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  "body": "\n## Definition\n\nLet $1 \\le m \\le n$, and let $T$ be an [[integral current|EG4AJVEFT2]] of dimension $m$ on $\\R^n$, with mass $\\mathbf{M}(T)$ and boundary $\\partial T$.\nThe current $T$ is *area-minimizing*, or *mass-minimizing*, if\n$$\\mathbf{M}(T) \\le \\mathbf{M}(S)$$\nfor every integral $m$-current $S$ on $\\R^n$ with $\\partial S = \\partial T$.\n\n## Remarks\n\nA local version is used for currents in an open set $U$.\nThere one compares $T$ with $S$ only when $T - S$ is the boundary of an integral current with compact support in $U$.\nFor a compact oriented $C^1$ submanifold, the mass of the associated [[current|360H7MD9RD]] is its area.\nSo an area-minimizing submanifold has least area among all oriented competitors with the same boundary, smooth or not.\n\nExample: let $D$ be the closed unit disc in the plane $x_3 = 0$ of $\\R^3$, oriented by $e_1 \\wedge e_2$, and let $\\omega = dx_1 \\wedge dx_2$, a closed [[differential form|0E3G0GRPZ6]] of comass $1$.\nFor every integral $2$-current $S$ with $\\partial S = \\partial [\\![D]\\!]$, one has $S - [\\![D]\\!] = \\partial R$ for an integral $3$-current $R$, since every integral cycle in $\\R^3$ bounds.\nThen $S(\\omega) - [\\![D]\\!](\\omega) = R(d\\omega) = 0$ by the definition of the boundary, as in [[Stokes' theorem|0AKRKVHW94]].\nSo $\\mathbf{M}(S) \\ge S(\\omega) = [\\![D]\\!](\\omega) = \\pi = \\mathbf{M}([\\![D]\\!])$, and $[\\![D]\\!]$ is area-minimizing.\n\nNon-example: the upper unit hemisphere, oriented so that its boundary is $\\partial [\\![D]\\!]$, has mass $2\\pi \u003e \\pi$, so it is not area-minimizing.\n\n## Explanations\n\n### L1 human\n\nA current is area-minimizing if no other integral current with the same boundary has smaller mass.\n",
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