example shows that there are many calibrations $\eta$, but the interesting question is: for which oriented planes $P=\operatorname{Span}\{e_1,\ldots,e_k\}$ does $\eta(e_1,\ldots,e_k)=1$? More importantly, can we find submanifolds $N$ so that this equality holds on each tangent space? This motivates the next definition.

\[calibsub.dfn\] Let $\eta$ be a calibration $k$-form on $(M,g)$. An oriented $k$-dimensional submanifold $N$ of $(M,g)$ is *calibrated* by $\eta$ if $\eta|_N=\operatorname{vol}_N$, i.e. if for all $p\in N$ we have $\eta(e_1,\ldots,e_k)=1$ for an oriented orthonormal basis $e_1,\ldots,e_k$ for $T_pN$.

Any oriented plane in $\R^n$ is calibrated. If we change coordinates so that the plane $P$ is $\{x\in\R^n\,:\,x_{k+1}=\ldots=x_n=0\}$ (with the obvious orientation) then $\eta=\d x_1\wedge\ldots\wedge \d x_k$ is a calibration and $P$ is calibrated by $\eta$.

Notice that the calibrated condition is now an algebraic condition on the tangent vectors to $N$, so being calibrated is a *first order nonlinear PDE*. We shall motivate these definitions further later, but for now we make the following observation.

\[calibmin.thm\] Let $N$ be a calibrated submanifold. Then $N$ is minimal and, moreover, if $F$ is any variation with compact support $\overline{S}$ then $\operatorname{Vol}(F(S,t))\geq\operatorname{Vol}(S)$; i.e. $N$ is volume-minimizing. In particular, if $N$ is compact then $N$ is volume-minimizing in its homology class.

Suppose that 