I investigate certain stationary charged perfect fluid solutions to the Einstein-Maxwell equations in general relativity. The classification of these solutions has led me to the following question:


Does hyperbolic 3-space admit a non-constant harmonic function (i.e., with vanishing Laplacian) that has a gradient of constant norm?


A non-constant affine function (i.e., with vanishing Hessian) would fit this description. However, hyperbolic space does not admit such a function.


T. Sakai: "On Riemannian manifolds admitting a function whose gradient is of constant norm", Kodai Math. J. 19 (1996), 39-51.




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Finally, a preprint of the paper is ready.
<https://www.researchgate.net/publication/291823768_ON_SOME_STRUCTURE_RESULTS_FOR_GODEL-TYPE_SPACETIMES_Preprint>



Harmonic function with gradient of constant norm in hyperbolic 3-space
The main paper you want to consult is 


É. Cartan, *Familles de surfaces isoparamétriques dans les espaces à courbure constante*,
Annali di Mat. 17 (1938), 177–191.


In this paper, Cartan considers the problem of studying the functions $f$ defined on an open set in a space $M$ of constant curvature that satisfy two equations of the form $|\nabla f|^2 = A(f)$ and $\Delta f = B(f)$ for some functions, $A>0$ and $B$, of one variable. He shows that the (hypersurface) level sets of $f$ are what he called *isoparametric*, i.e., they have all of their principal curvatures constant.


This had been considered somewhat earlier earlier by Levi-Civita for the case $n=3$ in flat space and by Segre for all $n$ (again, in flat space). They showed that, up to rigid motion and homothety, a connected isoparametric hypersurface must be an open subset of $S^k\times \mathbb{R}^{n-k-1}$ for some $0\le k\le n{-}1$. 


Cartan showed that, when the ambient sectional curvature is negative, i.e., in hyperbolic $n$-space, then, again, a connected isoparametric hypersurface must either be totally umbilic (i.e., an open subset of either a totally geodesic hyperplane, an equidistant hypersurface, a horosphere, or a hypersphere) or be an open subset of a hypersurface of the form $S^k\times H^{n-k-1}$, i.e., it must be a totally geodesic hypersurface or horosphere or else the set of points of constant distance from a totally geodesic submanifold of lower dimension. Thus, if you did have a solution of $|\nabla f|^2=1$ and $\Delta f = 0$ (even a local one in a neighborhood of $p$) in hyperbolic $n$-space, then $f-f(p)$ would be the oriented distance from the level set $f=f(p)$, and it is easy to calculate that no such function is harmonic, so you have a contradiction.


By the way, there is a much greater variety of isoparametric hypersurfaces in spaces of constant positive curvature. In fact, even though Cartan found many nontrivial examples, and there has been a lot of work since then constructing more examples and classifying them, the classification of these is still not complete.


