# Behnke–Stein theorem on Stein manifolds

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.

## References

1. Heinrich Behnke & Karl Stein (1948), "Entwicklung analytischer Funktionen auf Riemannschen Flächen", Mathematische Annalen 120: 430–461, doi:10.1007/BF01447838
2. Raghavan, Narasimhan (1960). "Imbedding of Holomorphically Complete Complex Spaces". American Journal of Mathematics 82 (4): 917–934. doi:10.2307/2372949.
3. Simha, R. R. (1989). "The Behnke-Stein Theorem for Open Riemann Surfaces". Proceedings of the American Mathematical Society 105 (4): 876–880. doi:10.2307/2047046.
4. Behnke, H.; Stein, K. (1939). "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität". Mathematische Annalen 116: 204–216. doi:10.1007/BF01597355.