Einstein metrics of cohomogeneity one with $S^{4m+3}$ as principal orbit

TL;DR

This paper constructs new non-compact Einstein metrics with cohomogeneity one on manifolds with $S^{4m+3}$ principal orbit, including Ricci-flat and negative Einstein metrics, some of which recover known special metrics and are asymptotically conical or hyperbolic.

Contribution

It introduces new families of Einstein metrics on vector bundles and Euclidean spaces with specific symmetry, including Ricci-flat and negative Einstein types, extending known metrics and analyzing their asymptotic behaviors.

Findings

Constructed Ricci-flat metrics including $ ext{Spin}(7)$ examples.

Developed negative Einstein metrics with asymptotically hyperbolic behavior.

Identified asymptotic limits as conical or hyperbolic geometries.

Abstract

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 $\mathbb{HP}^{m}$ 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 Cveti\v{c} 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).