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Uniqueness Of Volume-Minimizing Submanifolds Calibrated By The First Pontryagin Form, Daniel A. Grossman, Weiqing Gu

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One way to understand the geometry of the real Grassmann manifold G_k(R^k+n) parameterizing oriented k-dimensional subspaces of 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 p₁ on G_k(R^k+n) for all k,n ≥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 …