Desingularization of Coassociative 4-Folds with Conical Singularities

Abstract

Suppose that N is a compact coassociative 4-fold with a conical singularity in a 7-manifold M, with a G2 structure given by a closed 3-form. We construct a smooth family, {N′(t) : t ∈ (0,τ)} for some τ > 0, of compact, nonsingular, coassociative 4-folds in M which converge to N in the sense of currents, in geometric measure theory, as t → 0. This realisation of desingularizations of N is achieved by gluing in an asymptotically conical coassociative 4-fold in $${\mathbb{R}}^7$$ , dilated by t, then deforming the resulting compact 4-dimensional submanifold of M to the required coassociative 4-fold.