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  "body": "\n## Definition\n\nLet $M$ be a smooth $n$-dimensional [[manifold|F45VQ47GPV]] and let $T \u003e 0$.\nA *mean curvature flow* is a smooth map $F \\colon M \\times [0, T) \\to \\R^{n+1}$ such that:\n\n1. for each $t \\in [0, T)$, the map $F_t = F(\\cdot, t)$ is an immersion of $M$ as a hypersurface;\n2. $\\partial_t F(p, t) = \\vec{H}(p, t)$ for all $p \\in M$ and $t \\in [0, T)$.\n\nHere $\\vec{H}(p, t) = -H \\nu$ is the mean curvature vector of $F_t$ at $p$.\nThe vector $\\nu$ is a unit [[normal vector|0WQJJCCTC8]], and $H$ is the [[mean curvature|PA78NFEPCF]], the sum of the principal curvatures with respect to $\\nu$.\n\n## Remarks\n\nExample: let $F(p, t) = R(t)\\, p$ for $p$ in the unit sphere $S^n \\subset \\R^{n+1}$, with $R(t) = \\sqrt{R_0^2 - 2nt}$.\nThe sphere of radius $R$ has mean curvature vector of length $n/R$ pointing to the centre, so the flow condition is $R'(t) = -n/R(t)$.\nThis holds since $\\frac{d}{dt} R^2 = -2n$, and the flow exists until $T = R_0^2/(2n)$.\nNon-example: $R(t) = R_0 + t$ moves the sphere outward, against the mean curvature vector, so it fails clause 2.\nA plane or any minimal hypersurface, with $\\vec H = 0$, is a stationary flow.\nSign conventions for $H$ and $\\nu$ vary, but $\\vec H$ always points toward the centre of a round sphere.\n",
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