Projective Compactifications and Einstein metrics

Abstract

For complete affine manifolds we introduce a definition of compactification based on the projective differential geometry (i.e.\ geodesic path data) of the given connection. The definition of projective compactness involves a real parameter $\alpha$ called the order of projective compactness. For volume preserving connections, this order is captured by a notion of volume asymptotics that we define. These ideas apply to complete pseudo-Riemannian spaces, via the Levi-Civita connection, and thus provide a notion of compactification alternative to conformal compactification. For each order $\alpha$, we provide an asymptotic form of a metric which is sufficient for projective compactness of the given order, thus also providing many local examples. Distinguished classes of projectively compactified geometries of orders one and two are associated with Ricci-flat connections and non--Ricci--flat Einstein metrics, respectively. Conversely, these geometric conditions are shown to force the indicated order of projective compactness. These special compactifications are shown to correspond to normal solutions of classes of natural linear PDE (so-called first BGG equations), or equivalently holonomy reductions of projective Cartan/tractor connections. This enables the application of tools already available to reveal considerable information about the geometry of the boundary at infinity. Finally, we show that metrics admitting such special compactifications always have an asymptotic form as mentioned above.