On Complex Analysis in Several Variables

Abstract

In one dimension, every domain D ⊂ ℂ1 is a domain of holomorphy, that is, there is a holomorphic function f in D, f ∈ O(D) which cannot be holomorphically extended to larger domain because on the boundary ∂D of D f has a dense set of singularities. This fact was known already to Riemann and Weierstrass. For n > 1 the situation is quite different: Fritz Hartogs has constructed simple domains D ∈ ℂ2 which are not domains of holomorphy; every f ∈ O(D)can be extended to a larger convex domain \(\hat D\). Soon afterwards E.E. Levi discovered that domains of holomorphy in ℂ2, characterized by strictly plurisubharmonic exhausting functions φ have a convexity property.