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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^2$ be an open set and let $K \\in \\R$ be a constant.\n*Liouville's equation* on $\\Omega$ is the [[semilinear|RCNY89A4NY]] equation\n$$\\Delta u + K e^{2u} = 0$$\nfor a twice continuously differentiable function $u \\colon \\Omega \\to \\R$, where $\\Delta = \\partial_x^2 + \\partial_y^2$ is the [[Laplacian|0E0B6RA8K0]].\n\n## Remarks\n\nThe metric $e^{2u}(dx^2 + dy^2)$ has [[Gaussian curvature|SQVQ5DQT6W]] $-e^{-2u} \\Delta u$, so $u$ solves the equation exactly when this metric has constant curvature $K$.\nExample: for $K = 1$, $u(x, y) = \\log 2 - \\log(1 + r^2)$ with $r^2 = x^2 + y^2$ is a solution on $\\R^2$.\nIndeed, $\\Delta \\log(1 + r^2) = 4/(1 + r^2)^2$, so $\\Delta u = -4/(1 + r^2)^2 = -e^{2u}$.\nNon-example: for $K = 1$, the function $u = 0$ is not a solution, since $\\Delta u + e^{2u} = 1$.\nOther normalizations are common: the substitution $u = v/2$ gives $\\Delta v + 2K e^{v} = 0$, so the form $\\Delta v + \\lambda e^{v} = 0$ is equivalent.\nLocally, the solutions for $K \u003e 0$ are $u = \\log\\big( 2 \\abs{f'} / (1 + K \\abs{f}^2) \\big)$ with $f$ a holomorphic function of $z = x + iy$ whose derivative has no zeros.\n",
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