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  "body": "\n## Definition\n\nLet $M$ be a compact complex manifold of complex dimension $n$ with a Kähler form $\\omega_0$.\nThis is the form of a [[Hermitian metric|CK1DCPC3D3]] with $d\\omega_0 = 0$.\nFor a smooth real function $\\varphi$ on $M$ with $\\omega_\\varphi = \\omega_0 + i \\partial \\bar\\partial \\varphi \u003e 0$, let $S(\\varphi)$ be the [[scalar curvature|ZHX2VYQDZ3]] of the Kähler metric $\\omega_\\varphi$.\nLet $\\underline{S}$ be the average of $S(\\varphi)$ with respect to the volume form $\\omega_\\varphi^n$; it depends only on the Kähler class of $\\omega_0$.\nA smooth family $(\\varphi_t)_{0 \\le t \u003c T}$ of such functions is a solution of the *Calabi flow* if\n$$\\frac{\\partial \\varphi_t}{\\partial t} = S(\\varphi_t) - \\underline{S}.$$\n\n## Remarks\n\nExample: if $\\omega_0$ has constant scalar curvature, such as a Kähler–Einstein metric or a flat metric on a complex torus, then $S(0) = \\underline{S}$, and the constant family $\\varphi_t = 0$ solves the flow.\nNon-example: a family $\\omega_t = (1 + t)\\omega_0$ leaves the Kähler class of $\\omega_0$, so it is not of the form $\\omega_0 + i\\partial\\bar\\partial\\varphi_t$ and is not a Calabi flow.\nIn local coordinates the flow reads $\\partial g_{j\\bar k}/\\partial t = \\partial_j \\partial_{\\bar k} S$ for the metric coefficients $g_{j\\bar k}$.\nAlong the flow the Calabi energy $\\int_M (S - \\underline{S})^2\\, \\omega_\\varphi^n$ does not increase.\n",
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    "name": "Calabi flow",
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  "name": "Calabi flow",
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