From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique hyperbolic metric which can be considered as a canonical metric on the surface. I am wondering if similar results exist in the case of 3- or 4-manifolds where the manifolds admit Lorentzian metrics, metrics with signature $(-,+,+)$ or $(-,+,+,+)$ in the case of 3- or 4-manifold, respectively.

Let $M$ be a compact 3- or 4-manifolds, possibly with smooth boundaries, which admits Lorentzian metrics, and consider a diffeomorphism $f:M\longrightarrow M$. This will relate Lorentzian metrics $g_1$ and $g_2$ which are compatible with a given smooth structure on $M$:

$$f^*g_1=g_2$$

Where $f^*g_1$ denotes the pull-back of the metric $g_1$. If I denote the equivalence class of such Lorentzian metrics under diffeomorphism as $[g]$, there are two questions:

Is there a canonical Lorentzian metric in $[g]$ for the general 3- or 4-manifolds?

For the special case that the 4-manifold $M$ can smoothly be written as $M=\mathbb{R}\times X$, where $X$ is a compact 3-manifold possibly with boundaries, is there a canonical Lorentzian metric in $[g]$? By "smoothly", I mean there is a global diffeomorphism between $M$ and $\mathbb{R}\times X$;

A good reference on 3- or 4-manifolds containing related results is highly appreciated.

Riemannianmetrics on 3-manifolds. (See here: en.wikipedia.org/wiki/Geometrization_conjecture .) I'm not sure what's known in the Lorentzian setting, but there has been plenty of work on it; see, for instance, here -- math.univ-lille1.fr/~kassel/flat-lorentzian.pdf . $\endgroup$ – HJRW May 11 '18 at 10:22