The Stable 4-dimensional Geometry of the Real Grassmann Manifolds
In their fundamental paper Calibrated geometries in 1982 [HL], R. Harvey and B. Lawson, discussing future global applications, posed the problem of determining the subvarieties of the real Grassman manifolds GkRn of oriented k-planes through the origin in Rn which can be shown to be volume-minimizing in the homology classes by using as calibrations the invariant forms representing the universal Pontryagin classes.
We carry out this method here for the first Pontryagin form, using it at a calibration to determine volume-minimizing 4-dimensional cycles in all real Grassman manifolds. We also learn that their 4-dimensioncal geometry stabilizes immediately after the Grassmannian G4R8.
© 1998 Duke University Press
Gu, Weiqing. "The stable 4-dimensional geometry of the real Grassmann manifolds." Duke Math. J. 93 (1998), no. 1, 155–178.