New Inhomogeneous Nearly $\mathrm{G}_2$-metric on the Berger Space and a Sine Cone Desingularisation

Abstract

Nearly $\mathrm{G}_2$ manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, $\mathrm{Spin}(7)$. We study nearly parallel $\mathrm{G}_2$-structures invariant under a cohomogeneity-one action of $\mathrm{SO}(4)$, and construct the first inhomogeneous nearly parallel $\mathrm{G}_2$-structure on the Berger space $\mathrm{SO}(5)/\mathrm{SO}(3)$. The cone over this metric has full holonomy $\mathrm{Spin}(7)$. The existence of the solution is established by a rigorous computer-assisted shooting argument. We also establish a local desingularisation of a finite quotient of the sine cone over $S^3\times S^3$ by gluing in the asymptotically conical $C_7$ bubble constructed by Foscolo--Haskins--Nordstr\"om. We show that as the singular orbit of a family of nearly parallel $\mathrm{G}_2$ manifolds collapses, away from the singular orbit the desingularising family converges to the sine cone quotient, while the natural blow-up converges to the AC torsion-free $\mathrm{G}_2$-metric of $C_7$.