Does every globally hyperbolic spacetime admit a maximal slicing? A globally hyperbolic spacetime is (roughly speaking) one with the topology of $\Sigma \times \mathbb{R}$ and a physically reasonable causal structure (no closed causal curves, that sort of thing) and can be foliated by spacelike hypersurfaces. A spacelike hypersurface is maximal if its mean extrinsic curvature is zero (meaning that the shape operator $\chi (v)=\nabla_v n$, measuring the bending of the timeline unit normal to the hypersurface in directions within the surface, has eigenvalues that sum to zero).
My question is: can every globally hyperbolic spacetime be foliated by maximal spacelike hypersurfaces, or are there are conditions on when this is possible? (And if so, what are those conditions?)
 A: Here are some ideas, but I am not sure if they constitute the kind of answer you are looking for:
The short answer is no, not every globally hyperbolic spacetime admits a slicing by maximal spacelike hypersurfaces. I am not sure the concrete necessary conditions are known, when this is the case, but there are definitely some sufficient conditions you can look at.
More generally, what you are asking is a particular case of a foliation or a slicing of a spacetime by constant mean curvature (CMC) hypersurfaces. If you search for "CMC slicing" or "CMC foliation" you will find a lot of research in that area still going on. This is because CMC hypersurfaces make good initial conditions for solving Einstein's equations. I don't know the latest state of research in CMC foliations because it was some years back, that I worked in that area.
But if you want to get into that topic, I suggest to start with this old paper by Choquet-Bruhat and work your way to more recent publications following the chain of citations. A more recent account of the state of research in CMC foliations and some open problems might be found here.
If I remember correctly, there is an example by Bartnik here, which is a globally hyperbolic spacetime, that does not admit any CMC slice and then in particular no zero mean curvature foliation either.
If you are looking for a simple sufficient condition for a spacetime to admit a zero mean curvature foliation by spacelike hypersurfaces, you can consider the existence of a complete timelike, irrotational Killing vector field. A spacetime admitting such a Killing vector field is standard static and splits as $(N = \mathbb{R}\times M,g)$ with
$$ g = -\beta(x)\mathrm{d}t^2 + h, $$
where $\beta>0$ is a function on $M$ and $h$ is a Riemannian metric on $M$. You can easily check that all slices $\{t\}\times M$ in $N$ are spacelike hypersurfaces of zero mean curvature. I don't know if this condition is also necessary for the existence of a zero mean curvature foliation.
In the references above there are some other conditions from cosmology that people usually consider for spacetimes admitting CMC foliations.
