The collapsing sphere product of Poincaré-Einstein spaces

Any smooth even asymptotically hyperbolic space with boundary can be doubled in a natural way, producing a smooth conformal manifold without boundary. In the case of a Poincaré-Einstein metric the doubling gives rise to an almost Einstein manifold with hypersurface singularity. We show in this article that the doubling construction can be generalised by the so-called collapsing sphere product DℓM¯ alias Sℓ-doubling. The Sℓ-doubling of an AH Einstein space has various interesting geometric properties. In particular, DℓM¯ admits multiple almost Einstein structures with intersecting scale singularities and decomposable conformal holonomy. The pole of DℓM¯ is a totally umbilic submanifold. We also provide an explicit Ricci-flat Fefferman-Graham ambient space for DℓM¯.