Let $X$ be a smooth manifold of dimension $d$ and $M$ an oriented submanifold of dimension $p < d$ so that the multiples k⋅M are absolutely minimizing $p$-volume in their integral homology classes for all k∈Z . Is $M$ calibrated by some $p$-form w? (Thanks to Robert Bryant for correcting my initial question.)

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 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.

The question below is a good one. I not even know if one can "locally calibrate" some neighborhood of stable oriented minimal submanifold with oriented normal bundle. The local question may contain the important difficulties. A simple closed stable geodesics on a surface can run through areas of positive curvature, so even in that case, the local calibration is not found simply by pulling the length form of the geodesic back along normal coordinates.