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  "body": "\n## Statement\n\nLet $T$ be a torus and let $f \\colon T \\to \\R^3$ be a smooth [[immersion|6RW68JGBHC]].\nThen the [[Willmore energy|R7WFJGJBV4]] of $f$ satisfies\n$$\\mathcal{W}(f) = \\int_T H^2 \\, dA \\ge 2\\pi^2.$$\nLet $T_0$ be the torus of revolution whose generating circle has radius $1$ and center at distance $\\sqrt{2}$ from the axis.\nEquality holds if and only if $f(T)$ is the image of $T_0$ under a Möbius transformation of $\\R^3 \\cup \\set{\\infty}$.\n\n## Remarks\n\nWillmore stated the conjecture in 1965, and Marques and Neves proved it; their proof appeared in 2014.\nThe torus in the equality case is the stereographic image of the Clifford torus in the three-sphere.\nIf $f$ is not an embedding, a theorem of Li and Yau gives $\\mathcal{W}(f) \\ge 8\\pi \u003e 2\\pi^2$, so only embedded tori need work.\nFor those, Marques and Neves used conformal invariance to move the problem to the three-sphere and applied Almgren–Pitts min-max theory to a five-parameter family of surfaces built from the torus.\nA key step is that every closed embedded minimal surface of positive genus in the three-sphere has area at least $2\\pi^2$.\nEquality holds there only for the Clifford torus, up to isometry.\n\nFor every closed surface $\\mathcal{W} \\ge 4\\pi$, so the conjecture says that being a torus costs at least $2\\pi^2 - 4\\pi$ extra energy.\n\n## Explanations\n\n### L1 human\n\nThe Willmore energy measures how much a closed surface bends overall.\nAmong all surfaces it is smallest for a round sphere.\nThe conjecture, now a theorem, finds the least possible energy for a doughnut-shaped surface: $2\\pi^2$, reached by one particular fat doughnut and its Möbius images.\n\n### L2 human\n\nBecause $\\mathcal{W}$ is invariant under Möbius transformations, the minimizers come in whole families, and the natural setting is conformal geometry.\nThe proof is one of the main applications of min-max theory for minimal surfaces.\nIt belongs to a larger program of finding the least Willmore energy for each genus.\n",
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