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

Abstract: We prove that there exists at least one positive Einstein metric on $$\mathbb{HP}^{m+1}\sharp \overline{\mathbb{HP}}<sup>{m+1}$$ for $m\geq 2$. Based on the existence of the first Einstein metric, we give a criterion to check the existence of a second Einstein metric on $$\mathbb{HP}^{m+1}\sharp \overline{\mathbb{HP}}<sup>{m+1}$$. We also investigate the existence of cohomogeneity one positive Einstein metrics on $\mathbb{S}<sup>{4m+4}$ and prove the existence of a non-standard Einstein metric on $\mathbb{S}<sup>8$.