Cohomogeneity one coassociative submanifolds in the bundle of anti-self-dual 2-forms over the 4-sphere

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.

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Received 1 April 2015

Published 7 May 2018