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  "body": "\n## Summary\n\nThe Willmore flow is the gradient flow of the [[Willmore energy|R7WFJGJBV4]] for closed surfaces immersed in $\\R^3$.\nEach point of the surface moves along the normal with a speed given by a fourth-order expression in the surface.\nIn this way the energy decreases as fast as possible.\nIts stationary surfaces are the Willmore surfaces, such as round spheres.\nIt is a tool for finding surfaces that minimize bending energy.\n\n## Explanations\n\n### L1 human\n\nThe Willmore energy measures how much a closed surface bends, by integrating the square of its [[mean curvature|PA78NFEPCF]].\nA round sphere has the least energy, $4\\pi$, among all closed surfaces.\nThe Willmore flow deforms a surface step by step so that this energy drops.\nA bumpy sphere tends to smooth out toward a round shape, while a round sphere does not move at all.\nUnlike mean curvature flow, the flow does not simply shrink surfaces, because the energy does not change under scaling.\n",
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