Behnke–Stein theorem

Not to be confused with Behnke–Stein theorem on Stein manifolds.

In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence ${\displaystyle G_{k}\subset \mathbb {C} ^{n}}$ (i.e., ${\displaystyle G_{k}\subset G_{k+1}}$) of domains of holomorphy is again a domain of holomorphy. It was proved by Heinrich Behnke and Karl Stein in 1938.[1]

This is related to the fact that an increasing union of pseudoconvex domains is pseudoconvex and so it can be proven using that fact and the solution of the Levi problem. Though historically this theorem was in fact used to solve the Levi problem, and the theorem itself was proved using the Oka–Weil theorem. This theorem again holds for Stein manifolds, but it is not known if it holds for Stein space.[2]

References

1. Behnke, H.; Stein, K. (1939). "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität". Mathematische Annalen. 116: 204–216. doi:10.1007/BF01597355. S2CID 123982856.
2. Coltoiu, Mihnea (2009). "The Levi problem on Stein spaces with singularities. A survey". arXiv:0905.2343 [math.CV].

• This article incorporates material from Behnke-Stein theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
• Chirka, E.M. (2001) [1994], "Stein manifold", Encyclopedia of Mathematics, EMS Press