Cohomogeneity One Coassociative Submanifolds in the Bundle of Anti-self-dual 2-forms over the 4-sphere

TL;DR

This paper constructs explicit coassociative submanifolds within a specific G2-manifold, classifies homogeneous cases, and provides new examples including those with singularities and their desingularizations.

Contribution

It provides explicit cohomogeneity one examples of coassociative submanifolds in Bryant-Salamon’s G2-manifold and classifies homogeneous cases.

Findings

Only the zero section is a homogeneous coassociative submanifold.

Constructed many explicit cohomogeneity one coassociative submanifolds.

Produced examples with conical singularities and their desingularizations.

Abstract

Coassociative submanifolds are 4-dimensional calibrated submanifolds in $G_{2}$-manifolds. In this paper, we construct explicit examples of coassociative submanifolds in $\Lambda^{2}_{-} S^{4}$, which is the complete $G_{2}$-manifold constructed by Bryant and Salamon. Classifying the Lie groups which have 3- or 4-dimensional orbits, we show that the only homogeneous coassociative submanifold is the zero section of $\Lambda^{2}_{-} S^{4}$ up to the automorphisms and construct many cohomogeneity one examples explicitly. In particular, we obtain examples of non-compact coassociative submanifolds with conical singularities and their desingularizations.