Cohomogeneity one central K\"ahler metrics in dimension four

TL;DR

This paper classifies complete cohomogeneity one central K"ahler metrics in four dimensions with zero Ricci determinant, focusing on three unimodular Lie groups, providing a full classification for SU(2) and existence results for others.

Contribution

It offers a complete classification of such metrics for SU(2) and establishes existence results for E(2) and nil_3 based on solutions to an associated ODE system.

Findings

Complete classification for SU(2) case.

Existence results for E(2) and nil_3 cases.

Solutions characterized by specific ODE systems.

Abstract

A K\"ahler 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.