Variation formulae for the volume of coassociative submanifolds

We prove new variation formulae for the volume of coassociative submanifolds, expressed in terms of $G_2$ data. As a special case, we obtain a second variation formula for variations within the moduli space of coassociative submanifolds; this formula highlights the role of the ambient torsion and Ricci curvature. These results apply, for example, to coassociative fibrations. We illustrate our formulae with several examples, both homogeneous and non.