Uniqueness of volume-minimizing submanifolds calibrated by the first Pontryagin form

One way to understand the geometry of the real Grassmann manifold $G_k(\mathbf{R}^{k+n})$ parameterizing oriented $k$-dimensional subspaces of $\mathbf{R}^{k+n}$ is to understand the volume-minimizing subvarieties in each homology class. Some of these subvarieties can be determined by using a calibration. In previous work, one of the authors calculated the set of $4$-planes calibrated by the first Pontryagin form $p_1$ on $G_k(\mathbf{R}^{k+n})$for all $k,n\geq 4$, and identified a family of mutually congruent round $4$-spheres which are consequently homologically volume-minimizing. In the present work, we associate to the family of calibrated planes a Pfaffian system on the symmetry group $SO(k+n,\mathbf R)$, an analysis of which yields a uniqueness result; namely, that any connected submanifold of $G_k(\mathbf{R}^{k+n})$ calibrated by $p_1$ is contained in one of these $4$-spheres. A similar result holds for $p_1$-calibrated submanifolds of the quotient Grassmannian $G_k^\natural(\mathbf{R}^{k+n})$ of non-oriented $k$-planes.