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

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

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The author is grateful to McKenzie Wang for introducing the problem and his useful comment. The author would like to thank Cheng Yang for helpful discussions on dynamic system. The author would also like to thank Christoph Böhm and Lorenzo Foscolo for their helpful suggestions and remarks on this project. Lemma 6.5 is proven thanks to the inspiring discussion with Lorenzo Foscolo.

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Correspondence to Hanci Chi.

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Communicated by P. Chrusciel.

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Chi, H. Einstein Metrics of Cohomogeneity One with \({\mathbb {S}}^{4m+3}\) as Principal Orbit. Commun. Math. Phys. 386, 1011–1049 (2021).

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