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  "body": "\n## Definition\n\nLet $S \\subset \\R^3$ be a [[compact|0TVJZYB6SH]] orientable [[regular surface|6080X3FAD4]] without boundary.\nLet $H$ be its [[mean curvature|PA78NFEPCF]], $H = (k_1 + k_2)/2$ for the principal curvatures $k_1, k_2$, and let $dA$ be its area element.\nThe *Willmore energy* of $S$ is\n$$\\mathcal{W}(S) = \\int_S H^2 \\, dA.$$\n\n## Remarks\n\nFor the sphere of radius $r$, one has $H^2 = 1/r^2$ and area $4\\pi r^2$, so $\\mathcal{W} = 4\\pi$, for every $r$.\nFor the cylinder $x^2 + y^2 = r^2$, which is not compact, $H^2 = 1/(4r^2)$ is constant and the area is infinite, so the integral diverges.\n\nThe same formula defines $\\mathcal{W}$ for a smooth immersion of a compact surface into $\\R^3$; the value does not depend on the orientation, since $H^2$ does not.\nSince $H^2 - K = (k_1 - k_2)^2/4$ and $\\int_S K \\, dA = 2\\pi\\chi(S)$ by the [[Gauss–Bonnet theorem|ESGEHKPD60]],\n$$\\mathcal{W}(S) = 2\\pi\\chi(S) + \\frac{1}{4} \\int_S (k_1 - k_2)^2 \\, dA.$$\nSome authors call $\\int_S (H^2 - K)\\, dA$ the Willmore energy; it differs from $\\mathcal{W}$ by the topological constant $2\\pi\\chi(S)$.\nFor every such $S$, $\\mathcal{W}(S) \\ge 4\\pi$, with equality exactly for round spheres.\n\n## Explanations\n\n### L1 human\n\nThe Willmore energy is the total of the squared mean curvature over a closed surface.\n",
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