> Let $X$ be a smooth manifold of dimension $d$ and $M$ an oriented
> submanifold of dimension $p < d$ which (absolutely) minimizes
> $p$-volume in its integral homology class. Is $M$ calibrated by some
> $p$-form w?

 
Definitions: A $p$-form $w$ is called a *calibration* if it is closed and its evaluation on every geometric $p$-vector $v$ of norm 1 is at most 1 in norm, $|w(v)| \le 1$. One says a $p$-dimensional submanifold $M$ is *calibrated* (by a calibration $w$) if $w$ evaluates to 1 on each unit tangent $p$-vector to $M$. The word "geometric" above is used to distinguish primitive (or geometric) $p$-vectors from linear combinations of these; "geometric" means "rank one" in tensor language.
 
Note: It is elementary that if $M$ is calibrated by any $w$ then $M$ minimizes $p$-volume in its homology class, so I'm asking for a kind of converse. In the case $p+1=d$ the converse amounts to a continuum version of Max Flow/Min Cut which is discussed in John Sullivan's 1990 Princeton Ph.D. thesis. I tried asking [a form of this question][1] last year, but I did not use the term "calibration". I'm hoping with the correct language an expert will notice the question and be able to answer.

[1]: http://mathoverflow.net/questions/181419/co-dimension-one-minimizing-verifolds