Variation formulae for the volume of coassociative submanifolds

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