CALIBRATED SUBMANIFOLDS OF ℝ7 AND ℝ8 WITH SYMMETRIES

The principal theory of this paper comprises a technique for constructing associative, coassociative and Cayley submanifolds of Euclidean space with symmetries, using first-order ordinary differential equations. Explicit examples of U(1)-invariant associative cones in ℝ⁷ and SU(2)-invariant Cayley 4-folds in ℝ⁸ are then produced using this method. Further examples of associative 3-folds are presented, which are ruled, and other systems of differential equations defining calibrated submanifolds in ℝ⁷ and ℝ⁸ are given.