A sub-product construction of Poincare-Einstein metrics

Abstract

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 Poincare metrics we construct is not Einstein, and so this describes a class of non-conformally Einstein metrics for which the (Fefferman-Graham) obstruction tensor vanishes.

23 pages Minor correction to section 5. References updated

References in corpus (3)

• The Ambient Obstruction Tensor and Q-Curvature
• Almost conformally Einstein manifolds and obstructions
• A remark on conformal $\SU(p,q)$-holonomy

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• The ambient obstruction tensor and conformal holonomy
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• Four dimensional conformal C-spaces