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  "body": "\n## Problem\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\C^n$ be a domain.\nEvery [[domain of holomorphy|4487CSC0MB]] is [[pseudoconvex|AHYZMSBBNH]].\nIs every pseudoconvex domain $\\Omega \\subseteq \\C^n$ a domain of holomorphy?\n\n## Remarks\n\nThe answer is yes.\nKiyoshi Oka proved it for $n = 2$ in 1942.\nFor all $n$ it was proved in the early 1950s, independently by Oka, by Hans-Joachim Bremermann and by François Norguet.\nHans Grauert solved the analogue for complex manifolds in 1958: a complex manifold with a smooth strictly plurisubharmonic exhaustion function is a Stein manifold.\nA modern proof solves the equation $\\bar\\partial u = f$ with Hörmander's $L^2$ estimates for weights built from a plurisubharmonic exhaustion function.\nThe problem has no content for $n = 1$, where every domain is both pseudoconvex and a domain of holomorphy.\n\n## Explanations\n\n### L1 human\n\nIn several complex variables, not every domain is the natural home of some holomorphic function: on some domains, every holomorphic function extends to a larger domain.\nThe domains that are natural homes are the domains of holomorphy.\nLevi found a geometric condition that they must satisfy, a complex form of convexity of the boundary called pseudoconvexity.\nThe Levi problem asks whether this geometric condition is enough.\nIt is, and the proof was one of the main achievements of complex analysis in the twentieth century.\n",
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      "first_stated": "1910",
      "posed_by": "Eugenio Elia Levi"
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    "name": "Levi problem",
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        "url": "https://en.wikipedia.org/wiki/Stein_manifold"
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  "markdown": "https://tamnd.github.io/math2/node/47CVS9RJ9D.md",
  "msc": [
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/47C/47CVS9RJ9D.md",
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  "tag": "47CVS9RJ9D",
  "uri": "m2://47CVS9RJ9D",
  "url": "https://tamnd.github.io/math2/wiki/levi-problem/",
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