= Calibration implies volume minimization
{title2=$\operatorname{Vol}(\Sigma)\leq\operatorname{Vol}(\Sigma')$}
If $\Sigma$ is calibrated by $\varphi$ and $\Sigma'$ is homologous with the same boundary, <Stokes theorem> gives $\int_\Sigma\varphi=\int_{\Sigma'}\varphi$. The tangent-plane bound gives $\int_{\Sigma'}\varphi\leq\operatorname{Vol}(\Sigma')$, while calibration gives $\int_\Sigma\varphi=\operatorname{Vol}(\Sigma)$. Thus $\Sigma$ minimizes volume in its relative homology class and is a <minimal submanifold> wherever smooth.
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