# 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 $$k = 1$$.