The Levi Problem in $\mathbb C^n$: A Survey

TL;DR

This survey explores the Levi problem in complex n-dimensional space, examining domains of holomorphy, pseudoconvexity, and providing a comprehensive solution within this context.

Contribution

It offers a mostly self-contained presentation of the Levi problem specifically for domains in ^n, clarifying the relationship between pseudoconvexity and holomorphic functions.

Findings

Clarified the equivalence of pseudoconvexity and domains of holomorphy in ^n

Presented a comprehensive, self-contained solution to the Levi problem

Drawn parallels between complex and geometric convexity notions

Abstract

We discuss domains of holomorphy and several notions of pseudoconvexity (drawing parallels with the corresponding notions from geometric convexity), and present a mostly self-contained solution to the Levi problem. We restrict our attention to domains of $\mathbb C^n$.