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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold and let $T \u003e 0$.\nLet $(g(t))_{0 \\le t \u003c T}$ be a smooth family of [[Riemannian metrics|Q61VFHCH2J]] on $M$.\nIt solves the *Yamabe flow* if\n$$\\frac{\\partial g}{\\partial t} = -R_{g(t)}\\, g(t),$$\nwhere $R_{g(t)}$ is the [[scalar curvature|ZHX2VYQDZ3]] of $g(t)$.\n\nLet $M$ be compact.\nThe *normalized Yamabe flow* is\n$$\\frac{\\partial g}{\\partial t} = \\left(r_{g(t)} - R_{g(t)}\\right) g(t),$$\nwhere $r_{g(t)}$ is the average of $R_{g(t)}$ over $M$ with respect to the volume of $g(t)$.\nIt preserves the total volume.\n\n## Remarks\n\nExample: on the unit sphere $S^n$ with $n \\ge 2$ and round metric $g_0$, put $g(t) = \\rho(t) g_0$.\nThe scalar curvature of $g_0$ is $n(n-1)$, so that of $g(t)$ is $n(n-1)/\\rho(t)$.\nThe equation becomes $\\rho'(t) g_0 = -n(n-1) g_0$, which holds for $\\rho(t) = 1 - n(n-1)t$ with $0 \\le t \u003c 1/(n(n-1))$.\nNon-example: the family $g(t) = (1 + t) g_0$ on the flat torus fails, since the scalar curvature is $0$ but the metric changes.\nThe right side is a multiple of $g$.\nSo a solution has the form $g(t) = u(\\cdot, t)\\, g(0)$ for a positive function $u$ on $M \\times [0, T)$, and it stays in the conformal class of $g(0)$.\nIn dimension $2$ the Ricci curvature equals $\\tfrac{1}{2} R g$, so the Yamabe flow coincides with the Ricci flow $\\partial g/\\partial t = -2 \\operatorname{Ric}$.\n",
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    "name": "Yamabe flow",
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  "name": "Yamabe flow",
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