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
- ^ Heinrich Behnke & Karl Stein (1948), "Entwicklung analytischer Funktionen auf Riemannschen Flächen", Mathematische Annalen, 120: 430–461, doi: 10.1007/BF01447838, S2CID 122535410, Zbl 0038.23502
- ^ Raghavan, Narasimhan (1960). "Imbedding of Holomorphically Complete Complex Spaces". American Journal of Mathematics. 82 (4): 917–934. doi: 10.2307/2372949. JSTOR 2372949.
- ^ 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. JSTOR 2047046.
- ^ Behnke, H.; Stein, K. (1939). "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität". Mathematische Annalen. 116: 204–216. doi: 10.1007/BF01597355. S2CID 123982856.
 | This differential geometry-related article is a stub. You can help Wikipedia by expanding it. |