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  "body": "\n## Summary\n\nThe Willmore energy of a closed surface in space is the integral of the square of its mean curvature over the surface.\nIt measures how far the surface is from being round: it is at least $4\\pi$, with equality only for round spheres.\nIt does not change under scaling, and more generally under Möbius transformations of space.\nIt appears in the study of elastic membranes, such as cell membranes, where bending costs energy.\n\n## Explanations\n\n### L1 human\n\nThink of a closed surface as a thin elastic shell; bending it costs energy.\nThe Willmore energy adds up, over the whole surface, the square of how much it bends on average at each point.\nA round ball is the most relaxed shape, with energy $4\\pi$ whatever its size.\nA long thin sausage or a wrinkled surface has much larger energy.\n",
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