Almost Einstein and Poincare-Einstein manifolds in Riemannian signature

Abstract

An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are special cases. The governing equation is a conformally invariant overdetermined PDE on a function. Away from the zeros of this the almost Einstein structure is Einstein, while the zero set gives a scale singularity set which may be viewed as a conformal infinity for the Einstein metric. In this article we give a classification of the possible scale singularity spaces and derive geometric results which explicitly relate the intrinsic conformal geometry of these to the conformal structure of the ambient almost Einstein manifold. Classes of examples are constructed. A compatible generalisation of the constant scalar curvature condition is also developed. This includes almost Einstein as a special case, and when its curvature is suitably negative, is closely linked to the notion of an asymptotically hyperbolic structure.