Invariant Ricci-flat Metrics of Cohomogeneity One with Wallach Spaces as Principal Orbits

TL;DR

This paper constructs a family of complete Ricci-flat metrics with cohomogeneity one on vector bundles over projective planes, featuring mostly generic holonomy and asymptotically conical limits, expanding understanding of special geometric structures.

Contribution

It introduces a new 1-parameter family of Ricci-flat metrics on vector bundles over complex, quaternionic, and octonionic projective planes with Wallach space principal orbits, including a unique G2 holonomy metric.

Findings

Most metrics have generic holonomy.

All metrics are asymptotically conical.

Includes a unique G2 holonomy metric.

Abstract

We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete $G_2$ metric discovered in [BS89][GPP90]. It lies in the interior of the 1-parameter family on $\bigwedge_-^2\mathbb{CP}^2$. All the Ricci-flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric on $G/K$.