The mean curvature of first-order submanifolds in exceptional geometries with torsion (Q2227504)

In this paper, the authors derive formulas for the mean curvature of associative \(3\)-folds, coassociative \(4\)-folds, and Cayley \(4\)-folds in the general case where the ambient space has intrinsic torsion. Consequently, they are able to characterize those \(\mathrm{G}_2\)-structures (resp., \(\mathrm{Spin}(7)\)-structures) for which every associative \(3\)-fold (resp. coassociative \(4\)-fold, Cayley \(4\)-fold) is a minimal submanifold. In the process, they obtain new obstructions to the local existence of coassociative \(4\)-folds in \(\mathrm{G}_2\)-structures with torsion.