by the minimal surface equation. The amount of literature in the area is vast, with key results including the proofs of the Lawson [@BLawson], Willmore [@MNWillmore] and Yau [@IMNYau; @MNYau; @SongYau] Conjectures, and minimal surfaces have applications to major problems in geometry including the Positive Mass Theorem [@SYMass; @SYMass2], Penrose Inequality [@HIPenrose] and Poincaré Conjecture [@Perelman].

Introduction to calibrations
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As we have seen, minimal submanifolds are extremely important. However there are two key issues.

-   Minimal submanifolds are defined by a second order nonlinear PDE system – therefore they are hard to analyse.

-   Minimal submanifolds are only critical points for the volume functional, but we are often interested in minima for the volume functional – we need a way to determine when this occurs.

We can help resolve these issues using the notion of calibration and calibrated submanifolds, introduced by Harvey–Lawson [@HL] in 1982.

\[calibration.dfn\] A differential $k$-form $\eta$ on a Riemannian manifold $(M,g)$ is a *calibration* if

-   $\d\eta=0$ and

-   $\eta(e_1,\ldots,e_k)\leq 1$ for all unit tangent vectors $e_1,\ldots,e_k$ on $M$.

Any non-zero form with constant coefficients on $\R^n$ can be rescaled so that it is a calibration with at least one plane where equality holds.

This 