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, S2CID 122535410, Zbl 0038.23502
  2. ↑ Raghavan, Narasimhan (1960). "Imbedding of Holomorphically Complete Complex Spaces". American Journal of Mathematics. 82 (4): 917–934. doi:10.2307/2372949. JSTOR 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. JSTOR 2047046.
  4. ↑ Behnke, H.; Stein, K. (1939). "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität". Mathematische Annalen. 116: 204–216. doi:10.1007/BF01597355. S2CID 123982856.


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