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  "body": "\n## Summary\n\nA Levi-flat hypersurface is a real hypersurface in a [[complex manifold|JSGWAN2RJV]] whose [[Levi form|80MTNXWW80]] vanishes at every point.\nFor a smooth hypersurface this means that it is filled, near each point, by complex hypersurfaces stacked in a one-parameter family.\nLevi-flat hypersurfaces are the opposite extreme to strongly pseudoconvex ones, and CR functions on them need not extend holomorphically to either side.\n\n## Explanations\n\n### L1 human\n\nA real hypersurface in $\\C^n$ has one real dimension less than the space.\nMost such hypersurfaces bend in the complex directions, and the Levi form measures this bending.\nA Levi-flat hypersurface does not bend at all in these directions.\nThe model is $\\set{z \\in \\C^2 : \\operatorname{Im} z_1 = 0}$, which is the union of the complex lines $\\set{z_1 = t}$ for real $t$.\nSo it looks like a stack of complex lines indexed by a real number, like the pages of a book.\n",
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