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

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 $G/K$.