I refer to the living review article by Jörg Frauendiener, “Conformal Infinity”. I really need to read the whole thing again, so I won’t make too many comments now.

That being said the article contains an interesting discussion of the relevance of being able to smoothly move from the conformal null past to the conformal null future. This necessarily requires traversing spacelike infinity. Unfortunately, with the usual asymptotic set up the surfaces are not smooth at the join given by spacelike infinity. Jörg discusses `blowing up’ conformal spacelike infinity from a 2-sphere to a 2-dimensional cylinder.

This allows the effects of the conformal past and future to be distanced from each other and therefore gives additional control over the situation which might allow for the necessary smoothness.

The relationship to singularity theory comes from the a-boundary where such `blowing up’ action are considered necessary to give better descriptions of behaviour at the boundary of a manifold, e.g. at singularities or at infinity. The mathematical formalism of the a-boundary allows for the two structures, the cylinder and sphere, to be treated as the same.

Perhaps there is some interesting research to be done on the juncture between these two fields?