Coassociative cones that are ruled by 2-planes

Abstract

It is shown that coassociative cones in R^7 that are r-oriented and ruled by 2-planes are equivalent to CR-holomorphic curves in the oriented Grassmanian of 2-planes in R^7. The geometry of these CR-holomorphic curves is studied and related to holomorphic curves in S^6. This leads to an equivalence between associative cones on one side and the coassociative cones whose second fundamental form has an O(2) symmetry on the other. It also provides a number of methods for explicitly constructing coassociative 4-folds. One method leads to a family of coassociative 4-folds whose members are neither cones nor are ruled by 2-planes. This family directly generalizes the original family of examples provided by Harvey and Lawson when they introduced coassociative geometry.