dg.differential geometry – Non-calibrated area-minimising surface


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 $k = 1$.