Bach-Flat Kaehler Surfaces

TL;DR

This paper classifies Bach-flat Kähler surfaces, analyzing solutions on compact 4-manifolds, with particular focus on cases involving 3-dimensional CR manifolds, contributing to understanding special geometric structures.

Contribution

It provides a classification of Bach-flat Kähler surfaces on compact 4-manifolds and detailed analysis of each case, especially those involving CR manifolds.

Findings

Classification of solutions on compact 4-manifolds

Detailed results for each case in the classification

Insights into the role of CR manifolds in Bach-flat Kähler surfaces

Abstract

A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is a critical point for the L2-norm of the Weyl curvature. When the Riemannian 4-manifold in question is a Kaehler surface, we provide a rough classification of solutions, followed by detailed results regarding each case in the classification. The most mysterious case prominently involves 3-dimensional CR manifolds.