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] 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.
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