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

We prove that there exists at least one positive Einstein metric on ${\mathbb{H}\mathbb{P}}^{m+1}\mathrm{♯}{\overline{\mathbb{H}\mathbb{P}}}^{m+1}$ for $m\ge 2$. Based on the existence of the first Einstein metric, we give a criterion to check the existence of a second Einstein metric on ${\mathbb{H}\mathbb{P}}^{m+1}\mathrm{♯}{\overline{\mathbb{H}\mathbb{P}}}^{m+1}$. We also investigate the existence of cohomogeneity one positive Einstein metrics on ${\mathbb{S}}^{4m+4}$ and prove the existence of a non-standard Einstein metric on ${\mathbb{S}}^8$.