Moduli of coassociative submanifolds and semi-flat G2G2-manifolds

We show that the moduli space of deformations of a compact coassociative submanifold CC has a natural local embedding as a submanifold of H2(C,R)H2(C,R). We show that a G2G2-manifold with a T4T4-action of isometries such that the orbits are coassociative tori is locally equivalent to a minimal 3-manifold in R3,3R3,3 with positive induced metric where R3,3≅H2(T4,R)R3,3≅H2(T4,R). By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R3,3R3,3 and hence G2G2-metrics from a real form of the affine Toda equations. The relations to semi-flat special Lagrangian fibrations and the Monge–Ampère equation are explained.