Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature

An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincaré-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 function 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 there are two main results: we give a simple classification of the possible scale singularity spaces of almost Einstein manifolds; we derive geometric results which explicitly relate the intrinsic (conformal) geometry of the conformal infinity to the conformal structure of the ambient almost Einstein manifold. The latter includes new results for Poincaré-Einstein manifolds. 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.