Calibrated submanifold ($\phi {|}_{\Sigma }={\mathrm{vol}}_{\Sigma }$)

An oriented submanifold is calibrated by a calibration when its tangent planes attain the comass bound at every point. Closedness then compares its volume with homologous competitors by Stokes theorem.

If $\Sigma$ is calibrated by $\phi$ and ${\Sigma }^{\prime }$ is homologous with the same boundary, Stokes theorem gives ${\int }_{\Sigma }\phi ={\int }_{{\Sigma }^{\prime }}\phi$. The tangent-plane bound gives ${\int }_{{\Sigma }^{\prime }}\phi \le \mathrm{Vol}({\Sigma }^{\prime })$, while calibration gives ${\int }_{\Sigma }\phi =\mathrm{Vol}(\Sigma )$. Thus $\Sigma$ minimizes volume in its relative homology class and is a minimal submanifold wherever smooth.