Coclosed \(G_2\)-structures on \(\text {SU}(2)^2\)-invariant cohomogeneity one manifolds

Abstract

We consider two different \(\text {SU}(2)^2\)-invariant cohomogeneity one manifolds, one non-compact \(M=\mathbb {R}^4 \times S^3\) and one compact \(M=S^4 \times S^3\), and study the existence of coclosed \(\text {SU}(2)^2\)-invariant \(G_2\)-structures constructed from half-flat \(\text {SU}(3)\)-structures. For \(\mathbb {R}^4 \times S^3\), we prove the existence of a family of coclosed (but not necessarily torsion-free) \(G_2\)-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed \(G_2\)-structure constructed from a half-flat \(\text {SU}(3)\)-structure is in this family. For \(S^4 \times S^3\), we prove that there are no \(\text {SU}(2)^2\)-invariant coclosed \(G_2\)-structures constructed from half-flat \(\text {SU}(3)\)-structures.