A sub-product construction of Poincare-Einstein metrics

Details A sub-product construction of Poincare-Einstein metrics A sub-product construction of Poincare-Einstein metrics

2006-08-02

arxiv.org/abs/math/0608044v2

Int.J.Math.20:1263-1287,2009

Mathematics

Differential Geometry

23 pages Minor correction to section 5. References updated

Scientific paper

10.1142/S0129167X09005753

Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincar\'e-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.

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