A CLASS OF COMPACT POINCARÉ–EINSTEIN MANIFOLDS: PROPERTIES AND CONSTRUCTION

Abstract

We develop a geometric and explicit construction principle that generates classes of Poincaré–Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalization of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scale singularity set that is a conformal infinity for the Einstein metric. In particular, the construction may be applied to yield families of compact Poincaré–Einstein manifolds, as well as classes of almost Einstein manifolds that are compact without boundary. We obtain classification results which show that the construction essentially exhausts a class of almost Einstein (and Poincaré–Einstein) manifolds. We develop the general theory of fixed conformal structures admitting multiple compatible almost Einstein structures. We also show that, in a class of cases, these are canonically related to a family of constant mean curvature totally umbillic embedded hypersurfaces.