On the completeness of some Bianchi type A and related K\"ahler-Einstein metrics

TL;DR

This paper establishes the existence of complete K"ahler-Einstein metrics with specific symmetries on four-dimensional manifolds, and explores local Ricci-flat K"ahler metrics through generalized PDEs, including Bianchi type A metrics.

Contribution

It proves the existence of complete cohomogeneity one K"ahler-Einstein metrics under Euclidean group actions and analyzes local Ricci-flat K"ahler metrics via generalized PDEs, extending understanding of these geometric structures.

Findings

Existence of complete cohomogeneity one K"ahler-Einstein metrics with Euclidean symmetry.

Local existence of Ricci-flat K"ahler metrics described by generalized PDEs.

Characterization of completeness conditions for these metrics.

Abstract

We prove the existence of complete cohomogeneity one triaxial K\"ahler-Einstein metrics in dimension four under an action of the Euclidean group $E(2)$. We also demonstrate local existence of Ricci flat K\"ahler 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 K\"ahler-Einstein metrics on $4$-manifolds which includes all diagonal Bianchi type A metrics.