Einstein Metrics of Cohomogeneity One with ${\mathbb {S}}^{4m+3}$ as Principal Orbit

Abstract

In this article, we construct non-compact complete Einstein metrics on two infinite series of manifolds. The first series of manifolds are vector bundles with ${\mathbb {S}}^{4m+3}$ as principal orbit and ${{\mathbb {H}}}{\mathbb {P}}^{m}$ as singular orbit. The second series of manifolds are ${\mathbb {R}}^{4m+4}$ with the same principal orbit. For each case, a continuous 1-parameter family of complete Ricci-flat metrics and a continuous 2-parameter family of complete negative Einstein metrics are constructed. In particular, $\mathrm {Spin}(7)$ metrics ${\mathbb {A}}_8$ and ${\mathbb {B}}_8$ discovered by Cvetič et al. in 2004 are recovered in the Ricci-flat family. A Ricci flat metric with conical singularity is also constructed on ${\mathbb {R}}^{4m+4}$. Asymptotic limits of all Einstein metrics constructed are studied. Most of the Ricci-flat metrics are asymptotically locally conical (ALC). Asymptotically conical (AC) metrics are found on the boundary of the Ricci-flat family. All the negative Einstein metrics constructed are asymptotically hyperbolic (AH).