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  "body": "\n## Summary\n\nThe Ricci flow is the evolution equation $\\partial_t g = -2 \\operatorname{Ric}(g)$ for a family of [[Riemannian metrics|Q61VFHCH2J]] on a manifold.\nIt was introduced by Richard Hamilton in 1982.\nIt is a nonlinear, weakly parabolic system that behaves like a heat equation for the metric, smoothing out irregularities of the curvature.\nGrigori Perelman used the Ricci flow with surgery to prove the Poincaré conjecture and Thurston's geometrization conjecture.\n\n## Explanations\n\n### L1 human\n\nThink of a lumpy shape that you let relax, with each region changing its size according to how curved it is.\nUnder the Ricci flow, regions of positive [[Ricci curvature|4GTNM0WHDH]] shrink and regions of negative Ricci curvature expand.\nA round sphere stays round and shrinks to a point in finite time.\nThe hope, made true in many cases, is that a manifold flows toward a simple standard geometry, which then reveals its shape.\n",
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