Conformal Walker metrics and linear Fefferman-Graham equations

TL;DR

This paper develops a method to explicitly construct Ricci-flat ambient metrics for conformal classes with Walker metrics, focusing on cases where the Fefferman-Graham equations reduce to linear PDEs, thus expanding known examples in conformal geometry.

Contribution

The paper introduces a new method to find explicit ambient metrics for conformal classes with two-step nilpotent Schouten tensors, especially Walker metrics, where equations simplify to linear PDEs.

Findings

Explicit ambient metrics for certain Walker and pp-wave conformal structures.

Reduction of Fefferman-Graham equations to linear PDEs in specific cases.

New examples of Ricci-flat ambient metrics in conformal geometry.

Abstract

The conformal Fefferman-Graham ambient metric construction is one of the most fundamental constructions in conformal geometry. It embeds a manifold with a conformal structure into a pseudo-Riemannian manifold whose Ricci tensor vanishes up to a certain order along the original manifold. Despite the general existence result of such ambient metrics by Fefferman and Graham, not many examples of conformal structures with Ricci-flat ambient metrics are known. Motivated by previous examples, for which the Fefferman-Graham equations for the ambient metric to be Ricci-flat reduce to a system of linear PDEs, in the present article we develop a method to find ambient metrics for conformal classes of metrics with two-step nilpotent Schouten tensor. Using this method, for metrics for which the image of the Schouten tensor is invariant under parallel transport, i.e., certain types of Walker metrics, we obtain explicit ambient metrics. This includes certain left-invariant Walker metrics as well as pp-waves.