Cohomogeneity one central Kähler metrics in dimension four

Abstract

A Kähler metric is called central if the determinant of its Ricci endomorphism is constant. For the case in which this constant is zero, we study on $4$-manifolds the existence of complete metrics of this type which are cohomogeneity one for three unimodular $3$-dimensional Lie groups: $SU(2)$, the group of Euclidean plane motions $E(2)$ and a quotient by a discrete subgroup of the Heisenberg group ${\mathrm{nil}}_3$. We obtain a complete classification for $SU(2)$, and some existence results for the other two groups, in terms of specific solutions of an associated ODE system.

arXiv admin note: text overlap with arXiv:2007.06471