Behnke–Stein theorem on Stein manifolds

For Behnke–Stein theorem on domains of holomorphy, see Behnke–Stein theorem.

In mathematics, especially several complex variables, the Behnke–Stein theorem states that a connected, non-compact (open) Riemann surface is a Stein manifold.^[1] In other words, it states that there is a nonconstant single-valued holomorphic function (univalent function) on such a Riemann surface.^[2] It is a generalization of the Runge approximation theorem and was proved by Heinrich Behnke and Karl Stein in 1948.^[3]

Method of proof

The study of Riemann surfaces typically belongs to the field of one-variable complex analysis, but the proof method uses the approximation by the polyhedron domain used in the proof of the Behnke–Stein theorem on domains of holomorphy^[4] and the Oka–Weil theorem.