# 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 the neighbourhood $$U$$, and 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 $$\mathrm{dist}(\mathrm{spt} \, S,\mathrm{spt} \, T) \leq \epsilon$$, $$\mathrm{Area} \, (T + \partial S) \geq \mathrm{Area} \, T$$. Is there a neighbourhood of $$T$$ on which it admits a calibration? Here again I would be most interested in the case $$k = 1$$.

• I think all minimal surfaces are locally calibrated, probably a result of Robert Bryant. Commented Oct 9, 2020 at 15:28
• @Ben I'm aware of this result, but I believe it's slightly different from what I am asking here. If I am not mistaken it says something like "if $\Sigma$ is a (regular) minimal surface, then for every point $p \in \Sigma$ there is $r > 0$ so that $\Sigma \cap B_r(p)$ is calibrated." However the local calibration forms cannot be patched to a global $n$-form. Commented Oct 9, 2020 at 16:01
• I might be mistaken but I think you should get some milage out of $RP^2\subset RP^3$. More precisely, I think it should be true that if it were calibrated, then so would the double cover $S^2 \to RP^3$. But this is not even stable. Commented Oct 9, 2020 at 18:16
• $\mathbb{RP}^2$ obviously can't be calibrated because it's not orientable, so maybe one should require orientability in the question. But it can still be homologically minimizing: Let $M^3$ be the quotient of the standard product $S^2\times \mathbb{R}$ by the involution $(u,t)\mapsto (-u,-t)$. The image of $S^2\times\{0\}$ in $M$ is an $\mathbb{RP}^2$ that is minimizing in its homology class: Anything it its homology class must meet all the images of the lines ${u_0}\times\mathbb{R}$, and the projection $M\to\mathbb{RP}^2$ given by $[u,t]\mapsto [u]$ is clearly area decreasing on any surface. Commented Oct 9, 2020 at 21:18
• By the way, because you have changed the definition of 'calibrated' (you now only require that it be calibrated in an open neighborhood $U$), it is no longer true that a calibrated submanifold need be area-minimizing in its homology class in $M$. Also, I think that, in your 'more precise' question, you meant to write '$\mathrm{Area}(T+\partial S) \ge \mathrm{Area}(T)$' instead of '$\mathrm{Area}(T+\partial S) \le \mathrm{Area}(T)$', didn't you? Commented Oct 10, 2020 at 1:19

Actually, a better example along the lines Otis suggests would be the geodesic $$\mathbb{RP}^1\subset\mathbb{RP}^2$$. Of course, $$\mathbb{RP}^1$$ is orientable and it is homologically mass-minimizing, but it can't be calibrated on any open set $$U\subset\mathbb{RP}^2$$ containing $$\mathbb{RP}^1$$ because twice it is not even stable.

Of course, this also works for any $$\mathbb{RP}^{2n-1}\subset\mathbb{RP}^{2n}$$, and there are higher codimension examples of closed geodesics in (orientable) lens spaces that are homologically mass-minimizing but that cannot be calibrated on any open neighborhood of the geodesic. One can even foliate $$\mathbb{RP}^3$$ by homologically mass-minimizing geodesics that cannot be calibrated on any open neighborhood.

What one probably needs to assume, at least, is that every multiple of $$\Sigma$$ is homologically area-minimizing in some neighborhood before one could hope to construct a 'neighborhood' calibration.

Remark (10/12/20): I just recalled one example of possible interest for this question, since the OP is interested in what can happen in Euclidean space. A student of mine, Timothy Murdoch, in his PhD thesis "Twisted calibrations and the cone on the Veronese surface" (Rice University, 1988), showed that the $$3$$-dimensional cone in $$\mathbb{R}^5$$ on the Veronese surface in $$S^4$$ is area-minimizing, but, of course, it's not orientable. However, its 'double cover' is a cone on the $$2$$-sphere and so is orientable. I don't know whether this double cover is area-minimizing in $$\mathbb{R}^5$$ or not. It obviously cannot be calibrated, even if it is area-minimizing.

