Coassociative submanifolds in Joyce's generalised Kummer constructions

Abstract

This article constructs coassociative submanifolds in $G_2$-manifolds arising from Joyce's generalised Kummer construction. The novelty compared to previous constructions is that these submanifolds all lie within the critical region of the $G_2$-manifold in which the metric degenerates. This forces the volume of the coassociatives to shrink to zero when the orbifold-limit is approached.