solution of the Levi problem

The Levi problem is the problem of characterizing domains ofholomorphy by a local condition on the boundary that does not involveholomorphic functions themselves. This condition turned out tobe pseudoconvexity.

Theorem.

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

The forward direction (domain of holomorphy implies pseudoconvexity) isnot hard to prove and was known for a long time. The opposite directionis really what’s called the solution to the Levi problem.

References

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

单词	SolutionOfTheLeviProblem
释义	solution of the Levi problem The Levi problem is the problem of characterizing domains ofholomorphy by a local condition on the boundary that does not involveholomorphic functions themselves. This condition turned out tobe pseudoconvexity. Theorem. An open set $G\\subset{\\mathbb{C}}^{n}$ is a domain of holomorphy ifand only if $G$ is pseudoconvex. The forward direction (domain of holomorphy implies pseudoconvexity) isnot hard to prove and was known for a long time. The opposite directionis really what’s called the solution to the Levi problem. References 1 Steven G. Krantz.,AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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