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  "body": "\n## Definition\n\nLet $g$ be a smooth [[Riemannian metric|Q61VFHCH2J]] on the [[sphere|P9CM4GQ2T0]] $S^2$.\nA *sweepout* of $S^2$ is a continuous map $c \\colon [0, 1] \\times S^1 \\to S^2$ such that:\n\n1. each curve $c_t = c(t, \\cdot)$ is piecewise smooth, and $t \\mapsto c_t$ is continuous for the $C^0$ topology and for the [[length|0TH78S7GBZ]];\n2. the curves $c_0$ and $c_1$ are constant;\n3. the map $S^2 \\to S^2$ induced by $c$ on the quotient of $[0, 1] \\times S^1$ that collapses $\\set{0} \\times S^1$ and $\\set{1} \\times S^1$ to points has nonzero [[degree|TMFG2P912R]].\n\nThe *width* of $(S^2, g)$ is\n$$W(g) = \\inf_{c} \\ \\max_{t \\in [0, 1]} L(c_t),$$\nthe infimum over all sweepouts $c$ of the length of the longest curve.\nThe *Birkhoff min-max method* shows that $W(g) \u003e 0$ and that $W(g)$ is the length of a [[closed geodesic|4MCCP8GG4J]].\nThe geodesic is a limit of longest curves of a minimizing sequence of sweepouts.\nBefore taking the limit, a curve-shortening process is applied to all curves of each sweepout at once.\n\n## Remarks\n\nThe width is positive because curves of small length lie in small discs, so a sweepout by short curves would induce a map of degree $0$.\nBirkhoff's curve-shortening process replaces each curve by a broken geodesic through finitely many of its points, and repeats.\nIt acts continuously on all curves of a sweepout and does not increase lengths.\n\nExample: on the round unit sphere, the circles of latitude $c_t(s) = (\\sin \\pi t \\cos s, \\sin \\pi t \\sin s, -\\cos \\pi t)$ form a sweepout of degree $1$, whose longest curve is the equator, of length $2\\pi$.\nHere $W = 2\\pi$, the length of a great circle.\n\nNon-example: a family that starts and ends at constant curves but stays in the southern hemisphere, such as $c_t(s) = (\\tfrac12 \\sin \\pi t \\cos s, \\tfrac12 \\sin \\pi t \\sin s, -\\sqrt{1 - \\tfrac14 \\sin^2 \\pi t})$, is not a sweepout.\nThe induced map misses the north pole, so its degree is $0$, and clause 3 fails.\n",
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