It is somewhat ironic that we have troubles with Poisson's equation over cosmological distances, yet are so sure that there is an urgent need of dark matter and/or dark energy. Models of constant curvature universes require constant mass densities, so it appears that the gravitational potential is infinite. Yet, on the strength of symmetry considerations, the gravitational potential should be the same everywhere so that its Laplacian will vanish requiring zero density.
Einstein recognized something was wrong with Poisson's equation and modified it into an inhomogeneous Helmholtz equation by introducing his infamous cosmological constant.
This had the effect of producing asymptotic static solutions of constant potential and constant density, instead of having the Laplacian vanish with a non-vanishing density. But, it did not address Newton's warning that a uniform distribution would be unstable, or, in his words, balancing an infinitely many needles on their heads. A uniform distribution could not spontaneously evolve into a lumpy mass distribution without some supernatural entity kick-starting the process. In Newton's words: "The principle that all infinities are equal is a precarious one."
Newton's gravity becomes consistent when we demand that the mass density go to zero as we move away from the central region containing mass. Yet, Newton rejected this possibility because all matter would clump into a single body at the center.
In their ground breaking study of harmonic maps, over a half a century ago, Eells and Sampson studies harmonic mappings on the basis of manifold distortion through a diffusion equation. If curvature creates motion, the tendency will be to run down hill and iron out all sectional curvatures in the course of time. If the shortest distance between two points is a straight line, then the second law of thermodynamics, or a supernatural power, as Newton believed, would tend to iron out the sectional curvatures of the manifold resulting in a flat space in the infinite time limit.
So, instead of considering all sectional curvatures constant, equal to rho, the density, i.e.,
G"/G=rho,
we consider their difference as the cause of motion:
G*=D(G"-rho G),
which is a diffusion equation known as the Feynman-Kac equation, where D is the coefficient of diffusion.The primes denote space derivatives and the asterisk the time derivative.
Like Einstein's modified Poisson equation, we consider two regimes. One where the density is small and the system diffuses until a homogenous state is attained. The second regime is where the curvature is small, and there is exponential decay. The combined solution, given by the Feynman-Kac formula describes how a probability distribution continually broadens out while conserving the total probability.
The above equation would be analogous to Hamilton's equation for Ricci flows. On the left-hand side would be the derivative of the metric tensor, while on the right-hand side the negative of twice the corresponding eigenvalue of the Ricci tensor. In two-dimensions Ricci curvature is Gaussian curvature, while in higher dimensions there will be fewer Ricci curvatures than sectional curvatures. Ricci curvature can be understood as average of sectional curvatures in different directions.
And in the case of non-constant sectional curvature, as in the outer Schwarzschild metric, the eigenvalues of the Ricci tensor vanish giving the impression of no motion, or in Einstein's view, complete emptiness. Yet, the tidal stress components are finite, but unequal, and their action is to deform a sphere into an ellipsoid. So "emptiness" would be an incorrect description of what is actually going on. And, in regard to Hamilton's equation, there would be no motion because the eigenvalues of the Ricci tensor all vanish. It is for this reason that we have considered the individual sectional curvature, and not its average, in the Feynman-Kac equation above for the evolution of a probability distribution in a potential well given by V=DGrho/c^2, where we have reinstated Newton's gravitational constant G and the speed of light c. This part of the solution decays exponentially as exp{-Vt}..