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.



Canonical Metrics on 3- and 4-Manifolds
There is no canonical Lorentzian metric in $[g]$, because that would be a diffeomorphism invariant Lorentzian metric. The diffeomorphism group of any manifold has infinite dimension, and infinite dimensional stabilizer, so does not preserve any affine connection (by Bochner's lemma), or any rigid geometric structure. Recall that Bochner's lemma proves that the stabilizer of a point in the symmetry group of any affine connection is expressed as linear transformations in geodesic coordinates, so has dimension bounded by the square of the dimension of the manifold.


Edit:
I can't seem to find a nice proof of what I called Bochner's lemma (which is perhaps not due to Bochner). The OP asked for one. Here is a proof. Take a $C^{\infty}$ manifold $M$ with a $C^{\infty}$ connection $\nabla$ on its tangent bundle. The connection determines an exponential map. Any $C^{\infty}$ diffeomorphism $\phi \colon M \to M$ fixing the connection fixes the exponential map: $\phi(\exp\_p(tv))=\exp\_{\phi(p)}(t\phi'(p)v)$. Suppose now that $p$ is a fixed point of $\phi$. Then $\phi'(p) \colon T\_p M \to T\_p M$ is related to $\phi \colon M \to M$ by the equation $\phi(e^{tv})=e^{t\phi'(v)}$, where we write $e^v$ to mean $\exp\_p v$. Note that $v \mapsto e^v$ is a local diffeomorphism near $v=0$, with local inverse which we denote $\log$. Take linear coordinates $v=v^i e\_i$ on $T\_p M$ (i.e. take a basis $\{e\_i\}$ for $T\_p M$). Define coordinates $x^1,\dots,x^n$ on $M$ near $p$ by $x^i(q)=v^i$ if $v^ie\_i = \log q$. So in these coordinates, $e^v$ is expressed as the identity map, and our equation $\phi(\exp\_p(tv))=\exp\_p(t\phi'(p)v)$ becomes in coordinates $\phi(x)=\phi'(0)x$, i.e. $\phi$ is linear in these coordinates.


If in additional $\phi$ preserves a pseudo-Riemannian metric (for example one for which $\nabla$ is the Levi--Civita connection), then $\phi'(p)$ preserves the quadratic form in $T\_p M$ given by that connection. 


