I'm trying to understand the term "conformal compactification" which is often used in physics. I reckon that most places take this to mean a (sometimes specific) compact conformal completion. That is, a conformal compactification of a manifold $M$ is a compactification $\tilde{M}$ in which all conformal transformations defined locally extend globally. Firstly, is this correct?


Now for my main question. Obviously not all compactifications are conformal compactifications (take the Alexandroff compactification of Minkowski space for example). But what about the other direction? Are all conformal completions compact? I can't think of a counterexample, but then my intuition about conformal completions is pretty shaky. I hoped that some combination of Liouville's theorem and perhaps the Hopf-Rinow theorem would help, but I'm not sure it does. 


Finally how unique are conformal compactifications? I'd like to think that for any manifold there's only one conformal compactification of the same dimension, given by the usual Penrose construction. But I can't find any references to help me start to get some intuition.


If anyone has any hints or suggestions for good literature I'd be very pleased to hear them! Many thanks!



How unique is a conformal compactification?
For Lorentzian manifolds, the conformal completion need not be compact. A typical example is the universal covering of the $d$-dimensional anti-de Sitter space-time (the maximally symmetric solution of the vacuum Einstein equations with negative cosmological constant) - its conformal boundary is diffeomorphic to $\mathbb{R}\times\mathbb{S}^{d-2}$.


Several examples of conformal completions of different space-times (i.e. time oriented Lorentzian manifolds) such as above can be found in Chapter 5 of S. W. Hawking and G. F. R. Ellis, "The Large Scale Structure of Space-Time" (Cambridge, 1973). As for the uniqueness of the procedure of conformal completion, there is of course (at least) the freedom to multiply the metric in the conformal completion by a positive smooth function thereof. There is an extensive discussion on the ambiguities in the definition of a conformal completion by Robert Geroch in F. P. Esposito and L. Witten (eds.), "The Asymptotic Structure of Space-Time" (Plenum, 1977), pp. 1-105. The book "General Relativity" by R. M. Wald (Chicago University Press, 1984) also discusses conformal completions to some length in Chapter 11.


