Page "Stein manifold" Paragraph 20

* Being a Stein manifold is equivalent to being a ( complex ) strongly pseudoconvex manifold.

The latter means that it has a strongly pseudoconvex ( or plurisubharmonic ) exhaustive function, i. e. a smooth real function on ( which can be assumed to be a Morse function ) with, such that the subsets are compact in for every real number.

This is a solution to the so-called Levi problem, named after E. E. Levi ( 1911 ).

The function invites a generalization of Stein manifold to the idea of a corresponding class of compact complex manifolds with boundary called Stein domains.

A Stein domain is the preimage.

Some authors call such manifolds therefore strictly pseudoconvex manifolds.