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  "body": "\n## Summary\n\nThe Yamabe flow is an evolution equation for a [[Riemannian metric|Q61VFHCH2J]].\nThe metric changes in the direction of minus its [[scalar curvature|ZHX2VYQDZ3]] times itself.\nRichard Hamilton introduced it as a tool for the Yamabe problem.\nThis problem asks for a metric of constant scalar curvature in a given conformal class.\nThe flow keeps the metric in its conformal class.\nIts fixed points, after normalizing the volume, are the metrics of constant scalar curvature.\n\n## Explanations\n\n### L1 human\n\nA metric tells how to measure lengths on a curved space.\nThe Yamabe flow rescales lengths at each point: it shrinks them where the scalar curvature is positive and stretches them where it is negative.\nAngles never change, because the change at each point is a pure rescaling.\nThe hope is that the curvature evens out over time.\nFor example, the round sphere of dimension $n$ and radius $1$ shrinks: its metric at time $t$ is $1 - n(n-1)t$ times the original one, until it collapses at $t = 1/(n(n-1))$.\n",
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