A SUB-PRODUCT CONSTRUCTION OF POINCARÉ–EINSTEIN METRICS

Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré–Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conformal structure. The ambient metrics have holonomy that agrees with the conformal holonomy. In the generic case the ambient metric arises directly as a product of the metric cones over the original Einstein spaces. In general the conformal infinity of the Poincaré metric we construct is not Einstein, and so this describes a class of non-conformally Einstein metrics for which the (Fefferman–Graham) obstruction tensor vanishes.