Let $(M^{n+k},g)$ be a Riemannian manifold. Call a surface $\Sigma^n \subset M$ calibrated if there is a closed $n$-form $\omega$ defined on a neighbourhood $U \subset M$ of $\Sigma$ so that $\omega \lvert \Sigma = \mathrm{vol}_\Sigma$ and for any $p \in U$ and $n$-tuples $(X_1,\dots,X_n) \in T_p M$ of orthonormal vectors $\omega(X_1,\dots,X_n) \leq 1$. (This is slightly different from the usual definition, where usually $\omega$ is defined on $M$.) A simple argument shows that a calibrated surface $\Sigma$ is area-minimising in its homology class, and in particular a small perturbation of $\Sigma'$ of $\Sigma$ will have $\mathrm{Area}(\Sigma') \geq \mathrm{Area}(\Sigma)$. In particular a calibrated surface is minimal, that is stationary for the area functional, and has mean curvature $H_\Sigma = 0$.

There are many examples of calibrated area-minimising surfaces:

1. linear subspaces of $\mathbf{R}^n$,
2. minimal graphs of $u: \Omega \subset \mathbf{R}^n \to \mathbf{R}$, where $\Omega$ is an open domain in $\mathbf{R}^n$,
3. special Lagrangian submanifolds $\Sigma \subset M$ in Calabi-Yau manifolds, that is Lagrangian submanifolds so that $\mathrm{Im} \, \Omega \lvert \Sigma = 0$ where $\Omega$ is the holomorphic volume form,
4. holomorphic subvarieties of $\mathbf{C}^n$,
5. area-minimising cones with an isolated singularity at the origin, for example the Simons cone $\mathbf{C}_S = \{ (X,Y) \in \mathbf{R}^n \times \mathbf{R}^n \mid \lvert X \rvert = \lvert Y \rvert \}$. (I believe these are calibrated because of the Hardt-Simon foliations.)

However I cannot think of any examples of area-minimising surfaces which are not calibrated.

Question: What are they? I am especially interested in the codimension one case, where $\Sigma^n \subset M^{n+1}$. In which settings, or under which hypotheses, is an area-minimising surface not be calibrated?

Remark: I can formulate a more technically precise question, at the price of using some terms from geometric measure theory. Let $B \subset \mathbf{R}^{n+k}$ be the unit ball, and $T \in \mathbf{I}_n(B)$ be an integral current with $\partial T = 0$ in $B$. Suppose that $T$ is area-minimising in the sense that for some $\epsilon > 0$ and all currents $S \in \mathbf{I}_{n+1}(B)$ with $\mathrm{spt} \, S \subset \subset B$ and $\mathrm{Vol} \, S \leq \epsilon$, $\mathrm{Area} \, (T + \partial S) \leq \mathrm{Area} \, T$. Is there a neighbourhood of $T$ on which it admits a calibration? Here again I am most interested in the case $k = 1$.