On the completeness of some Bianchi type A and related Kähler-Einstein metrics

英文摘要：

We prove the existence of complete cohomogeneity one triaxial Kähler-Einstein metrics in dimension four under an action of the Euclidean group $E(2)$. We also demonstrate local existence of Ricci flat Kähler metrics of a related type that are given via generalized PDEs, and determine, under mild conditions, whether they are complete. The common framework for both metric types is a frame-dependent system of Lie bracket relations and generalized PDEs yielding a class of Kähler-Einstein metrics on $4$-manifolds which includes all diagonal Bianchi type A metrics.