ 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 SolutionOfTheLeviProblem 2013-03-22 14:31:11 2013-03-22 14:31:11 jirka (4157) jirka (4157) 6 jirka (4157) Theorem msc 32T05 msc 32E40 Pseudoconvex DomainOfHolomorphy Levi problem