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  "body": "\n## Summary\n\nThe Calabi flow is an evolution of Kähler metrics on a compact complex manifold that stays inside one Kähler class.\nIt moves the Kähler potential with speed equal to the difference between the [[scalar curvature|ZHX2VYQDZ3]] and its average.\nEugenio Calabi introduced it in his work on extremal Kähler metrics.\nAlong the flow the Calabi energy, the squared $L^2$ norm of that difference, does not increase.\nIts fixed points are the Kähler metrics of constant scalar curvature, so it is a tool for finding such metrics.\n\n## Explanations\n\n### L1 human\n\nA Kähler metric is a way to measure lengths on a complex manifold that fits well with the complex structure.\nThe Calabi flow changes such a metric step by step, pushing it where the scalar curvature is above average and pulling it where it is below average.\nThe aim is to reach a metric whose scalar curvature is the same at every point.\nThe equation is of fourth order in the space variables, so it is harder to study than second-order flows such as the Ricci flow.\nA flat metric on a complex torus does not move at all, since its scalar curvature is $0$ everywhere.\n",
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