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

TL;DR

This paper establishes the existence of positive Einstein metrics on certain quaternionic projective space connected sums and investigates cohomogeneity one Einstein metrics on spheres, including a new metric on S^8.

Contribution

It proves the existence of at least one positive Einstein metric on bb7bbb7bb^{m+1}\u00a0#bb7bbb7bb^{m+1} and provides criteria for a second such metric, also exploring Einstein metrics on spheres.

Findings

Existence of positive Einstein metrics on bb7bbb7bb^{m+1}a0#bb7bbb7bb^{m+1} for m  2.

Criteria for the existence of a second Einstein metric on the same space.

Existence of a non-standard Einstein metric on b7bb^8.

Abstract

We prove that there exists at least one positive Einstein metric on $\mathbb{HP}^{m+1}\sharp \overline{\mathbb{HP}}^{m+1}$ 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 $\mathbb{HP}^{m+1}\sharp \overline{\mathbb{HP}}^{m+1}$. 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 $\mathbb{S}^8$.