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  "body": "\n## Statement\n\nLet $\\gamma_0$ be a smooth embedded closed plane curve that encloses area $A_0$.\nThen the [[curve-shortening flow|DVGFRRYK10]] with initial curve $\\gamma_0$ has a unique smooth solution $(\\gamma_t)_{0 \\le t \u003c T}$ with $T = A_0 / (2\\pi)$, and:\n\n1. each $\\gamma_t$ is embedded;\n2. there is $t_0 \u003c T$ such that $\\gamma_t$ bounds a strictly [[convex|02AS45T3H9]] region for all $t \\in [t_0, T)$;\n3. as $t \\to T$, the curves $\\gamma_t$ converge to a point $p$. Rescale $\\gamma_t$ by the dilation about $p$ that makes the enclosed area equal to $\\pi$. These rescaled curves converge smoothly to the unit [[circle|060W5CBZP6]] centered at $p$.\n\n## Remarks\n\nGage and Hamilton proved in 1986 that convex curves stay convex and shrink to a round point.\nGrayson proved in 1987 that every embedded curve becomes convex before it becomes singular, which reduces the general case to the convex one.\nThe value of $T$ comes from the identity $A'(t) = -\\int_{\\gamma_t} \\kappa \\, ds = -2\\pi$ for embedded curves, since the total [[curvature|7JQS6VKFEZ]] of a simple closed curve is $2\\pi$.\nEmbeddedness is needed: an immersed figure-eight curve, or a curve with a small loop, can develop a singularity before it shrinks to a point.\n\nExample: a circle of radius $r_0$ encloses area $\\pi r_0^2$ and vanishes at time $r_0^2 / 2 = A_0 / (2\\pi)$, in agreement with the formula.\n\n## Explanations\n\n### L1 human\n\nLet a simple closed curve in the plane move by curvature, so sharply bent parts move fastest.\nHowever twisted the curve is at the start, it never crosses itself, it becomes convex, and then it shrinks to a point while looking more and more like a circle.\nThe time it lives is exactly its area divided by $2\\pi$.\n\n### L2 human\n\nThe theorem is a model for later results on geometric flows, such as Huisken's theorem on convex hypersurfaces under mean curvature flow and Hamilton's work on Ricci flow.\nGrayson's step is the hard part, because it must rule out singularities for nonconvex curves.\nLater proofs by Huisken and by Andrews and Bryan compare distances along the curve with distances in the plane.\nThe flow can also be used to prove geometric inequalities and the existence of closed geodesics on surfaces.\n",
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