Volume-minimizing cycles in Grassmann manifolds (Q1902029)

Employing the method of calibrated geometries introduced by \textit{R. Harvey} and \textit{H. B. Lawson jun.} [Acta Math. 148, 47-157 (1982; Zbl 0584.53021)], the authors identify complete sets of volume-minimizing cycles in low-dimensional Grassmann manifolds and compute the homology unit mass ball in \(H_p(G_k \mathbb{R}^N)\) for \(N\leq 8\), except for \(H_6(G_2 \mathbb{R}^8)\), \(H_8(G_4 \mathbb{R}^8)\), and \(G_3 \mathbb{R}^8\). The minimizing cycles are calibrated by Kähler, quaternionic, Euler, and Pontryagin forms. The use of the first Pontryagin form as a calibration constitutes the major technical advance of this paper.