Moduli of coassociative submanifolds and semi-flat G2-manifolds

We show that the moduli space of deformations of a compact coassociative submanifold C has a natural local embedding as a submanifold of H-2(C, R). We show that a G(2)-manifold with a T-4-action of isometries such that the orbits are coassociative tori is locally equivalent to a minimal 3-manifold in R-3,R-3 with positive induced metric where R-3,R-3 congruent to H-2(T-4, R). By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R-3,R-3 and hence G(2)-metrics from a real form of the affine Toda equations. The relations to semi-flat special Lagrangian fibrations and the Monge-Ampere equation are explained. (C) 2010 Elsevier B.V. All rights reserved.