Moduli of Coassociative Submanifolds and Semi-Flat Coassociative Fibrations

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We study the natural structure on the moduli space of deformations of compact coassociative submanifolds. We show that a G2-manifold with a T⁴-action of isomorphisms such that the orbits are coassociative tori is locally equivalent to a minimal 3-manifold in R³,³ = H²(T⁴,R) with positive induced metric. By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R³,³ and hence G2-metrics from equations similar to a set of affine Toda equations. The relation to semi-flat special Lagrangian fibrations and the Monge-Ampère equation are explained.