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

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