CALIBRATED SUBMANIFOLDS OF R7 AND R8 WITH SYMMETRIES

The principal theory of this paper comprises a technique forconstructing associative, coassociative and Cayley submanifoldsof Euclidean space with symmetries, using first-order ordinarydifferential equations. Explicit examples of U(1)-invariantassociative cones in ⁷ and SU(2)-invariant Cayley 4-folds in⁸ are then produced using this method. Further examples of associative3-folds are presented, which are ruled, and other systems ofdifferential equations defining calibrated submanifolds in ⁷and ⁸ are given.