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  "body": "\n## Statement\n\nLet $(M, g)$ be a complete, asymptotically flat, three-dimensional [[Riemannian manifold|7CBT4SM83Y]] with nonnegative [[scalar curvature|ZHX2VYQDZ3]].\nAssume that the boundary $\\partial M$ is an outermost [[minimal surface|41X0YW2R9N]], possibly with several components.\nThis means that $\\partial M$ is a closed minimal surface and that no other closed minimal surface in $M$ encloses it.\nLet $A$ be the total area of $\\partial M$ and let $m$ be the ADM mass of $(M, g)$.\nThen\n$$m \\ge \\sqrt{\\frac{A}{16\\pi}}.$$\nEquality holds if and only if $(M, g)$ is isometric to the spatial [[Schwarzschild metric|6Y50T68WKY]] of mass $m$ outside its horizon.\n\n## Remarks\n\nHuisken and Ilmanen proved the inequality in 2001 when $A$ is the area of a largest component of $\\partial M$, using a weak form of [[inverse mean curvature flow|5XVEF5DKAG]].\nAlong this flow the Hawking mass of the evolving surface does not decrease, a monotonicity first observed by Geroch.\nThe Hawking mass equals $\\sqrt{A/16\\pi}$ on the horizon and tends to at most $m$ at infinity.\nBray proved the version with total area in 2001 by a conformal flow of metrics.\nFor the spatial Schwarzschild metric of mass $m$, the horizon is the sphere $r = 2m$ of area $16\\pi m^2$, so both sides equal $m$.\nNonnegative scalar curvature is needed: without it the mass can even be negative.\n\n## Explanations\n\n### L1 human\n\nIn general relativity, a black hole has a horizon with a certain area.\nThe inequality says that the total mass of the space is at least a fixed multiple of the square root of that area.\nSo black holes cannot carry more horizon than their mass allows.\nThe Schwarzschild black hole is the extreme case where the two sides are equal.\n\n### L2 human\n\nThe statement is the time-symmetric case of a conjecture of Penrose; the scalar curvature condition encodes nonnegative energy density.\nIt strengthens the [[positive mass theorem|PYRPXHNQTB]], which gives $m \\ge 0$ when there is no horizon.\nThe two proofs use different flows: inverse mean curvature flow of surfaces, and a flow of conformal metrics.\nThe general case of the Penrose conjecture, without time symmetry, is still open.\n",
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