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Positive Einstein metrics with
$\mathbb {S}^{4m+3}$ as the principal orbit
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 5 / May 2024
- Published online by Cambridge University Press:
- 08 April 2024, pp. 1004-1040
- Print publication:
- May 2024
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- Article
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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 
$\mathbb {S}^8$.