Explicitly, here is the example: Think of $$\mathbb{R}^5$$ as $$S^2_0(\mathbb{R}^3)$$, the traceless $$3$$-by-$$3$$ matrices with real entries endowed with the quadratic form $$\langle a,b\rangle = \mathrm{tr}(ab)$$, which is invariant under $$\mathrm{SO}(3)$$ with the irreducible action $$A\cdot a = AaA^{-1}$$ for $$A\in\mathrm{SO}(3)$$ and $$a\in S^2_0(\mathbb{R}^3)$$. Then the Veronese cone $$C\subset S^2_0(\mathbb{R}^3)$$ is the set of matrices $$a$$ with eigenvalues $$t^2,t^2, -2t^2$$ for some $$t\ge0$$. It is a cone on an $$\mathrm{SO}(3)$$-homogeneous minimal surface $$\mathbb{RP}^2\subset S^4$$ known as the Veronese surface. (Note that $$C$$ and $$-C$$ intersect only at the origin.) $$C$$ is smooth except at the origin, and, if you define the 'double cover' by counting each smooth point as two points with different orientations, then the double cover is homeomorphic to $$\mathbb{R}^3$$, parametrized by the quadratic map $$s:\mathbb{R}^3\to S^2_0(\mathbb{R}^3)$$ defined by $$s(x) = |x|^2\, I_3 - 3\,x\,x^T\quad\text{for}\ x\in\mathbb{R}^3.$$ Tim showed that, if you take the (literal) Riemannian double cover of $$S^2_0(\mathbb{R}^3)\setminus (-C)$$, then the double cover of $$C\setminus\{0\}\simeq \mathbb{R}^3\setminus\{0\}$$ can be calibrated in the ambient double cover as a Riemannian manifold.

• This is very informative. I am especially surprised by your remark on foliations of $\mathbf{R} P^3$ by mass-minimising geodesics, as it almost seemed to contradict a theorem of Rummler-Sullivan cited in Haefliger's Some remarks on foliations with minimal leaves. I remembered this as saying that the leaves of a minimal foliation (of some $X$) are necessarily calibrated, but checking back it only gives a form $\omega$ on $X$ which is relatively closed. A difference I admit had gone over my head... Commented Oct 11, 2020 at 17:49
• I'll add one small question. Do you expect this phenomenon, that an area-minimising surface has a non-minimising multiple to possibly also arise when working in Euclidean space? Commented Oct 11, 2020 at 17:53
• Yes, this is much better than what I had suggested, thanks Robert! Is there any evidence that the "all multiples are minimizing" condition does imply the existence of a calibration (even, say for geodesics on a surface)? Commented Oct 11, 2020 at 18:25
• @LeoMoos: Response to first comment: The fibers of the Hopf map $\pi:S^3\to S^2$ are geodesics, so this is a foliation with minimal leaves that cannot be calibrated. It can't even be calibrated locally, since the only possible candidate 1-form (up to sign) isn't closed. But this is a problem only for curves. If the leaves have dimension $k>1$ and are minimal, then every point has an open neighborhood $U$ on which there is a calibration that callibrates the leaves within $U$. (However, the 'obvious' candidate $k$-form is only closed when the orthogonal plane field is integrable.) Commented Oct 11, 2020 at 18:56
• @LeoMoos: Response to second comment: I don' expect it in Euclidean space, but I don't immediately see how to rule it out. Commented Oct 11, 2020 at 19:42

If $$M$$ is orientable and the codimension is $$1$$, the problem has been settled by Federer in

MR0348598 Reviewed Federer, Herbert Real flat chains, cochains and variational problems. Indiana Univ. Math. J. 24 (1974/75), 351–407. (Reviewer: F. J. Almgren Jr.) 49F22

The conclusion is that every codimension 1 homological area-minimizer on an orientable manifold is calibrated by a weakly closed measurable calibration form.

It is easy to construct smooth area-minimizing hypersurfaces so that any calibration form defined globally must be singular by the results of Yongsheng Zhang. Thus, do not expect a smooth calibration form.

For all codimensions larger than $$1$$, far more examples of non-calibrated area minimizers are available, even on manifolds with torsion-free homology in all dimensions. For instance, you can check out Almgren's example in Section 5.11 of the above paper. See also examples due to Lawson on a flat torus.

MR0460708 Reviewed Lawson, H. Blaine, Jr. The stable homology of a flat torus. Math. Scand. 36 (1975), 49–73. (Reviewer: Pierre Lelong) 32C30