Calibrated Submanifolds
A p-dimensional submanifold Σ of M is said to be a calibrated submanifold with respect to φ (or simply φ-calibrated) if TΣ lies in G(φ).
A famous one line argument shows that calibrated p-submanifolds minimize volume within their homology class. Indeed, suppose that Σ is calibrated, and Σ ′ is a p submanifold in the same homology class. Then
where the first equality holds because Σ is calibrated, the second equality is Stokes' theorem (as φ is closed), and the third equality holds because φ is a calibration.
Read more about this topic: Calibrated Geometry