Behnke–Stein theorem

The Behnke–Stein theorem states that every noncompact Riemann surface is a Stein manifold. It makes one-dimensional complex analysis a setting where holomorphic functions can separate points and provide local coordinates.

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Stein manifold: A complex manifold with sufficiently many global holomorphic functions to separate points and provide local coordinates. The theorem identifies every noncompact Riemann surface as a member of this class.
Holomorphic convexity: A property requiring holomorphic hulls of compact sets to remain compact. It is one of the defining global conditions satisfied by Stein manifolds.
Heinrich Behnke: A German mathematician who helped develop the theory of several complex variables and Riemann surfaces. His work with Stein gave the theorem its first name.
Compact Riemann surface: A Riemann surface that is compact as a topological space. Positive-dimensional compact complex manifolds are not Stein, unlike the surfaces covered by the theorem.
Oka–Grauert principle: A theorem equating topological and holomorphic classification of vector bundles on Stein manifolds. Steinness lets complex-analytic bundle problems on these surfaces be controlled by topology.
Holomorphic separability: The property that distinct points can be distinguished by global holomorphic functions. The theorem guarantees this separation for any two points on a noncompact surface.
Karl Stein: A German mathematician known for foundational contributions to complex analysis and Stein spaces. His collaboration with Behnke established the result for noncompact Riemann surfaces.
Maximum modulus principle: A principle stating that a nonconstant holomorphic function cannot attain a local maximum of its modulus. On compact Riemann surfaces it forces every holomorphic function to be constant, preventing point separation.
Runge's theorem: A theorem giving conditions under which holomorphic functions can be approximated by rational functions. Runge approximation is a central one-variable tool in establishing global holomorphic flexibility.
Holomorphic convex hull: The set of points bounded by every holomorphic function as it is bounded on a given compact set. Compactness of these hulls expresses the global control required for Steinness.