Einstein metrics of cohomogeneity one with $S^{4m+3}$ 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 ${\mathbb{HP}}^m$ 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).

Remark 3.6 added. Details for Proposition 5.2 and Lemma 5.6 added