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  "body": "\n## Statement\n\nLet $X$ be a [[Stein manifold|HMCD0JBFST]] of complex dimension $n$.\nThen there is a [[proper|TM528Q96ZD]] injective [[holomorphic map|3MQTWKBBBT]] $F \\colon X \\to \\C^{2n+1}$ whose differential is injective at every point.\nSo $X$ is biholomorphic to a closed complex submanifold of $\\C^{2n+1}$.\n\n## Remarks\n\nThe theorem is due to Remmert, Bishop and Narasimhan.\nThe proof starts from the many global holomorphic functions that Cartan's theorems A and B provide on a Stein manifold.\nWith them one builds a proper injective holomorphic map into some $\\C^N$ with injective differential.\nThen one lowers $N$ by generic linear projections, as in the proof of the Whitney embedding theorem.\nThe converse holds: every closed complex submanifold of $\\C^N$ is Stein, because the coordinate functions separate points and give compact holomorphically convex hulls.\nSo the Stein manifolds are exactly the closed complex submanifolds of complex Euclidean spaces.\nExample: a noncompact Riemann surface, which is Stein, embeds properly into $\\C^3$.\nThe dimension $2n + 1$ is not optimal.\nFor $n \\ge 2$, Eliashberg and Gromov, with a final step by Schürmann, showed that $\\C^N$ with $N = \\floor{3n/2} + 1$ suffices, and this value is optimal.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that every Stein manifold can be drawn inside some $\\C^N$ by holomorphic functions, without self-intersections.\nGoing off to infinity in the manifold then means going off to infinity in $\\C^N$.\nSo the abstract definition of a Stein manifold, by conditions on its holomorphic functions, describes the same objects as the concrete closed submanifolds of $\\C^N$.\nIt is the complex-analytic analogue of the Whitney embedding theorem for smooth manifolds.\n\n### L2 human\n\nEmbeddings reduce questions about Stein manifolds to questions about submanifolds of $\\C^N$.\nFor example, $\\abs{F}^2$ is a smooth strictly plurisubharmonic exhaustion function, which gives the easy half of Grauert's characterization of Stein manifolds.\nWhether every open Riemann surface embeds properly in $\\C^2$ is an open problem.\n",
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