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  "body": "\n## Definition\n\nLet $S^1 = \\R / \\Z$ and let $T \u003e 0$.\nLet $\\gamma \\colon S^1 \\times [0, T) \\to \\R^2$ be continuous and smooth on $S^1 \\times (0, T)$.\nAssume that for each $t \\in (0, T)$ the curve $\\gamma_t = \\gamma(\\cdot, t)$ is an immersed plane curve.\nLet $\\kappa(u, t)$ be the signed [[curvature|7JQS6VKFEZ]] of $\\gamma_t$ at $u$.\nLet $N(u, t)$ be the unit normal obtained by turning the unit tangent by $+90^\\circ$.\nThe family $\\gamma$ is a solution of the *curve-shortening flow* with initial curve $\\gamma_0$ if\n$$\\frac{\\partial \\gamma}{\\partial t}(u, t) = \\kappa(u, t)\\, N(u, t) \\qquad \\text{for all } u \\in S^1, \\ t \\in (0, T).$$\n\n## Remarks\n\nEquivalently, $\\partial_t \\gamma = \\partial_s^2 \\gamma$, where $s$ is the arc length parameter of $\\gamma_t$; this is a nonlinear partial differential equation of heat type.\nThe normal velocity is what matters: adding a tangential velocity only changes the parametrization of the curves.\nThe length $L(t)$ of $\\gamma_t$ satisfies $L'(t) = -\\int_{\\gamma_t} \\kappa^2 \\, ds$, and the enclosed area $A(t)$ of an embedded counterclockwise curve satisfies $A'(t) = -2\\pi$.\n\nExample: the circles $\\gamma(u, t) = \\sqrt{r_0^2 - 2t}\\,(\\cos 2\\pi u, \\sin 2\\pi u)$ for $0 \\le t \u003c r_0^2 / 2$ form a solution.\nFor a counterclockwise circle of radius $r$, $\\kappa = 1/r$ and $N$ points to the center, and indeed $\\partial_t r = -1/r$.\n\nNon-example: the shrinking circles $\\gamma(u, t) = (r_0 - t)(\\cos 2\\pi u, \\sin 2\\pi u)$ move with normal speed $1$ instead of $1/r$, so they do not solve the equation.\n",
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