Geometry of the sphere of calibrations of degree two.
In the space of \(p\)-forms \(\Lambda^p(\mathbb{R}^n)\) a Minkowski norm (the comass norm) \(| | \phi| |^*=\sup\{\langle \phi,\omega\rangle \mid \omega \in G^+_{p,n} \subset \Lambda_p (\mathbb{R}^n)\}\) is defined, where \(G^+_{p,n}\) is the Grassmann manifold of unit simple \(p\)-vectors. The authors construct a bijection between the set of extreme points of the comass-unit sphere \(S^*_{2,2k} \subset \Lambda^2(\mathbb{R}^{2k})\) and the manifold of orthogonal complex structures in \(\mathbb{R}^{2k}\). This bijection is used for obtaining a classification of faces of the sphere \(S^*_{2,2k}\). These results can be used for studying minimal submanifolds in \(\mathbb{R}^n\).
scientific article; zbMATH DE number 1777767