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  "body": "\n## Summary\n\nLiouville's equation is the [[semilinear|RCNY89A4NY]] elliptic equation $\\Delta u + K e^{2u} = 0$ for a function $u$ of two real variables, with $K$ a constant.\nIt says that the metric $e^{2u}(dx^2 + dy^2)$ has constant [[Gaussian curvature|SQVQ5DQT6W]] $K$.\nJoseph Liouville found its general local solution in terms of holomorphic functions.\nEquations of this exponential type also appear in models of self-gravitating gases and of vortices.\n\n## Explanations\n\n### L1 human\n\nTake a flat piece of the plane and rescale lengths by a factor $e^{u}$ that changes from point to point.\nThe rescaled surface can look curved, like a piece of a sphere, and its curvature depends on how $u$ varies.\nLiouville's equation asks for the rescalings that give the same curvature everywhere.\nFor example, $u = \\log\\big(2/(1 + x^2 + y^2)\\big)$ turns the plane into the round sphere of radius one with a point removed, which has curvature $1$.\n",
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