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  "body": "\n## Summary\n\nAlmgren–Pitts min-max theory constructs closed [[minimal hypersurfaces|41X0YW2R9N]] in a closed Riemannian manifold by a min-max over families of hypersurfaces that sweep out the manifold.\nIt is the higher-dimensional analogue of the [[Birkhoff min-max method|1R5X9HW6D3]] for closed geodesics.\nFrederick Almgren set up the topology of spaces of cycles, and Jon Pitts proved the regularity of the min-max hypersurface in low dimensions.\nFernando Marques and André Neves developed the theory further and used it to prove the Willmore conjecture.\nIt is also the main tool behind [[Yau's conjecture on infinitely many minimal surfaces|VAP45DFXW3]].\n\n## Explanations\n\n### L1 human\n\nFill a closed curved space with a family of soap films that starts as nothing, grows to fill the whole space, and shrinks back to nothing.\nAt some moment the film in the family has the largest area.\nChoose the family so that this largest area is as small as possible.\nThe film at the critical moment cannot be pushed down further, and it settles into a minimal hypersurface, a film whose mean curvature is zero.\nFor a round $3$-sphere, sweeping by parallel $2$-spheres gives the equatorial $2$-sphere as the min-max surface.\n",
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