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Behnke–Stein theorem on Stein manifolds

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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. In other words, it states that there is a nonconstant single-valued holomorphic function (univalent function) on such a Riemann surface. It is a generalization of the Runge approximation theorem and was proved by Heinrich Behnke and Karl Stein in 1948.

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 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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