Domain of Holomorphy - Equivalent Conditions

Equivalent Conditions

For a domain the following conditions are equivalent:

1. is a domain of holomorphy
2. is holomorphically convex
3. is pseudoconvex
4. is Levi convex - for every sequence of analytic compact surfaces such that for some set we have ( cannot be "touched from inside" by a sequence of analytic surfaces)
5. has local Levi property - for every point there exist a neighbourhood of and holomorphic on such that cannot be extended to any neighbourhood of

Implications are standard results. The main difficulty lies in proving, i.e. constructing a global holomoprhic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem and was first solved by Kiyoshi Oka, and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of -problem).